Global Aspects of Complex Geometry

Catanese · Esnault · Huckleberry · Hulek · Peternell (Eds.) Global Aspects of Complex Geometry Fabrizio Catanese · Hél...

Author: Fabrizio Catanese | Hélène Esnault | Alan T. Huckleberry | Klaus Hulek | Thomas Peternell

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Catanese · Esnault · Huckleberry · Hulek · Peternell (Eds.) Global Aspects of Complex Geometry

Fabrizio Catanese · Hélène Esnault Alan T. Huckleberry · Klaus Hulek Thomas Peternell (Eds.)

Global Aspects of Complex Geometry With 15 Figures

123

Fabrizio Catanese

Hélène Esnault

Lehrstuhl Mathematik VIII Mathematisches Institut Universität Bayreuth 95440 Bayreuth, Germany e-mail: [email protected]

Mathematik Universität Duisburg-Essen 45117 Essen, Germany e-mail: [email protected]

Alan T. Huckleberry

Klaus Hulek

Institut für Mathematik Universität Bochum Universitätsstraße 150 44801 Bochum, Germany e-mail: [email protected]

FB Mathematik Institut für Mathematik Universität Hannover Welfengarten 1, 30167 Hannover, Germany e-mail: [email protected]

Thomas Peternell Lehrstuhl Mathematik I Mathematisches Institut Universität Bayreuth 95440 Bayreuth, Germany e-mail: [email protected]

Library of Congress Control Number: 2006929537

Mathematics Subject Classification (2000): 14-02, 14F05, 14F42, 14H60, 14J10, 14J32, 14J40, 14J45 32-02, 32M05, 32M25, 32Q15

ISBN-10 3-540-35479-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-35479-6 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Production and Data conversion: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

44/3100/YL

543210

Preface

Over the period 2000–2006 the Deutsche Forschungsgemeinschaft sponsored a special Schwerpunkt programme, entitled “Global Methods in Complex Geometry”. The articles of this volume grew out of this programme and document some of the scientific activity performed in the realm of the Schwerpunkt. They also aim at giving a broader overview of recent developments in various directions of Complex Geometry such as • Low-dimensional geometry: surfaces of general type, Fano threefolds, Calabi-Yau threefolds; • moduli spaces and families of varieties over curves; • Hodge theory, motivic cohomology and characteristic p-geometry; • moment maps and group actions on flag manifolds; • geometry of singular varieties: vector fields, equisingular families and vector bundles; • geometry of rational curves and pseudo-effective line bundles. The articles are devoted to a broad spectrum of topics, which range from purely algebraic to complex-analytic aspects of our subject. The participants of the Schwerpunkt would like to thank the Deutsche Forschungsgemeinschaft for its generous support. Bayreuth, Essen, Bochum, Hannover, June 2006 Fabrizio Catanese, Hélène Esnault, Alan Huckleberry, Klaus Hulek, Thomas Peternell

Contents

Complex Surfaces of General Type: Some Recent Progress Ingrid C. Bauer, Fabrizio Catanese, Roberto Pignatelli . . . . . . . . . . . . . . .

1

Characteristic 0 and p Analogies, and some Motivic Cohomology Manuel Blickle, Hélène Esnault, Kay R¨ ulling . . . . . . . . . . . . . . . . . . . . . . . . 59 Vector Bundles and Torsion Free Sheaves on Degenerations of Elliptic Curves Lesya Bodnarchuk, Igor Burban, Yuriy Drozd, Gert-Martin Greuel . . . . . 83 Indices of Vector Fields and 1-Forms on Singular Varieties W. Ebeling, S. M. Gusein-Zade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Equisingular Families of Projective Curves Gert-Martin Greuel, Christoph Lossen, Eugenii Shustin . . . . . . . . . . . . . . . 171 Critical Points of the Square of the Momentum Map Peter Heinzner, Henrik St¨ otzel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Actions on Flag Manifolds: Related Cycle Spaces Alan Huckleberry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Modularity of Calabi-Yau Varieties Klaus Hulek, Remke Kloosterman, Matthias Sch¨ utt . . . . . . . . . . . . . . . . . . . 271 Some Recent Developments in the Classification Theory of Higher Dimensional Manifolds Priska Jahnke, Thomas Peternell, Ivo Radloff . . . . . . . . . . . . . . . . . . . . . . . 311 Existence of Rational Curves on Algebraic Varieties, Minimal Rational Tangents, and Applications Stefan Kebekus, Luis Sol´ a Conde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

VIII

Contents

Special Families of Curves, of Abelian Varieties, and of Certain Minimal Manifolds over Curves Martin M¨ oller, Eckart Viehweg, Kang Zuo . . . . . . . . . . . . . . . . . . . . . . . . . . 417 Hodge Theory and Algebraic Cycles Stefan J. M¨ uller-Stach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 K¨ ahler Geometry of Moduli Spaces of Holomorphic Vector Bundles Georg Schumacher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

Complex Surfaces of General Type: Some Recent Progress Ingrid C. Bauer1 , Fabrizio Catanese2 , and Roberto Pignatelli3, 1 2 3

Mathematisches Institut, Lehrstuhl Mathematik VIII, Universit¨ atstraße 30, D-95447 Bayreuth, Germany [email protected] Mathematisches Institut, Lehrstuhl Mathematik VIII, Universit¨ atstraße 30, D-95447 Bayreuth, Germany [email protected] Dipartimento di Matematica, Universit` a di Trento, via Sommarive 14, I-38050 Povo (TN), Italy [email protected]

Introduction In this article we shall give an overview of some recent developments in the theory of complex algebraic surfaces of general type. After the rough or Enriques-Kodaira classification of complex (algebraic) surfaces, dividing compact complex surfaces in four classes according to their Kodaira dimension −∞, 0, 1, 2, the first three classes nowadays are quite well understood, whereas even after decades of very active research on the third class, the class of surfaces of general type, there is still a huge number of very hard questions left open. Of course, we made some selection, which is based on the research interest of the authors and we claim in no way completeness of our treatment. We apologize in advance for omitting various very interesting and active areas in the theory of surfaces of general type as well as for not being able to mention all the results and developments which are important in the topics we have chosen. Complex surfaces of general type come up with certain (topological, birational) invariants, topological as for example the topological Euler number e and the self intersection number of the canonical divisor K 2 of a minimal surface, which are linked by several (in-) equalities. In the first chapter we will summarize the classically known inequalities, which force surfaces of general type in a certain region of the plane having K 2 and e as coordinates, and we shall briefly comment on the so-called geography problem, whether,

The present work was performed in the realm of the SCHWERPUNKT “Globale Methoden in der komplexen Geometrie”, and was also supported by a VIGONIDAAD Program. A first draft of this article took origin from the lectures by the second author at the G.A.C. Luminy Meeting, october 2005: thanks to the organizers!

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Ingrid C. Bauer, Fabrizio Catanese, and Roberto Pignatelli

given numerical invariants lying in the admissible range, i.e., fulfilling the required inequalities, does there exist a surfaces having these invariants. We shall however more broadly consider the three classical invariants K 2 , pg , q, which determine the other invariants χ := 1 − q + pg , e = 12χ − K 2 . An important new inequality, which Severi tried without success to establish, and which has been attacked for many years with partial results by several authors, asserts that a surface of maximal Albanese dimension satisfies the inequality K 2 ≥ 4χ. We will report on Pardini’s surprisingly simple proof of this so-called Severi’s conjecture (cf. [Par05]). The study of the pluricanonical maps is an essential technique in the classification of surface of general type. The main results concerning the m-canonical maps with m ≥ 3 go back to an earlier period and we refer to [Cat87b] for a report on them. We will report in the second chapter on recent developments concerning the bicanonical map; we would like to mention Ciliberto’s survey (cf. [Cil97]) on this topic for the state of art ten years ago. Here instead, we combine a discussion of this topic with the closely intertwined problem of classification of surfaces with low values of the numerical invariants. In the third chapter we report on surfaces of general type with geometric genus pg equal to four, a class of surfaces whose investigation was started by Federigo Enriques (cf. chapter VIII of his book ’Le superficie algebriche’, [Enr49]). By Gieseker’s theorem we know that for fixed K 2 and χ there exists a quasi projective coarse moduli space MK 2 ,χ for the birational equivalence classes of surfaces of general type. It is a very challenging problem to understand the geometry of these moduli spaces even for low values of the invariants. The case pg = 4 is studied via the behaviour of the canonical map. While it is still possible to divide the moduli space into various locally closed strata according to the behaviour of the canonical map, it is very hard to decide how these strata patch together. Using certain presentations of Gorenstein rings of codimension 4 introduced by M. Reid and D. Dicks, which arrange the defining equations as Pfaffians of certain matrices with many symmetries in such a way that these equations behave well under deformation, it is possible to exhibit explicit deformations, which allow to “connect” certain irreducible components of the moduli space. Inspired by a construction of A. Beauville of a surface with K 2 = 8, pg = q = 0, the second author defined Beauville surfaces as surfaces which are rigid and which admit an unramified covering which is isomorphic to a product of curves of genus at least 2. In this case the moduli space of surfaces orientedly homeomorphic to a given surface consists either of a unique real point, or of a pair of complex conjugate points corresponding to complex conjugate surfaces.

Complex Surfaces of General Type: Some Recent Progress

3

These surfaces, and the more general surfaces isogenous to a product, not only provide cheap counterexamples to the Friedman-Morgan speculation (which will be treated more extensively in the sixth section of this article), but provide also a wide class of surfaces quite manageable in order to test conjectures, and offer also counterexamples to various problems. The ease with which one can handle these surfaces is based on the fact that these surfaces are determined by “discrete” combinatorial data. Beauville surfaces, their relations to group theory and to Grothendieck’s theory of ’Dessins d’enfants’ will be discussed in the fourth chapter. It is a very difficult and very intriguing problem to decide whether two algebraic surfaces, which are not deformation equivalent, are in fact diffeomorphic. The theory of Lefschetz fibrations provides an algebraic tool to prove that two surfaces are diffeomorphic. By a theorem of Kas (which holds also in the symplectic context) two Lefschetz fibrations are diffeomorphic if and only if their corresponding factorizations of the identity in the mapping class group are equivalent under the equivalence relation generated by Hurwitz moves and by simultaneous conjugation. We outline the theory, which was used with success in [CW04] in chapter five, which we end with a brief report on the status of two very old conjectures by Chisini concerning cuspidal curves and algebraic braids. As already mentioned before, one of the fundamental problems in the theory of surfaces of general type is to understand their moduli spaces, in particular the connected components which parametrize the deformation equivalence classes of minimal surfaces of general type. By a classical result of Ehresmann, two deformation equivalent algebraic varieties are diffeomorphic. The other direction, i.e., whether two diffeomorphic minimal surfaces of general type are indeed in the same connected component of the moduli space, was an open problem since the eighties. We discuss in the last chapter the various counterexamples to the Friedman-Morgan speculation, who expected a positive answer to the question (unlike the second author, cf. [Kat83]). Moreover, we briefly report on another equivalence relation introduced by the second author, the so-called quasi étale-deformation (Q.E.D.) equivalence relation, i.e., the equivalence relation generated by birational equivalence, by quasi étale morphisms and by deformation equivalence. For curves and surfaces of special type two varieties are Q.E.D. equivalent if and only if they have the same Kodaira dimension, whereas there are infinitely many surfaces of general type, which are pairwise not Q.E.D. equivalent.

4


1 Old and New Inequalities 1.1 Invariants of Surfaces n Let X be a compact complex manifold and let ΩX be its canonical bundle, i.e., the line bundle of holomorphic n−forms (usually denoted by ωX , since it is a dualizing sheaf in the sense of Serre duality). A corresponding canonical divisor is usually denoted by KX . To X one associates its canonical ring ⊗m R(X) := ⊕m≥0 H 0 (ωX ).

The trascendency degree over C of this ring leads to • the Kodaira dimension κ(X) := tr(R(X)) − 1, if R(X) = C, otherwise κ(X) := −∞. The Kodaira dimension is invariant under deformation (by Siu’s theorem [Siu02], generalizing Iitaka’s theorem for surfaces) and can assume the values −∞, 0, . . . , n = dim X . Definition 1. X is said to be of general type if the Kodaira dimension is maximal, κ(X) = dim X. We are interested in the case of surfaces, i.e., of manifolds of dimension 2, of general type. The three principal invariants under deformations for the study of these surfaces are • the self intersection of the canonical class KS2 of a minimal model, • the geometric genus pg := h0 (ωX ) and • the irregularity q := h1 (OS ) = h0 (ΩS1 ). The equality h1 (OS ) = h0 (ΩS1 ) follows by Hodge theory since every algebraic surface is projective. The invariants we have introduced, with the exception of KS2 , are not only deformation invariants but also birational invariants. Definition 2. A smooth surface S is called minimal (or a minimal model) iff it does not contain any exceptional curve E of the first kind (i.e. E ∼ = P1 , 2 E = −1). Every surface can be obtained by a minimal one (its “minimal model”) after a finite sequence of blowing ups of smooth points; this model is moreover unique if κ(S) ≥ 0 (see III.4.4, III.4.5 and III.4.6 of [BHPV04]). Thus, every birational class of surfaces of general type contains exactly one minimal surface, and one classifies surfaces of general type by studying their minimal models. To each minimal surface of general type we will associate its numerical • type (KS2 , pg , q),


5

a triple of integers given by the three invariants introduced above. In fact these determine all other classical invariants, as • the Euler-Poincaré characteristic of the trivial sheaf χ(OS ) = 1 − q + pg ; 2 • the topological Euler characteristic e(S) = c2 (S) S ) − KS ; 12χ(O = ⊗m 2 K ) = χ(OS ) + m . • the plurigenera Pm (S) := h0 (ωX S 2 The epression for c2 is a classical theorem of M. Noether, and the expression for the plurigenera follows by Riemann-Roch and by Mumford’s vanishing theorem. By the theorems on pluricanonical maps (cf. [Bom73]), minimal surfaces S of general type with fixed invariants are birationally mapped to normal surfaces X in a fixed projective space of dimension P5 (S) − 1. X is uniquely determined, is called the canonical model of S, and is obtained contracting to points all the (-2)-curves of S (curves E ∼ = P1 , with E 2 = −2). Let us recall Gieseker’s theorem Theorem 1 (Gieseker [Gie77]). There exists a quasi-projective coarse moduli scheme for canonical models of surfaces of general type S with fixed KS2 and c2 (S). In particular, we can consider the subscheme MKS2 ,pg ,q corresponding to minimal surfaces of general type of type (KS2 , pg , q). By the above theorem, it is a quasi projective scheme, in particular, it has finitely many irreducible components. It is a dream ever since to completely describe MKS2 ,pg ,q for as many types as possible. 1.2 Classical Inequalities and Geography Obviously the first question is: for which values of (KS2 , pg , q) is MKS2 ,pg ,q non empty? For example, it is clear that pg (S) and q(S) are always nonnegative, since they are dimensions of vector spaces. In fact much more is known. In the following table we collect the well known classical inequalities holding among the invariants of minimal surfaces of general type: KS2 (N ) KS2 (D) if q (M Y ) KS2

≥1 χ≥1 ≥ 2pg − 4 or the weaker KS2 ≥ 2χ(OS ) − 6 > 0, KS2 ≥ 2pg or the weaker if q > 0, KS2 ≥ 2χ(OS ) ≤ 9χ

We have labeled by (N)= Noether, (D) = Debarre, (MY) = Miyaoka-Yau the rows, corresponding to the names of the inequalities ([Deb82], [Deb83], [Miy77], [Yau78], see also [BHPV04], chap. 7).

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Fig. 1. The geography of minimal surfaces of general type

In figure 1 we have drawn the limit lines (i.e., where equality holds) of the various inequalities in the (χ, KS2 )-plane. The above listed inequalities show that the pair of invariants χ, KS2 of a minimal surface of general type gives a point with integral coordinates in the convex region limited by the “bold” piecewise linear curve. Moreover, if q > 0 this point cannot be at the “right” of the line D. We drew one more line in our picture, labeled by S. This is the Severi line K 2 = 4χ, i.e., the equality case of the Severi inequality K 2 ≥ 4χ ⇔ K 2 ≥ 12 e, which will be discussed in detail at the end of this section. 1.3 Surfaces Fibred over a Curve An important method for the study of surfaces of general type is to consider relatively minimal fibrations of surfaces over curves f : S → B. Definition 3. A fibration f : S → B is a surjective morphism with connected fibres. We are interested in the case of fibrations of surfaces to curves, meaning that in this paper S and B will always be smooth compact complex manifolds of respective dimensions 2 and 1. The fibration is said to be relatively minimal if f does not contract any rational curve of self intersection −1 to a point.


7

One denotes • by b the genus of the base curve B; • by g the genus of a general fibre. To avoid confusion, let us point out that a fibration is called rational or irrational according to the genus b of the base being 0 or > 0. On the other hand, the genus of the fibration is the genus g of the fibre. For example, if we say f is a genus 2 rational fibration, we intend that g = 2 and b = 0. The classical way of saying: a genus b pencil of curves of genus g is however still the most convenient way to describe a fibration. To a relatively minimal fibration f one associates ∨ • its relative canonical bundle ωS|B := ωS ⊗ f ∗ (ωB ) and ⊗n • the sheaves (∀n ≥ 0) Vn := f∗ (ωS|B ).

The sheaves Vn are vector bundles (i.e., locally free sheaves) with very nice properties. Theorem 2 (Fujita [Fuj78a], [Fuj78b])). The vector bundles Vn are semipositive, i.e., every locally free quotient of it has nonnegative degree. To be more precise, V1 is a direct sum of an ample vector bundle with q(S) − b copies of the trivial bundle and with some undecomposable stable degree 0 vector bundle without global sections. Zucconi [Zuc97] proved moreover that if one of those stable bundles has rank 1, then it is a torsion line bundle. For n ≥ 2 we have: Theorem 3 (Esnault-Viehweg [EV90]). ∀n ≥ 2 the vector bundle Vn is ample unless f has constant moduli, which means that all the smooth fibres are isomorphic. ⊗n = 0 ∀n ≥ 2 by Since R1 f∗ ωS|B = OB by relative duality, and R1 f∗ ωS|B the assumption of relative minimality, one can compute the Euler characteristic of Vn by Riemann-Roch, and consequently its degree. We introduce the following invariants of the fibration f :

• the self intersection of the relative canonical divisor Kf2 := ωS|B · ωS|B = KS2 − 8(g − 1)(b − 1), • the Euler characteristic of the relative canonical divisor χf = χ(ωS|B ) = χ(OS ) − (g − 1)(b − 1), • its slope λ(f ) := Kf2 /χf .

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The slope is clearly defined only for χf = 0, or equivalently (as we will see soon) if the fibration is not a holomorphic bundle. The above mentioned computation gives deg Vn = χf +

n(n − 1) 2 Kf 2

and since by Fujita’s theorem these numbers are nonnegative this gives the two inequalities Kf2 ≥ 0 and χf ≥ 0 respectively known as Arakelov’s inequality (cf. [Ara71]) and Beauville’s inequality (cf. [Bea82]). In fact, we have the following list of inequalities (A) (B) (ZS) (N N ) (X)

Kf2 ≥ 0 , i.e., KS2 ≥ 8(g − 1)(b − 1), χf ≥ 0 , i.e., χ(OS ) ≥ (g − 1)(b − 1), c2 (S) ≥ 4(b − 1)(g − 1), q ≤ b + g, 4 − g4 ≤ λ(f ) ≤ 12.

Here the meaning of the labeling is the following: (A) = Arakelov’s inequality, (B) = Beauville’ inequality, (X) = Xiao’s inequality (also known as slope inequality), (NN) = no name’s inequality, (ZS) = Zeuthen-Segre. A proof of those inequalities can be found in [Bea82] with the exception of the slope inequality, proved in [Xia87] (see also [CH88] in the semistable case). The equality cases of the first 4 inequalities are well described: • • • •

if equality holds in (A), f has constant moduli; equality holds in (B) ⇔ f has constant moduli and is smooth; for g ≥ 2, equality holds in (ZS) ⇔ f is smooth; q = b + g ⇔ f is birationally equivalent to the projection of a product B × F to the first factor.

In particular, we see that the slope is defined whenever the fibration is not a holomorphic bundle, since the denominator χf vanishes iff equality holds in Beauville’s inequality. An important consequence is the following Theorem 4 (Beauville). If X is a minimal surface of general type, then pg ≥ 2q − 4. Moreover, if pg = 2q − 4, then S is a product of a curve of genus 2 with a curve of genus q − 2. Note for later use (see next section) the following Corollary 1. If pg = q (i.e., if χ(OS ) = 1), then pg = q ≤ 4. Moreover, minimal surfaces of general type with pg = q = 4 are exactly the products of two genus 2 curves. Proof of theorem 4. The standard wedge product on 1−forms induces a natural map ∧ : Λ2 H 0 (ΩS1 ) → H 0 (ΩS2 )


9

Recall that q = dim H 0 (ΩS1 ), pg = dim H 0 (ΩS2 ). Let us assume pg ≤ 2q − 4. By a dimension count, if pg ≤ 2q − 4, the projective linear subspace of P(Λ2 H 0 (ΩS1 )) corresponding to the kernel of the above map must intersect the Pl¨ ucker embedding of the Grasmannian G2 (H 0 (ΩS1 )) (which has dimension 2q − 4), and therefore there are two linearly independent 1−forms ω1 and ω2 such that the following holomorphic two form is identically zero: ω1 ∧ ω2 ≡ 0. By the theorem of Castelnuovo-De Franchis there is a fibration f : S → B 1 ) with base of genus b ≥ 2, and two holomorphic 1−forms α1 , α2 ∈ H 0 (ΩB ∗ such that f αi = ωi . Since S is of general type, also g ≥ 2. Then χf ≥ 0 ⇒ χ(OS ) ≥ (b − 1)(g − 1) = (b − 2)(g − 2) + b + g − 3 ≥ q − 3. So we have 1 − q + pg ≥ q − 3 ⇔ pg ≥ 2q − 4. If pg = 2q − 4, all inequalities are equalities and then, since q = b + g and (b − 2)(g − 2) = 0, S is a product of two curves of genus at least 2, and one of the two must have genus exactly 2. 1.4 Severi’s Inequality We recall that the Albanese variety Alb(X) of a compact Kähler manifold X is the cokernel of the natural map : H1 (X, Z) → H 0 (Ω 1 (X))∨ defined by integrating 1−forms on 1-cycles. The Albanese morphism α : X → Alb(X) is defined (up to translations in Alb(X)) by fixing a point p0 ∈ X, and by p associating to each point p ∈ X the class in Alb(X) of p0 , where the integral is taken along any path between p0 and p. Recall that, if X is projective (as any surface of general type), Alb(X) is an abelian variety (of dimension q). The Albanese morphism is a powerful tool for studying irregular surfaces (q > 0) and in particular: Definition 4. A variety X is called of maximal Albanese dimension if the image of the Albanese morphism has the same dimension as X. This is the general case for surfaces, since otherwise the Albanese morphism is a fibration onto a smooth curve of genus q. We see then that for surfaces maximal Albanese dimension is equivalent to the non existence of a genus q pencil. We can now state the theorem known as Severi’s inequality

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Theorem 5 (Pardini [Par05]). If S is a smooth complex minimal surface of maximal Albanese dimension, then KS2 ≥ 4χ. This theorem was proved only very recently by R. Pardini, but it has a long story, which we briefly sketch in the following. Severi’s Conjecture The inequality takes its name from F. Severi, since he was the first to claim the result in the 30’s [Sev32]. His proof turned out to be wrong, as was pointed out in [Cat83], since it was based on the assertion that a surface with irregularity q either contains an irrational genus q fibration, or the sections of H 0 (ΩS1 ) have no common zero. Counterexamples were given in [Cat84], where there were constructed bidouble covers S → X of any algebraic surface with, among other properties, q(S) = q(X). If X has no irrational pencils, since the Albanese map of S factors through the cover, then also S has no irrational pencils. But any ramification point of the cover is a base point for H 0 (ΩS1 ). Therefore Severi’s inequality was posed in [Cat83] as Severi’s conjecture, a conjecture on surfaces of general type, since for surfaces with κ(S) ≤ 1 it is a straightforward consequence of the Enriques-Kodaira classification. It had also been posed as a conjecture by M. Reid (conj. 4 in [Rei79]) who proved the weaker KS2 ≥ 3χ. Proofs in Special Cases In the 80’s, Xiao’s work on surfaces fibred over a curve was mainly motivated by Severi’s conjecture. In [Xia87] he proved the slope inequality and Severi’s conjecture for surfaces having an irrational pencil. In the 90’s Konno [Kon96] proved the conjecture in the special case of even surfaces, i.e., surfaces whose canonical class is 2− divisible in the Picard group. Finally, at the end of the 90’s, Manetti [Man03] could prove the inequality for surfaces of general type whose canonical bundle is ample. Manetti’s Proof Manetti considers the tautological line bundle L of the P1 −bundle π : P(ΩS1 ) → S; standard computations give 3(KS2 − 4χ) = L2 · (L + π ∗ KS ). Then, using the fact that ΩS1 is generically globally generated, he can write the right hand side of the above equation as 2KS E + (L + π ∗ KS )C for an effective 1−cycle C in P(ΩS1 ), and where E is the maximal effective divisor


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in S such that h0 (ΩS1 (−E)) = h0 (ΩS1 ). Thus the problem is reduced to the nonnegativity of the term (L + π ∗ KS )C. This is obvious if Ω 1 (KS ) is nef, but in general it requires a very detailed and complicated analysis of the 1−cycle C. In fact, Pardini’s proof not only does not require the ampleness of the canonical divisor, but is much easier than Manetti’s. We should however mention that Manetti’s argument leads to a very detailed description of the equality case, showing that a surface of general type of maximal Albanese dimension, lying on the Severi line (K 2 = 4χ), and having ample canonical class, has irregularity q = 2 and is a double cover of a principally polarized Abelian surface, branched on a divisor D algebraically equivalent to 2Θ. Up to now there is no similar description of the limit case without the assumption that K be ample. Pardini’s Proof Pardini’s idea is to construct a sequence of genus gd fibrations fd : Yd → P1 such that lim gd = +∞ and lim λ(fd ) = KS2 /χ(OS ). d→∞

d→∞

Then, taking the limit of the left-hand side of the slope inequality, one gets the desired inequality KS2 /χ(OS ) ≥ 4. To construct these fibrations, she considers the Cartesian diagram p

S α

/S α

Alb(S)

·d

/ Alb(S) ,

where d : Alb(S) → Alb(S) is multiplication by d. One observes that S is connected since we have a surjection π1 (S) → π1 (Alb(S)) = H1 (S, Z). Clearly, KS2 = d2q KS2 , χ(OS ) = d2q χ(OS ). ∗ Let L be a very ample divisor on Alb(S) and set H := α∗ L, H := α L. 2 Then p∗ H ∼num d2 H , whence H = d2q−4 H 2 and KS H = d2q−2 KS H. Let now D1 , D2 ∈ |H | be two general curves and define C1 := D1 + D2 ∈ |H1 + H2 |. Moreover, choose C2 ∈ |2H | sufficiently general such that C1 and C2 intersect transversally. C1 and C2 define a rational pencil fd : Yd → P1 , where Yd is the blow up of S at C1 ∩ C2 . The singular fibre induced by C1 guarantees that fd is not a holomorphic bundle, whence the slope λ(fd ) is well defined. For the invariants of Yd we get KY2d = KS2 − 4H = d2q KS2 − 4d2q−4 H 2 2

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χ(Yd ) = χ(S ) = d2q χ(S) gd = 1 + KS H + 2H = 1 + d2q−2 KS H + 2d2q−4 H 2 2

and therefore limd gd = +∞ as requested. Moreover, Kf2d = KY2d + 8(gd − 1) and χ(fd ) = χ(Yd ) + (gd − 1) and we see that both invariants are polynomials in d of degree 2q, whose leading terms are respectively KS2 and χ(OS ). In particular, lim λ(fd ) = lim Kf2 /χf = KS2 /χ(OS )

d→∞

d→∞

2 Surfaces with χ = 1 and the Bicanonical Map 2.1 The Bicanonical Map The behaviour of the m − th canonical map of S (i.e., the rational map associated to |mKS |) is an essential tool in the theory of surfaces of general type. As we mentioned in the introduction, the cases where m ≥ 3 are solved since long (cf. the survey [Cat87b]). The canonical map (m = 1) was first studied by Beauville in [Bea79], but there remain still many unresolved questions. The case m = 2 was particularly studied in the last years, and we have the impression that we are very close to a complete understanding. In order to fix the starting point, we summarize the results of several authors ([Fra91], [Rei88], [Cat81], [CC91], [CC93], [Xia85a]) in the following Theorem 6. Let S be a minimal surface of general type. Then • the bicanonical map is generically finite unless pg = 0 and K 2 = 1; • if KS2 ≥ 5 or pg ≥ 1, the bicanonical map is a morphism. Note that, if pg = 0 and K 2 = 1, then P2 = 2 and the bicanonical map is a rational (b = 0) fibration. In all known examples this is a genus 4 fibration, although at the moment it is only proven that its genus is 3 or 4 (see [CP05]). These surfaces are usually called numerical Godeaux surfaces. Numerical Godeaux surfaces with torsion (in the Picard group) of cardinality at least 3 are classified in [Rei78], a family with torsion Z/2 was constructed in [Bar84]. Up to last year only sporadic examples of surfaces with trivial torsion were known, but recently Schreyer [Sch05] has announced the construction of a family of the expected dimension (= 8) using a new approach based on homological algebra. The above theorem says that in all other cases the bicanonical map maps S to a surface, and it is a morphism (except for finitely many families). In the last years many people studied the degree of this map, in particular, trying to classify the surfaces such that the bicanonical map is not birational.


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The Standard Case It is well known that the bicanonical map of a smooth curve of general type (i.e., of genus at least 2) fails to be birational if and only if the curve has genus 2. This exception induces a “standard exception” to the birationality of the bicanonical map in dimension 2. Definition 5. A surface S of general type presents the standard case (for the non birationality of the bicanonical map) if there exists a dominant rational map onto a curve f : S B whose general fibre is irreducible of genus 2. In fact, if S presents the standard case, then the restriction of the bicanonical map of S to a general fibre factors through the bicanonical map of the fibre itself and therefore cannot be birational. The subschemes of the moduli space corresponding to surfaces presenting the standard case are not empty for infinitely many moduli spaces, and Persson [Per81] constructed many interesting surfaces considering double covers of ruled surfaces branched on relative sextics, thereby filling a big region of the convex region represented in figure 1. Bombieri ([Bom73]) showed that the standard case gives almost all exceptions to the birationality of the bicanonical map. More precisely, combining his results with those of Reider ([Rei88]) we know now that a minimal surface of general type with K 2 ≥ 10 either presents the standard case, or its bicanonical map is birational. In particular, the exceptions to the birationality of the bicanonical map not presenting the standard case belong to finitely many families and many authors are trying since then to classify them (see [Cil97] for a survey updated until ’96). Du Val’s Double Planes In the same paper [Bom73] Bombieri constructed a surface of type (K 2 , pg , q) = (9, 6, 0) not presenting the standard case. His example can be easily described as a hypersurface F14 of degree 14 in the weighted projective space P(1, 1, 2, 7), and from this description it follows rightaway that it bicanonical map is a double cover of the weighted projective space P(1, 1, 2) (isomorphic to a quadric cone in P3 ). This example is in fact a special case of a more general “geometric” situation studied first by du Val. Let S be a minimal regular surface with pg ≥ 2, such that the general canonical curve is irreducible, smooth and hyperelliptic. Since the restriction of the bicanonical map ϕ2K to a canonical curve factors through the canonical map of the curve itself, ϕ2K cannot be birational. Du Val [Duv52] gave a list of such surfaces obtained as double covers of rational surfaces. A generalization (see [CML00], [Bor03]) leads to the following:

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Definition 6 (du Val’s double planes). A smooth surface S is a du Val double plane if it is birational to D) a double cover of P2 branched over a smooth curve of degree 8; Dn ) a double cover of P2 branched over the union of a curve of degree 10 + n, n ≤ 6, with n distinct lines through a point p, such that the essential singularities of the branch curve are the following: – p is a singular point of multiplicity 2n + 2, – there is a singular point of type [5, 5] on each line, – possibly there are some quadruple points and some points of type [3, 3]; B) a double cover of the Hirzebruch surface F2 whose branch curve can be decomposed as C0 + G , G ∈ |7C0 + 14Γ | (where |Γ | is the ruling of F2 and C0 is the section with self intersection −2), whose only essential singularities are [3, 3] points that are tangent to a fibre. Recall that a singular point of type [d, d] is a singular point of multiplicity d having a further singular point of multiplicity d infinitely near to the first one. In other words, if we blow-up the singular point, the strict transform of the curve has one more singular point of multiplicity d lying on the exceptional divisor. Remark 1. In the definition of the du Val’s double planes of type Dn and B we only care about the essential singularities of the branch curve (as usual in the theory of double covers) since adding a simple singularity to the branch curve does not affect the properties of the resulting surface we are interested in. On the contrary, in the definition of the double planes of type D, we assume the branch curve to be smooth. In fact, if we take a double cover of P2 branched over a curve of degree 8 with a double point, the pull back of the pencil of lines through this point to the surface defines a pencil of curves of genus 2 (through a singular point of the surface), so the resulting surface presents the standard case. Note that this example shows that one can degenerate surfaces presenting a nonstandard case to surfaces presenting the standard case (just take a family of smooth plane curves of degree 8 degenerating to a singular one and consider the corresponding family of double covers). Borrelli proved that this list is “complete” in the following sense Theorem 7 (Borrelli [Bor03]). If S is a minimal surface of general type, not presenting the standard case, whose bicanonical map factors through a degree 2 rational map onto a rational or ruled surface: then S is the smooth minimal model of a du Val double plane. In particular, either q = 0 or pg = q = 1. The “Classification” The standard case and the du Val’s double planes do not give all possible surfaces of general type with nonbirational bicanonical map, but the remaining exceptions are really few.


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What is known about these is summarized in the following Theorem 8. Let S be a smooth minimal surface of general type whose bicanonical map ϕ2K is not birational. Then one of the following cases occur: i) S presents the standard case; ii) S is the smooth minimal model of a du Val double plane; iii) S is a surface of type (1, 1, 0) (for these automatically deg ϕ2K = 4 and S is a complete intersection of two sextics in P(1, 2, 2, 3, 3)); iv) S is of type (2, 1, 0) with Picard group having torsion Z/2Z (for these automatically deg ϕ2K = 4 and its double cover corresponding to the torsion class is a complete intersection of two quartics in P(1, 1, 1, 2, 2)); v) ϕ2K is 2 : 1 onto a K3 surface and pg = 1, q = 0, 2 ≤ K 2 ≤ 8; vi) S is of type (6, 3, 3) or of type (4, 2, 2) (for these both cases automatically ϕ2K has degree 2 and we have a nonstandard case) vii)S has pg = q ≤ 1. All these cases with the exception of pg = q ≤ 1 are now rather clear. The history of this theorem is rather complicated and combines the efforts of several authors. We try to reconstruct its more important steps here, giving some more details on each class. One of the first results in this direction is due to Xiao Gang [Xia90], giving, in the nonstandard case, and under the assumption that the degree d of the bicanonical map is at least 3, a list of the possible values of d and of the possible places in the Enriques classification of the bicanonical image Σ. In 1997, Ciliberto, Francia and Mendes Lopes [CFML97] gave a complete classification of the case pg ≥ 4, essentially confirming du Val’s list. Then, Ciliberto and Mendes Lopes, with contributions of the second author and Borrelli, worked in the next years to extend the classification to pg ≥ 2 (see [CCML98], [CML00], [CML02a], [CML02b], [Bor02]). The case pg = 1 and q = 0, giving cases iii), iv) and v) is classified in [Bor03]. In fact, cases iii) and iv) resulted already from the analysis of [Xia90] where it is proven that, if deg ϕ2K ≥ 3, then either S is of type (1, 1, 0) or of type (2, 1, 0), or with pg = q ≤ 2. The description given in iii) and iv) of the first two cases comes from the papers [Cat79] and [CD89], where all surfaces of respective types (1, 1, 0) and (2, 1, 0) are classified. In particular, it is shown that all surfaces of type (1, 1, 0) are as in iii). Remark 2. Surfaces of type (2, 1, 0) without torsion in homology, also sometimes called Catanese-Debarre surfaces, offer the following interesting phenomenon: there is an irreducible component of the moduli space such that 1) for the general surface the bicanonical map is birational, while there are subvarieties for which the bicanonical map can respectively be 2) of degree 2 onto a K3- quartic surface, 3) of degree 2 onto a rational quartic surface, 4) of degree 4 onto a smooth quadric surface.

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The surfaces in v) are usually called Todorov surfaces, since they were introduced in [Tod81]. The subspaces of the moduli spaces corresponding to them is described in [Mor88]. Finally, the largely open case vii) is very strongly related with the problem, of independent interest, of the classication of surfaces of general type with pg = q. We shall describe in the next subsection what is currently known on these surfaces, showing in particular that we have a very precise description of the two cases in vi). 2.2 Surfaces with pg = q These are the surfaces corresponding to the “vertical” piece of the bold line in figure 1. In particular, 1 ≤ K 2 ≤ 9. Surfaces with pg = q ≥ 4 This case is clear, by corollary 1 of Beauville’s theorem 4. If pg = q ≥ 4, then S is a product of two genus 2 curves and pg = q = 4. We recall that then K 2 = 8 and clearly the bicanonical map has degree 4, and we have a standard case. Surfaces with pg = q = 3 These surfaces have been first studied in [CCML98], and a complete classification has been recently achieved independently by Pirola [Pir02] and HaconPardini [HP02]. The result is the following Theorem 9. A minimal surface of general type with pg = q = 3 has K 2 = 6 or K 2 = 8 and, more precisely, • if K 2 = 6, S is the symmetric square of a genus 3 curve; • otherwise S = C2 × C3 /τ , where Cg denotes a curve of genus g and τ is an involution of product type acting on C2 as an elliptic involution (i.e., with elliptic quotient), and on C3 as a fixed point free involution. In particular, the moduli space of minimal surfaces of general type with pg = q = 3 is the disjoint union of M6,3,3 and M8,3,3 , which are both irreducible of respective dimension 6 and 5. We sketch the idea of the proof. By Debarre’s inequality (in the “stronger” form: q > 0 ⇒ KS2 ≥ 2pg ), pg = q = 3 implies K 2 ≥ 6. As in the proof of Beauville’s theorem, consider now the map


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∧ : Λ2 (H 0 (ΩS1 )) → H 0 (ΩS2 ). Since pg = q = 3, it is a linear map between two three dimensional spaces. If this is not an isomorphism, then (since every vector in Λ2 C3 is decomposable) there are two nontrivial 1−forms ω1 and ω2 with ω1 ∧ ω2 ≡ 0. This yields then (by Castelnuovo-De Franchis) a pencil f : S → B with b, g ≥ 2. Then, by Beauville’s inequality, b = g = 2 and the fibration is a holomorphic bundle (this forces KS2 = 8). Therefore f is induced by a map π1 (B) → Aut(F ) (where F is a smooth fibre), whose kernel induces an unramified cover ϕ : C → B Galois with group G. Since q = 3, the quotient of F by the group G has genus 1. By Hurwitz’s formula one easily sees that, if φ : F → F/G is branched in one point, then |G| ≤ 4, hence G is Abelian, contradicting that φ is ramified. Again Hurwitz’s formula shows that φ is branched in 2 points and G ∼ = Z/2. Otherwise ∧ is an isomorphism, and therefore S does not have any pencil f : S → B with b ≥ 2. In particular α(S) is a surface, a divisor Θ in Alb(S). Pirola noticed that Θ must be ample, else it would have an elliptic fibration and therefore an irrational pencil with base of genus b ≥ 2. This implies, by Lefschetz’s hyperplane theorem, that the induced map 1 1 ) → H 1 (ΩS1 ) is injective: since, for any class η ∈ H 1 (ΩAlb(S) ) H 1 (ΩAlb(S) c1 (Θ) ∧ η ∧ η¯ > 0. In particular, h1 (ΩS1 ) ≥ 9 and this (since by Hodge theory 12χ − K 2 = c2 = 2pg − 4q + 2 + h1 (Ω 1 )) implies KS2 ≤ 7. The case KS2 = 6 was already settled in [CCML98], where it is first shown that the degree of the scheme of base points is K 2 − 6, and then that in the case KS2 = 6 α is an embedding. More precisely it is shown that its image is a theta divisor in a principally polarized abelian threefold and therefore S is the symmetric square of a genus 3 curve. What remains to prove is M7,3,3 = ∅, and this is done in [Pir02] by a careful study of the paracanonical system. The fact that the bicanonical map has degree 2 is an easy consequence of the adjunction formula by which KS is the pull back of Θ, and of the fact that the sections of OA (2Θ) are invariant, as well as Θ, for the symmetry of A sending x → −x. Surfaces with pg = q = 2 This case is still far from being classified. Ciliberto and Mendes Lopes [CML02a] classified all surfaces with pg = q = 2 and non-birational bicanonical map (not presenting the standard case). Their result, corresponding to the subcase (4, 2, 2) of case vi) of theorem 8, is the following Theorem 10. If S is a minimal surface of general type with pg = q = 2 and non-birational bicanonical map not presenting the standard case, then S is a double cover of a principally polarized abelian surface (A, Θ), with Θ

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irreducible. The double cover S → A is branched along a divisor B ∈ |2Θ|, having at most double points. In particular KS2 = 4. Note that, again by Debarre’s inequality, pg = q = 2 ⇒ K 2 ≥ 4, so Ciliberto and Mendes Lopes’ surfaces belong to the limit case. Their theorem solves completely the problem of the non birationality of the bicanonical map in this case, but of course a complete classification of minimal surfaces of general type with pg = q = 2 would be interesting by itself. Results in this direction have been recently obtained by F. Zucconi; to explain them we need to give the following definition. Definition 7. A surface S is said to be isogenous to a (higher) product if S admits an unramified finite covering which is biholomorphic to a product of two curves of respective genera at least 2. We have already seen surfaces isogeneous to a product in our analysis of surfaces with pg = q, namely all surfaces in M8,4,4 and all surfaces in M8,3,3 . Zucconi’s theorem is the following Theorem 11 (2.9 in [Zuc03]). There are two classes of minimal surfaces of general type with pg = q = 2 whose Albanese image is a surface and having an irrational pencil, and they are both isogenous to a higher product. More precisely, either they have a double cover which is a product of two genus 2 curves or they are a quotient of the product of two genus 3 curves by an action of Z/2Z. In both cases Zucconi describes precisely the group action as a diagonal action induced by actions on the two curves. The interested reader will find all details in Zucconi’s paper. Zucconi managed also to remove the hypothesis on the Albanese map, by use of a special class of surfaces isogenous to a higher product, the generalized hyperelliptic surfaces introduced in [Cat00]. Definition 8. Let C1 and C2 be two smooth curves, G a finite group with two injections respectively in Aut(C1 ) and Aut(C2 ). Then the quotient surface S = C1 × C2 /G by the diagonal action is said to be a generalized hyperelliptic surface if • the projection C1 → C1 /G is unramified; • C2 /G is rational. Then Zucconi proved Theorem 12. If S has pg = q = 2, and the image of the Albanese map is a curve, then S is a generalized hyperelliptic surface. What remains to be classified is the class of surfaces with pg = q = 2 having no irrational pencils. Chen and Hacon, in a preprint, constructed an example of surfaces with pg = q = 2, K 2 = 5 and Albanese morphism of degree 3.


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Surfaces with pg = q = 1 In this case Debarre’s inequality gives only K 2 ≥ 2. The Albanese morphism is a map onto an elliptic curve, in particular, all these surfaces have a fibration with base of genus b = 1. We summarize in the following statement what is known about these surfaces. Theorem 13. • M2,1,1 is unirational (by this we mean: irreducible and unirational) of dimension 7. The Albanese map of all these surfaces is a genus 2 fibration. • M3,1,1 has 4 connected components, all unirational of dimension 5. The Albanese map is a genus 3 fibration for the surfaces in one of those components, and a genus 2 fibration in all other cases. • M4,1,1 , M5,1,1 and M8,1,1 are non empty. Actually much more can be said, and we try to be more precise in the following. First the most mysterious cases. It remains unsettled the existence of surfaces of general type with pg = q = 1 and K 2 = 6, 7, 9. In a recent preprint by Rito appears the construction of a surface with pg = q = 1 and K 2 = 6 as a double cover of a Kummer surface modifying slightly (i.e., adding a singular point to the branch curve) Todorov’s construction of a surface with pg = 1, q = 0 and K 2 = 8 in [Tod81]. Its construction makes use of the computer program MAGMA (to find a branch curve with the right singularities). Second, the “partially understood” cases. Examples of surfaces with pg = q = 1 and K 2 = 4, 5 were constructed by the second author as bidouble covers in [Cat99]. In both cases the Albanese map turns out to be a genus 2 fibration, so they present the standard case. The case K 2 = 8 was studied by Polizzi [Pol06], who considered the cases of surfaces having bicanonical map of degree 2. He could prove that all these surfaces are isogenous to a product and that they form three components of the moduli space, one of dimension 5 and two of dimension 4. All these surfaces do not contain any genus 2 pencil and they are in fact du Val double planes. Finally, the cases K 2 = 2, 3 are completely classified. The first to be settled was K 2 = 2, done by the second author in [Cat81], representing all those surfaces as double covers of the symmetric square of their Albanese curve. The case K 2 = 3 was first studied in [CC91] where it was shown, among other things, that the Albanese map could be either a genus 2 or a genus 3 fibration. The case g = 3 was then classified in [CC93], showing that it gives a unirational family of dimension 5. Note that, if there is surface with pg = q = 1, K 2 ≤ 3 and nonbirational bicanonical map not presenting the standard case, it must belong to this family.

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The question whether such a surface exists is still open. Recently Polizzi [Pol05] has shown that a general surface in this component has birational bicanonical map, but this is not true for all of them, since Xiao [Xia85b] has found a subfamily of dimension 1 having a genus 2 pencil. The classification of the case K 2 = 3 was completed in [CP05] classifying all those surfaces having Albanese fibres of genus 2. The main tool for this classification is a new method for studying fibrations f : S → B of genus 2, and fibrations of genus 3 with general fibre non hyperelliptic, basically giving generators and relations of their relative canonical algebra R(f ) = ⊕Vn , seen as a sheaf of algebras over B. Let us recall the vector bundles Vn introduced in the previous section as ⊗n . Roughly speaking then, R(f ) is a bundle whose fibres are the Vn = f∗ ωS|B canonical rings of the fibres of f . We state here only the theorem for genus 2 fibrations, since it is the one used in order to complete this classification. Theorem 14. A genus 2 fibration f : S → B is determined by the following 5 data • • • • •

the base curve B; the rank 2 vector bundle V1 := f∗ ωS|B over B; an effective divisor τ on B; a class ξ ∈ Ext1OB (S 2 (V1 ), Oτ )/(AutOB (Oτ ) yielding V2 ; letting A be the subring of the relative canonical algebra generated by V2 , V3+ the (+1) eigenbundle for the hyperelliptic involution on the fibres, and defining A˜6 := Hom((V3+ )2 , A6 ) (where A6 is the image in A of S 3 (V2 )), the last datum is an element w ∈ P(H 0 (A˜6 )).

Moreover, deg V1 = χ(OS ) − (b − 1), deg τ = KS2 − 2χ(OS ) − 10(b − 1). We want to explain here the geometry behind this theorem, which at a first glance can appear slightly technical. The vector bundles Vn yield the degree n part of the canonical ring of each fibre. So each of these vector bundles induces a rational map, the relative ncanonical map, from S to the corresponding projective bundle P(Vn ), mapping each fibre via its n-canonical map. The multiplication map of degree 1 forms give a morphism of sheaves S 2 (V1 ) → V2 which fits into an exact sequence 0 → S 2 (V1 ) → V2 → Oτ → 0 for an effective divisor τ on B supported on the image of the “bad” fibres (those which are not 2-connected, i.e., the fibres that can be decomposed as A + B with A, B effective divisors such that A · B = 1). ξ is the class of this extension. Therefore ξ yields V2 and determines the relative bicanonical map. Since the bicanonical map of a genus 2 curve is


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a double cover of a conic (branched in 6 points), this map has degree 2 onto a conic subbundle C of the P2 -bundle P(V2 ). We have C = P roj(A). In [CP05] it is proven that the relative bicanonical map is a morphism contracting at most some rational curves with self-intersection (−2), which implies that the branch curve has no “essential” singularities. In fact the 5th datum w determines the branch curve divC (w). In fact, sections of S 3 (V2 ), or of a twist of it, are “equations” of divisors in P(V2 ) which cut a cubic curve on each fibre. Taking the quotient by the subsheaf corresponding to the equations vanishing on the conic bundle C, one gets an equation for a divisor on the conic bundle, which cuts 6 points (intersection of a cubic and a conic) on a general fibre, and gives our branch curve. Let us come back to the case pg = q = 1, K 2 = 3 and g = 2. One needs to construct a suitable genus 2 fibration over an elliptic curve, with, in the sense of theorem 14, deg V1 = deg τ = 1. This is done in [CP05], by studying vector bundles on elliptic curves, and three different families are found. Let us finally mention that P(V1 ) is the symmetric square of B. In fact, our double cover is birational to a double cover of it. The behaviour of this double cover was described in [CC91], characterising these surfaces as double covers of the symmetric product of an elliptic curve with branch locus belonging to a certain algebraic system with prescribed singularities. This new method shows then, rather surprisingly, that this algebraic system is not connected. Surfaces with pg = q = 0 The class of surfaces with pg = q = 0 is one of the most complicated and intriguing classes of surfaces of general type. By the standard inequalities we have: 1 ≤ KS2 ≤ 9. We have already mentioned the case K 2 = 1, of the numerical Godeaux surfaces, the only case for which the bicanonical map is not finite, so let us restrict to KS2 ≥ 2. These surfaces are very far from being classified. From the point of view of the bicanonical system, this case was object of an intensive analysis by Mendes Lopes and Pardini in the last years. What it is known on the degree of the bicanonical map can be summarized in the following Theorem 15 ([MLP05], [MLP02]). Let S be a surface with pg = q = 0. Then • • • •

if K 2 = 9 ⇒ deg ϕ2K = 1, if K 2 = 7, 8 ⇒ deg ϕ2K = 1 or 2, if K 2 = 5, 6 ⇒ deg ϕ2K = 1, 2 or 4, if K 2 = 3, 4 ⇒ deg ϕ2K ≤ 5 and if moreover ϕ2K is a morphism, then deg ϕ2K = 1, 2 or 4,

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• if K 2 = 2 then obviously, since the image of the bicanonical map is P2 , then the bicanonical map is non birational, and obviously we have: deg ϕ2K ≤ 8, equality holding if and only if ϕ2K is a morphism. Let us recall that, by Reider’s theorem, the bicanonical map is a morphism as soon as K 2 ≥ 5. In fact the bicanonical map of all known examples of surfaces with pg = q = 0 and K 2 ≥ 2 is a morphism. So one could suspect4 that the bicanonical map is always a morphism whenever K 2 ≥ 2. Langer ([Lan00]) has proven that the bicanonical map of a minimal surface of general type with pg = q = 0 and KS2 = 4 has no fixed part. Mendes Lopes and Pardini gave also a description of some of these surfaces having non birational bicanonical map, in particular for K 2 ≥ 6. We summarize here some of their results Theorem 16 ( [MLP03], [MLP01], [MLP04a], [MLP04b]). Let S be a minimal surface of general type with pg = q = 0 whose bicanonical map is not birational. Then the image of the bicanonical map is a rational surface unless K 2 = 3, deg ϕ2K = 2, and the image is an Enriques sextic. These last surfaces form an irreducible and unirational family of dimension 6 of the moduli space. Moreover, • if K 2 = 8, S has an isotrivial genus 3 rational fibration whose general fibre is hyperelliptic with 6 double fibres; • if K 2 = 7, S has a genus 3 rational fibration whose general fibre is hyperelliptic with 5 double fibres and a fibre with reducible support, consisting of two components; • if K 2 = 6 and deg ϕ2K = 2, S has a genus 3 rational fibration whose general fibre is hyperelliptic with 4 or 5 double fibres; • if K 2 = 6 and deg ϕ2K = 4, S is a Burniat surface. Remark 3. Surfaces with pg = q = 0 and K 2 = 3, 4 were constructed by Keum ([Keu88]) and Naie ([Nai94]). For K 2 = 3 the degree of the bicanonical map can be equal to 2 and to 4, and it is an open question if it can be birational. Concerning the classification of surfaces with pg = q = 0 there has been recent progress. We would like to mention that in [BCG05b] a complete classification of surfaces with pg = q = 0 isogenous to a product (this forces K 2 = 8) is given. All the known surfaces with K 2 = 8, pg = q = 0 have the bidisk as universal covering. In the case pg = q = 0 and K 2 = 9 all the surfaces in question are, by Yau’s theorem ([Yau77] and [Yau78]), quotient of the complex unit ball in C2 by a discrete group Γ acting freely. The first effective example of such 4

Added in proof: Mendes Lopes and Pardini give in math.AG/0602633 an example of surfaces with K 2 = 2, pg = q = 0 such that |2KS | has base points.


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surfaces, called fake projective planes since they have the same Betti numbers as the projective plane, was given by Mumford ([Mum79]) using 2-adic uniformization. Other examples were given later in [IK98], while recently Keum ([Keu05]) gave an explicit geometric construction as a cyclic cover of a particular Dolgachev surface. In a recent preprint ([PY05]) G. Prasad and S.K. Yeung, using the arithmeticity of Γ , give twelve rather explicit lists of fake √ projective planes, each corresponding to an imaginary quadratic field Q( −a) and a prime p which ramifies in it. The interesting geometric features of these examples are that: i) all these groups Γ are indeed contained in SU (2, 1), hence the canonical divisor K is divisible by 3, ii) all these surfaces have a nontrivial first homology group H1 (S, Z).

3 Surfaces with pg = 4 In Enriques’ book on algebraic surfaces [Enr49] much emphasis was put on the effective construction of surfaces whose canonical map is birational, particularly for surfaces with pg = 4, where the canonical image is a surface in P3 . Later on, in particular in the last thirty years, many authors studied surfaces with pg = 4, with particular interest in the construction of surfaces with pg = 4, birational canonical map and K 2 as high as possible. If the canonical map of a minimal surface of general type S with pg = 4 is birational, then the standard inequalities give 5 ≤ K 2 ≤ 45. Nowadays we know examples, by the contribution of several authors, for every value of KS2 in the range 5 ≤ KS2 ≤ 28 (cf. e.g. [Cil81], [Cat99]). An example with KS2 = 31 has been recently obtained in [Lie03], although the example constructed has a big fixed part of the canonical system so that its canonical image has “only” degree 12. Moreover, the first two authors together with F. Grunewald have constructed a canonical surface in the projective 3space with K 2 = 45. This surface is obtained as a Galois covering of the plane with group (Z/5Z)2 , branched over a configuration of lines introduced by Hirzebruch (cf. [BCG05c]). In this case we have a rigid surface such that its canonical system has a fixed part. Obviously also in this case classification is the biggest challenge: for which values of KS2 is it possible to classify all possible minimal surfaces of general type with pg = 4? And more ambitiously: for which values of KS2 , q it is possible to completely describe the moduli space MK 2 ,4,q ? 3.1 K 2 = 4, 5 The cases K 2 = 4, 5 were already treated by Enriques ([Enr49], section 2, chapter VIII, pp.268–271), and the corresponding moduli spaces were completely understood already in the 70’s.

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We briefly recall these results. By Debarre’s inequality, all these surfaces are regular (in fact, this is true for K 2 ≤ 7). The canonical map of surfaces with K 2 = 4 and pg = 4 is a morphism of degree 2 onto an irreducible quadric in P3 , i.e., either a smooth quadric or a quadric cone. The general surface is a double cover of a smooth quadric branched over a smooth complete intersection with a sextic surface. A detailed analysis of the corresponding moduli space can be found in [Hor76], where the following is proven. Theorem 17. M4,4,0 is irreducible, unirational of dimension 42, its singular locus is irreducible of codimension 1 and corresponds exactly to the surfaces whose canonical image is a quadric cone. We have two classes of minimal surfaces of general type with K 2 = 4: let us say surfaces of type I (double covers of a smooth quadric) and II (double covers of a quadric cone). Correspondingly we have a stratification of M4,4,0 as a union of two locally closed strata, both irreducible, which we denote simply by I and II, of respective dimension 42 and 41. To draw a picture of this moduli space we need the following notation: Definition 9. Let A and B be two (locally closed) irreducible strata of a moduli space MK 2 ,pg ,q . If we write “A → B”, it means that there is a flat family with base a small disc ∆ε ⊂ C, whose central fibre is of type B and whose general fibre is of type A. In other words it means that the closure of the stratum A intersects the stratum B. With this notation a picture of M4,4,0 is the following: 42

I

41

II

Note that at the left of each stratum stands the dimension of the corresponding irreducible stratum. The case K 2 = 5 is slightly more complicated, and completely described in [Hor75]: the canonical map is either a birational morphism to a quintic in P3 (type I), or a rational map of degree 2 onto an irreducible quadric, which can be as in the previous case either smooth (type IIa ) or a quadric cone (type IIb ). Theorem 18. M5,4,0 has two irreducible components, both unirational of dimension 40, intersecting in a 39 dimensional subvariety.


25

Here is the picture for M5,4,0 40

39

IIa I >> | >> | | >> | >> || ~|| IIb

The moduli space has two irreducible unirational components of dimension 40 whose general point corresponds to surfaces with canonical image respectively a quintic or a smooth quadric. The surfaces whose canonical image is a quadric cone form a 39-dimensional subvariety of this moduli space, the intersection of the two irreducible components. 3.2 K 2 = 6 This case is much more complicated and surfaces with K 2 = 6 and pg = 4 were completely classified by Horikawa in [Hor78], obtaining a stratification of M6,4,0 in 11 strata. We will not enter here the details of this classification. A complete understanding of M6,4,0 is still missing, since it is not clear how exactly these 11 strata “glue” toghether. Theorem 19 ([Hor78], [BCP04]). M6,4,0 has 4 irreducible components, all unirational, one of dimension 39, the others of dimension 38. Moreover, the number of connected components of M6,4,0 is at most two. The main question on this moduli space remains the following: Question 1. Is M6,4,0 connected? Here is a partial picture:

39

37

IVa1 Ia {{ xx { x { x {{ xx |xx }{{ IVa2 IVb1

36

IVb2

38

IIIa { { {{ {{ }{{ V1 IIIb o || || | | }|| V2

II

Ib

This picture is partial because up to now it is not known whether all possible arrows are drawn. More precisely, M6,4,0 is connected if and only if one of the two following degenerations is possible: Ia → V1 or Ia → V2 .

26


This picture was done by Horikawa in [Hor78] with the exception of the horizontal line IIIb ← II, recently obtained in [BCP04]. We are going to explain how this arrow was obtained. We need to construct a flat family of surfaces whose central fibre is of type IIIb and whose general fibre is of type II. The difficulty lies in the obvious fact that, by reasons of dimension, the general surface of type IIIb cannot deform to a surface of type II. First however we need to explain what surfaces of type II and of type IIIb are. Surfaces of type II are defined as follows Definition 10. A minimal surface of general type with pg = 4 and K 2 = 6 is of type II if the canonical map has degree 3. It is immediate, that the canonical image is a quadric cone. In fact, one can see that their canonical models X are hypersurfaces of degree 9 in the weighted projective space P(1, 1, 2, 3), so that the canonical divisor of X is divisible by 2 as a Weil divisor. Definition 11. A minimal surface of general type with pg = 4 and K 2 = 6 is of type IIIb if it has no genus 2 pencil and the canonical system has a fixed component. Horikawa gave a very concrete description of this class (theorem 5.2 in [Hor78]) showing, among other things, that the canonical map is a double cover of a quadric cone. In both cases there is a pencil L on S, the strict transform of the ruling of the quadric cone, such that KS − 2L is effective. Computing intersection numbers one sees (cf. also [MLP00] for case II) that in both cases we have a decomposition KS = 2L+Z where Z is a fundamental cycle (Z 2 = −2, KZ = 0) and L is a genus 3 pencil with one simple base point. The main difference between the two cases is that in case IIIb all curves in the pencil are hyperelliptic, while in case II the general curve is nonhyperelliptic. The idea is then to construct a family preserving this decomposition and therefore the genus 3 pencil. Since Z is a fundamental cycle, one can consider the canonical model X of S, the surface with rational double points obtained contracting all fundamental cycles. The canonical class of X is then ˜ L ˜ being the image of L on X. 2−divisible (as a Weil divisor), since KX = 2L, ˜ the name semicanonWe consider the semicanonical ring R = R(X, L): ical being selfexplanatory, since the subring generated by the homogeneous elements of even degree is exactly the canonical ring of X (and of S). In order to compute the ring R we use the hyperplane section principle ˜ [Rei90a], first computing the quotient ring R(X, L)/(x 0 ), where x0 is a general homogeneous element of degree 1, i.e., corresponding to a general element C ˜ of the pencil L.


27

It is not difficult to prove that this ring equals the ring 3 3m R(C, P ) := ]P )) H 0 (C, OC ([ 2 2 m (we refer for the definition of the ring structure to [BCP04]), which is a suitable subring of the ring R(C, P ), where P is the base point of L (and a Weierstrass point for the hyperelliptic curve C). Then the strategy is the following: i) take any hyperelliptic genus 3 curve C, a Weierstraß point P on it, and compute the ring R(C, 32 P ) ii) “deform” it, adding an element of degree 1 in a flat way, to get the semicanonical ring of a surface of type IIIb iii) construct a flat family of rings (say with parameter t), whose central fibre (t = 0) is the ring constructed in ii), and whose general fibre (t = 0) is a hypersurface ring of type II Step i) is easy, steps ii) and iii) depend on the result of step i), concerning R(C, 32 P ). It turns out that R(C, 32 P ) is a Gorenstein ring of codimension 4, which can be expressed in a nice way: Proposition 1. Let C be a hyperelliptic curve of genus 3, p ∈ C a Weierstraß point. Then R(C, 32 p) ∼ = C[x, y, z, w, v, u]/I, where deg(x, y, z, w, v, u) = (1, 2, 3, 4, 5, 6) and the ideal I is generated by the 4 × 4 Pfaffians of the skewsymmetric ’extra-symmetric’ matrix ⎞ ⎛ 0 0 z v y x ⎜ 0w u z y ⎟ ⎟ ⎜ ⎜ 0 P˜9 u v ⎟ ⎟, M =⎜ ⎜ 0 w2 zw⎟ ⎟ ⎜ ⎝ 0 0⎠ −sym 0 where P˜9 is homogeneous of degree 9 in the variables x, y, z, w. The graded matrix M has the nice property to be “extrasymmetric”. Extrasymmetric matrices. Extrasymmetric matrices were introduced by Miles Reid and Duncan Dicks ([Rei90a], [Rei89]). Let A be a polynomial ring and let M be a skew ’extrasymmetric’ matrix of the form ⎛ ⎞ 0 ab c d e ⎜ 0f g h d⎟ ⎜ ⎟ ⎜ 0 i g c⎟ ⎜ ⎟ ⎜ 0 pf pb ⎟ ⎜ ⎟ ⎝ 0 pa⎠ −sym 0

28


where a, b, c, d, e, f, g, h, i, p ∈ A. Then 6 of the 15 (4 × 4) Pfaffians belong to the ideal generated by the other 9; moreover, if the entries are general enough, the ideal generated by these pfaffians has exactly 16 independent syzygies, which can all be written explicitely as functions of the entries of the matrix. This implies that, if we have a ring presented in this form, and the ring has no further syzygies, deforming the entries of the matrix (preserving the symmetries), we obtain automatically a a flat deformation of the ring. We mention that recent studies lead to generalizations of this “format”: here we need only this special “original” case. By proposition 1 R(C, 32 P ) is presented by an extrasymmetric matrix. Therefore we are reduced to add “x0 ” (step ii)) and “t” (step iii)) such that the obtained matrix is still extrasymmetric and has homogeneous entries and Pfaffians. Moreover, we have to take care that for t = 0 we obtain a fibration by hyperelliptic curves, whereas for t = 0 the general curve of the fibration has to be non hyperelliptic. A crucial ingredient is The zero of degree zero. Notice that the second entry of the first row of M is equal to 0, and it corresponds to a homogeneous element of degree 0. This allows to substitute in this entry a parameter t. For t = 0 the upshot is that from the 9 Pfaffians we can eliminate the variables w, v, u and we are left with the variables x0 , x1 , y, z and with a single equation of degree 9: that is, we have a semicanonical ring of type II, and we have succeeded! We obtain thus the following result: Theorem 20. Consider the ring C[x0 , x1 , y, z, w, v, u] with variables of respective degrees (1, 1, 2, 3, 4, 5, 6). Consider a family of skew extrasymmetric matrices, with parameter t ⎞ ⎛ 0 t z v y x1 ⎜ 0 w u P3 y ⎟ ⎟ ⎜ ⎜ 0 P9 u v ⎟ ⎟. ⎜ Mt = ⎜ 0 wP4 zP4 ⎟ ⎟ ⎜ ⎝ 0 tP4 ⎠ −sym 0 where the Pi ’s are homogeneous of degree i in the first 5 variables of the ring, and let Jt be the ideal generated by the 4 × 4 pfaffians of Mt . Then, for general choice of the polynomials Pi , C[x0 , x1 , y, z, w, v, u]/Jt is, for t = 0, the semicanonical ring of a surface of type IIIb , and for t = 0 the semicanonical ring of a surface of type II. The surfaces of type IIIb whose semicanonical ring can be presented as in the above theorem form a codimension 2 subscheme of the corresponding 38-dimensional stratum of M6,4,0 , lying in the intersection with the 38dimensional stratum II.


29

It is still unclear whether this is the whole intersection of the closures of these two strata. A priori one can only say that this intersection has at most dimension 37, and what we found has only dimension 36. Therefore there remains the following: Question 2. Exactly which surfaces of type IIIb lie in the closure of the stratum II? The above question is of course related to the singularity type of the local moduli space. 3.3 K 2 = 7 This is the last case for which there is a complete classification. In [Bau01] the first author gives a very precise description of surfaces with pg = 4, K 2 = 7 according to the behaviour of the canonical map, allowing to show that the moduli space M7,4,0 has three irreducible components M36 , M36 and M38 of respective dimensions 36, 36 and 38. Moreover, it is shown that the two irreducible components of dimension 36 intersect, whereas it is up to now not yet clear whether the component of dimension 38 is indeed a connected component or it intersects M36 . We encounter here a very similar situation as for K 2 = 6. There are two families, one in M36 , the other in M38 , where the first consists of surfaces admitting a non hyperelliptic genus 3 pencil, whereas the surfaces in the other family admit a hyperelliptic genus 3 pencil. In fact, the first and last author have been able to calculate the relative canonical algebra for the hyperelliptic case, which is Gorenstein of codimension 6. For this very high codimension there are yet no flexible formats known to organize the equations. Hopefully it will be possible to understand the deformations of this family.

4 Surfaces Isogeneous to a Product, Beauville Surfaces and the Absolute Galois Group Surfaces isogenous to a (higher) product were introduced and extensively studied by the second author in [Cat00], where it is proven that any surface S isogenous to a higher product has a unique minimal realization as a quotient S = (C1 × C2 )/G. Here C1 and C2 are smooth algebraic curves of genus at least 2 and G is a finite group acting freely, and with the property that no element acts trivially on one of the factors Ci . Moreover, it was shown that the topology of a surface isogenous to a product determines its deformation class up to complex conjugation. The following result contains a correction to Theorem 4.14 of [Cat00] (cf. theorem 3.3 of [Cat03]).

30


Theorem 21. Let S = (C1 × C2 )/G be a surface isogenous to a product. Then any surface S with the same topological Euler number and the same fundamental group as S is diffeomorphic to S. If moreover S is orientedly ¯ In other diffeomorphic to S, then S is deformation equivalent to S or to S. top dif f is either irreducible words, the corresponding moduli space MS = MS and connected or it contains two connected components which are exchanged by complex conjugation. This class of surfaces and their higher dimensional analogues provide a wide specimen of examples where one can test or disprove several conjectures and questions (cf. e.g. [Cat03], [BC04], [BCG05a], compare also the next section). ¯ Moreover, the absolute Galois group Aut(Q/Q) acts on the moduli spaces of this class of surfaces: we shall outline a direct connection with Grothendieck’s dream of “dessins d’enfants”. In the following we shall concentrate on a class of surfaces isogenous to a product, namely the rigid ones. We recall that an algebraic variety X is rigid if and only if it does not have any non trivial deformations (e.g., the projective space is rigid). There is another (stronger) notion of rigidity, which is the following Definition 12. An algebraic variety X is called strongly rigid if any other variety homotopically equivalent to X is either biholomorphic or antibiholomorphic to X. Remark 4. 1) It is nowadays wellknown that smooth compact quotients of symmetric spaces are rigid (cf. [CV60]). 2) Mostow (cf. [Mos73]) proved that indeed locally symmetric spaces of complex dimension ≥ 2 are strongly rigid, in the sense that any homotopy equivalence is induced by a unique isometry. These varieties are of general type and the moduli space of varieties of general type is defined over Z, and naturally the absolute Galois group ¯ Gal(Q/Q) acts on the set of their connected components. So, in our special ¯ case, Gal(Q/Q) acts on the isolated points which parametrize rigid varieties. In particular, rigid varieties are defined over a number field and work of Shimura gives a possible way of computing explicitly their fields of definition. By this reason these varieties were named Shimura varieties (cf. Deligne’s Bourbaki seminar [Del71]). A quite general question is Question 3. What are the fields of definition of rigid varieties? What is the ¯ Gal(Q/Q)-orbit of the point in the moduli space corresponding to a rigid variety? Much simpler examples of rigid varieties were found by the second author (cf. [Cat00]).


31

Beauville Surfaces Inspired by a construction of A. Beauville of a surface with K 2 = 8, pg = q = 0 (cf. [Bea78]) as a quotient of the product of two Fermat curves of degree 5 by the action of the group Z/5Z, in [Cat00] the following definition was given Definition 13. A Beauville surface is a compact complex surface S which 1) is rigid, i.e., it has no nontrivial deformation, 2) is isogenous to a higher product, i.e., it is a quotient S = (C1 × C2 )/G of a product of curves of resp. genera ≥ 2 by the free action of a finite group G. Notice that, given a surface isogenous to a product, we obtain always three more, exchanging C1 with its conjugate curve C¯1 , or C2 with C¯2 : but only if we conjugate both C1 , C2 we obtain an orientedly diffeomorphic surface. These four surfaces could however be all biholomorphic to each other. If S is a Beauville surface and X is orientedly diffeomorphic to S, then ¯ theorem 21 implies: X ∼ = S or X ∼ = S. In other words, the corresponding subset of the moduli space MS consists of one or two points (if we insist on keeping the orientation fixed, else we may get up to four points). Definition 14. C is a triangle curve if there is a finite group G acting effectively on C and satisfying the properties i) C/G ∼ = P1C , and ii) f : C → P1C ∼ = C/G has {0, 1, ∞} as branch set. Remark 5. The rigidity of a Beauville surface is equivalent to the condition that (Ci , G0 ) is a triangle curve, for i = 1, 2 (G0 ⊂ G is the subgroup of index ≤ 2 which does not exchange the two factors). Recall now the classical Theorem 22. (Riemann’s Existence Theorem) There is a natural bijection between: 1) Equivalence classes of holomorphic mappings f : C → P1C , of degree n and with Branch set Bf ⊂ B, (where C is a compact Riemann surface, and f : C → P1C , f : C → P1C are said to be equivalent if there is a biholomorphism g : C → C such that f = f ◦ g). 2) Conjugacy classes of monodromy homomorphisms µ : π1 (P1C −B) → Sn (here, Sn is the symmetric group in n letters, and µ ∼ = µ iff there is an −1 , (∀γ). element τ ∈ Sn with µ(γ) = τ µ (γ)τ Moreover: 3) C is connected if and only if the subgroup Im(µ) acts transitively on {1, 2, . . . n}. 4) f is a polynomial if and only if ∞ ∈ B, the monodromy at ∞ is a cyclical permutation, and g(C) = 0.

32


Remark 6. 1) Assume that ∞ ∈ B, so {∞, b1 , . . . bd } = B: then π1 (P1C − B) is a free group generated by γ1 , . . . γd and µ is completely determined by the local monodromies τi := µ(γi ). Grothendieck’s enthusiasm was raised by the following result, where Belyi ([Bel79])made a very clever and very simple use of some explicit polynomials, m+r m r now called the Belyi polynomials, of the form (m+r) mm rr z (z − 1) in order to ¯ reduce the number of critical values of an algebraic function defined over Q. Theorem 23. (Belyi) An algebraic curve C can be defined over Q if and only if there exists a holomorphic map f : C → P1C with branch set only {0, 1, ∞}. The word ”dessin d’ enfant” = child’s drawing is due to the fact that the monodromy of f is determined by the ’dessin d’ enfant’ f −1 ([0, 1]), a bipartite graph (the vertices have label 0 or 1 according to their image) such that at each vertex one has a cyclical order of the edges incident in the vertex (this property holds because the graph is contained in a complex curve C thus we choose the corresponding cyclical counterclockwise order). It is clear that the triangle curves correspond to a certain class of ’dessins d’ enfants’, those which admit a group action with quotient the interval [0, 1]. Let us parenthetically observe that Gabino Gonzalez was recently able to extend Belyi’s theorem to the case of complex surfaces (in terms of Lefschetz maps with three critical values) (cf. [Gon04]). Grothendieck ([Gro97] )proposed to look at the ’dessins d’ enfants’ in order to get representations of the absolute Galois group Gal(Q, Q). We just explained that a Beauville surface is defined over Q, and that the Galois group Gal(Q, Q) operates on the discrete subset of the moduli space MS corresponding to Beauville surfaces. This action is rather strictly related to the action on the ’dessins d’ enfants’, but in this case, by theorem 21, the Galois group Gal(Q, Q) may transform a Beauville surface into another one with a non isomorphic fundamental group. Phenomena of this kind were already observed by J.P. Serre (cf. [Ser64]): here the idea is not to consider this as a pathology, but as a source of information, and to actually try to understand the representation of the Galois group Gal(Q, Q) on the class of groups which are fundamental groups of Beauville surfaces (and of their higher dimensional analogues). It looks therefore interesting to investigate these surfaces and to address the following problems: Question 4. Existence and classification of Beauville surfaces, i.e., a) which finite groups G can occur? b) classify all possible Beauville surfaces for a given finite group G. Question 5. Is the Beauville surface S biholomorphic to its complex conjugate ¯ surface S?


33

Is S real (i.e., does there exist a biholomorphic map σ : S → S¯ with σ = id)? 2

Another motivation to find these surfaces was also given by the following FRIEDMAN-MORGAN’S SPECULATION ( [FM88] 1987): DEF ⇐⇒ DIFF (Differentiable equivalence and deformation equivalence coincide for surfaces). A series of counterexamples were given by several authors and the simplest examples were given using non rigid surfaces isogenous to a product. Great part of the next section will be devoted to these equivalence relations. In order to reduce the description of Beauville surfaces to some group theoretic statement, we need to recall that surfaces isogenous to a higher product belong to two types: • S is of unmixed type if the action of G does not mix the two factors, i.e., it is the product action of respective actions of G on C1 , resp. C2 . • S is of mixed type, i.e., C1 is isomorphic to C2 , and the subgroup G0 of transformations in G which do not mix the factors has index precisely 2 in G. The datum of a Beauville surface can be completely described group theoretically, since it is equivalent to the datum of two triangle curves with isomorphic groups. Definition 15. Let G be a finite group. 1) A quadruple v = (a1 , c1 ; a2 , c2 ) of elements of G is an unmixed Beauville structure for G if and only if (i) the pairs a1 , c1 , and a2 , c2 both generate G, (ii) Σ(a1 , c1 ) ∩ Σ(a2 , c2 ) = {1G }, where Σ(a, c) :=

∞

{gai g −1 , gci g −1 , g(ac)i g −1 }.

g∈G i=0

We write U(G) for the set of unmixed Beauville structures on G. 2) A mixed Beauville quadruple for G is a quadruple M = (G0 ; a, c; g) consisting of a subgroup G0 of index 2 in G, of elements a, c ∈ G0 and of an element g ∈ G such that i) G0 is generated by a, c, ii) g ∈ / G0 , / Σ(a, c). iii) for every γ ∈ G0 we have gγgγ ∈ iv) Σ(a, c) ∩ Σ(gag −1 , gcg −1 ) = {1G }. We write M(G) for the set of mixed Beauville quadruples on the group G. Remark 7. We consider here finite groups G having a pair (a, c) of generators. Setting (r, s, t) := (ord(a), ord(c), ord(ac)), such a group is a quotient of the triangle group T (r, s, t) := x, y | xr = y s = (xy)t = 1.

(1)

34


It is now easy to explain how to get a surface from the above data, if we remember Riemann’s Existence Theorem which we recalled just above. We take as base point ∞ ∈ P1C and consider B := {−1, 0, 1}. We choose the following generators α, β of π1 (P1C − B, ∞) (γ := (α · β)−1 ):

Let now G be a finite group and v = (a1 , c1 ; a2 , c2 ) ∈ U(G). We get surjective homomorphisms π1 (P1C − B, ∞) → G,

α → ai , γ → ci

(2)

and Galois coverings λi : C(ai , ci ) → P1C ramified only in {−1, 0, 1} with ramification indices equal to the orders of ai , bi , ci and with group G (by Riemann’s existence theorem). Remark 8. 1) Condition (1), ii) ensures that the action of G on C(a1 , c1 ) × C(a2 , c2 ) is free. −1 −1 −1 2) Let be ι(a1 , c1 ; a2 , c2 ) = (a−1 1 , c1 ; a2 , c2 ). Then S(ι(v)) = S(v) (note −1 −1 in fact that α ¯ = α , γ¯ = γ ). 3) One can verify that the required conditions automatically imply: g(C(a1 , c1 )) ≥ 2 and g(C(a2 , c2 )) ≥ 2. One has: Proposition 2. Let G be a finite group and v = (a1 , c1 ; a2 , c2 ) ∈ U(G). Assume that {ord(a1 ), ord(c1 ), ord(a1 c1 )} = {ord(a2 ), ord(c2 ), ord(a2 c2 )} and that ord(ai ) < ord(ai ci ) < ord(ci ). Then S(v) ∼ = S(v) if and only if there are inner automorphisms φ1 , φ2 of G and an automorphism ψ ∈ Aut(G) −1 such that, setting ψj := ψ ◦ φj , we have ψ1 (a1 ) = a−1 1 , ψ1 (c1 ) = c1 , and −1 −1 ψ2 (a2 ) = a2 , ψ2 (c2 ) = c2 . In particular, under the above assumption, S(v) is isomorphic to S(v) if and only if S(v) has a real structure. Remark 9. Dropping the assumption on the orders of ai , ci , we can define a finite permutation group AU (G)such that for v, v ∈ U(G) we have: S(v) ∼ = S(v ) if and only if v is in the AU (G)-orbit of v . Remark 10. If G is abelian, v ∈ U(G). Then S(v) always has a real structure. We have the following results (cf. [BCG05a] for some of these, others have not yet been published):


35

Theorem 24. 1) An abelian group G admits an unmixed Beauville structure iff G ∼ = (Z/n)2 , (n, 6) = 1. 2) The following groups admit unmixed Beauville structures: a) the alternating group An for large n, b) the symmetric group Sn for n ∈ N with n ≥ 7 c) SL(2, Fp ), PSL(2, Fp ) for p = 2, 3, 5. With the help of the computer algebra program MAGMA all finite simple nonabelian groups of order ≤ 50000 were checked and unmixed Beaville structures were found on all of them, with the exception of A5 (where it can’t be found!). This led to the following Conjecture 1. ([BCG05a]) All finite simple nonabelian groups except A5 admit an unmixed Beauville structure. This conjecture was also checked for some bigger simple groups like the Mathieu groups M12, M22 and also matrix groups of size bigger then 2. Call now (r, s, t) ∈ N3 hyperbolic if 1 1 1 + + < 1. r s t In this case the triangle group T (r, s, t) is hyperbolic. These studies led also to the following suggestive: Conjecture 2. ([BCG05a]) Let (r, s, t), (r , s , t ) be two hyperbolic types. Then almost all alternating groups An have an unmixed Beauville structure v = (a1 , c1 ; a2 , c2 ) where (a1 , c1 ) has type (r, s, t) and (a2 , c2 ) has type (r , s , t ). The above conjectures are variations of a conjecture of Higman (proved by B. Everitt (2000), [Eve00]) asserting that every hyperbolic triangle group surjects onto almost all alternating groups. Concrete explicit examples of rigid surfaces not biholomorphic to their complex conjugate were also given: Theorem 25. The following groups admit unmixed Beauville structures v such that S(v) is not biholomorpic to S(v): 1. the symmetric group Sn for n ≥ 7 2. the alternating group An for n ≥ 16 and n ≡ 0 mod 4, n ≡ 1 mod 3, n ≡ 3, 4 mod 7. And also new examples of real points of moduli spaces which do not correspond to real surfaces: Theorem 26. Let p > 5 be a prime with p ≡ 1 mod 4, p ≡ 2, 4 mod 5, p ≡ 5 mod 13 and p ≡ 4 mod 11. Set n := 3p + 1. Then there is an unmixed Beauville surface S with group An which is biholomorphic to the complex ¯ but is not real. conjugate surface S,

36


For mixed Beauville surfaces the situation is more complicated, as already the following suggests. Theorem 27. 1) If a group G admits a mixed Beauville structure, then the subgroup G0 is non abelian. 2) No group of order ≤ 512 admits a mixed Beauville structure. A general construction of finite groups admitting a mixed Beauville structure was given in [BCG05a]. Let H be a non-trivial group, and let Θ : H × H → H × H be the automorphism defined by Θ(g, h) := (h, g) (g, h ∈ H). We consider the semidirect product (3) H[4] := (H × H) Z/4Z where the generator 1 of Z/4Z acts through Θ on H × H. Since Θ2 is the identity we find H[2] := H × H × (2Z/4Z) ∼ = H × H × Z/2Z

(4)

as a subgroup of index 2 in H[4] . We have now Lemma 1. Let H be a non-trivial group and let a1 , c1 , a2 , c2 be elements of H. Assume that 1. the orders of a1 , c1 are even, 2. a21 , a1 c1 , c21 generate H, 3. a2 , c2 also generate H, 4.(ord(a1 ) · ord(c1 ) · ord(a1 c1 ), ord(a2 ) · ord(c2 ) · ord(a2 c2 )) = 1. Set G := H[4] , G0 := H[2] as above and a := (a1 , a2 , 2), c := (c1 , c2 , 2). Then (G0 ; a, c) is a mixed Beauville structure on G. Proof. It is easy to see that a, c generate G0 := H[2] . The crucial observation is that (1H , 1H , 2) ∈ / Σ(a, c).

(5)

In fact, if this were not correct, it would have to be conjugate of a power of a, c or b. Since the orders of a1 , b1 , c1 are even, we obtain a contradiction. Suppose that h = (x, y, z) ∈ Σ(a, c) satisfies ord(x) = ord(y): then our condition 4 implies that x = y = 1H and (5) shows h = 1H[4] . Let now g ∈ H[4] , g ∈ / H[2] and γ ∈ G0 = H[2] be given. Then gγ = (x, y, ±1) for appropriate x, y ∈ H. We find (gγ)2 = (xy, yx, 2) and the orders of the first two components of (gγ)2 are the same, contradicting the above remark. Therefore the third condition is satisfied.


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We come now to the fourth condition of a mixed Beauville quadruple. / H[2] be given, for instance (1H , 1H , 1). Conjugation with Let g ∈ H[4] , g ∈ g interchanges then the first two components of an element h ∈ H[4] . Our hypothesis 4 implies the result. As an application we find the following examples Theorem 28. Let p be a prime with p ≡ 3 mod 4 and p ≡ 1 mod 5 and consider the group H := SL(2, Fp ). Then H[4] admits a mixed Beauville structure u such that S(u) is not biholomorphic to S(u). Remark 11. Note that the smallest prime satifying the above congruences is p = 11 and we get that G has order equal to 6969600. Question 6. : which is the minimal order of a group admitting a mixed Beauville structure?

5 Lefschetz Pencils and Braid Monodromies 5.1 Braids and the Mapping Class Group The elegant definition by E. Artin of the braid group (cf. [Art26], [Art65]) supplies a powerful tool, even if difficult to handle, for the study of the differential topology of algebraic varieties, in particular of algebraic surfaces. Remark 12. We observe that the subsets {w1 , . . . , wn } ⊂ C of n distinct points in C are in one to one correspondence with monic polynomials P (z) ∈ C[z] of degree n with non vanishing discriminant δ(P ). Definition 16. The group Bn := π1 (C[z]n \{P |δ(P ) = 0}), i.e., the fundamental group of the space of polynomials of degree n having n distinct roots, is called Artin’s braid group.

n Usually, one takes as base point the polynomial P (Z) = ( i=1 (z − i)) ∈ C[z]n (or the set {1, . . . , n}). To a closed (continuous) path α : [0, 1] → (C[z]n \{P |δ(P ) = 0}) one associates the subset {(z, t) ∈ C × R | α(t)(z) := αt (z) = 0} of R3 . Figure 2 below shows two realizations of the same braid. Remark 13. Obviously there is a lifting of α to Cn , the space of n-tuples of roots of polynomials of degree n and there are (continuous) functions wi (t) n such that wi (0) = i and αt (z) = i=1 (z − wi (t)). Then to each braid is associated a naturally defined permutation τ ∈ Sn given by τ (i) := wi (1).

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Fig. 2. Relation aba = bab on braids

A very powerful generalization of Artin’s braid group was given by M. Dehn (cf. [Deh38], we refer also to the book [Bir74]). Definition 17. Let M be a differentiable manifold, then the mapping class group (or Dehn group) of M is the group M ap(M) := π0 (Dif f (M)) = Dif f (M)/Dif f 0 (M), where Dif f 0 (M) is the subgroup of diffeomorphisms of M isotopic to the identity. Remark 14. If M is oriented then we often tacitly take Dif f + (M), the group of orientation preserving diffeomorphisms of M instead of Dif f (M), in the definition of the mapping class group. But it is more accurate to distinguish in this case M ap+ (M) from M ap(M). If M is a curve of genus g, then its mapping class group will be denoted by M apg . The relation between the above two definitions is the following: Theorem 29. The braid group Bn is isomorphic to the group π0 (M ap∞ (C\{1, . . . n})), where M ap∞ (C\{1, . . . n}) is the group of diffeomorphisms which are the identity outside the circle with center 0 and radius 2n. Therefore Artin’s standard generators σi of Bn (i = 1, . . . n − 1) can be represented by so-called half-twists. Definition 18. The half-twist σj is the diffeomorphism of C\{1, . . . n} isotopic to the homeomorphism given by: - rotation of 180 degrees on the circle with center j + 12 and radius 12 , - on a circle with the same center and radius 2+t 4 the map σj is the identity if t ≥ 1 and rotation of 180(1 − t) degrees, if t ≤ 1.


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Fig. 3. A geometric base of π1 (C − {1, . . . n})

Now, it is obvious that Bn acts on the free group π1 (C\{1, . . . n}), which has a geometric basis (we take as base point the complex number p := −2ni) γ1 , . . . γn as explained in figure 3. This action is called the Hurwitz action of the braid group and has the following algebraic description • σi (γi ) = γi+1 −1 • σi (γi γi+1 ) = γi γi+1 , whence σi (γi+1 ) = γi+1 γi γi+1 • σi (γj ) = γj for j = i, i + 1. Observe that the product γ1 γ2 . . . γn is left invariant under this action. Definition 19. We consider a group G and its cartesian product Gn . The map associating to each (g1 , g2 , . . . , gn ) the product g := g1 g2 . . . , gn ∈ G gives a partition of Gn , whose subsets are called factorizations of an element g ∈ G. Bn acts on Gn leaving invariant the partitions, and its orbits are called Hurwitz equivalence classes of factorizations. Definition 20. (cf. figure 4 below) Let C be a compact Riemann surface. Then a positive Dehn twist Tα with respect to a simple closed curve α on C is an isotopy class of a diffeomorphism h of C which is equal to the identity outside an annular neighbourhood of α, while inside the annulus h rotates one boundary of the annulus by 360 degrees to the right and damps the rotation down to the identity at the other boundary. If one considers a hyperelliptic Riemann surface given as a branched cover of P1C \{1, . . . n} one sees that the Artin half twist σj lifts to the Dehn twists on the loop which is the inverse image of the segment [j, j + 1]. Dehn’s fundamental result is the following Theorem 30. The mapping class group M apg is generated by Dehn twists. Explicit presentations of M apg have been given by Hatcher and Thurston ([HT80]), which have been improved by Wajnryb ([Waj83]) who obtained a simpler presentation of the mapping class group (cf. also [Waj99]).

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Fig. 4. At the left, a half twist; at the right: its lift: the Dehn twist T and its action on the segment D

5.2 Lefschetz Fibrations The method introduced by Lefschetz for the study of the topology of algebraic varieties is the topological analogue of the method of hyperplane sections and projections of the classical italian algebraic geometers. It was classically used to describe the homotopy and homology groups of algebraic varieties. In the 70’s Moisezon and Kas realized, after the work of Smale, that Lefschetz fibrations could be used to investigate the differential topology of algebraic varieties, especially of algebraic surfaces. For instance, it is an extremely difficult problem to decide whether two algebraic surfaces which are not deformation equivalent are in fact diffeomorphic, even in the case where they are simply connected. Here, the theory of Lefschetz fibrations offers a method to prove that two surfaces are diffeomorphic ([Kas80]). Definition 21. Let M be a compact differentiable (or even symplectic) manifold of dimension 4 A Lefschetz fibration is a differentiable map f : M → P1C which a) is of maximal rank except for a finite number of critical points p1 , . . . pm which have distinct critical values b1 , . . . bm ∈ P1C , b) has the property that around pi there are complex coordinates (x, y) ∈ C2 such that locally f = x2 − y 2 + const. (in the symplectic case, in the given coordinates the symplectic form ω of M has to correspond to the natural symplectic structure on C2 ). Remark 15. 1) A similar definition can be given if M is a manifold with boundary, replacing P1C by a disc D ⊂ C.


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2) An important theorem of Donaldson ([Don99]) asserts that for symplectic manifolds there exists (as for the case of projective manifolds) a Lefschetz pencil, i.e., a Lefschetz fibration f : M → P1C on a symplectic blow up M of M (cf. [MS98]). 3) A Lefschetz fibration with fibres genus g curves and with critical values b1 , . . . bm ∈ P1C , once a geometric basis γ1 , γ2 , . . . , γm of π1 (P1C \{b1 , . . . , bm }) is chosen, determines a factorization of the identity in the mapping class group M apg τ1 ◦ τ2 ◦ · · · ◦ τm = Id as a product of Dehn twists. We are now ready to state the theorem of Kas (cf. [Kas80]). Theorem 31. Two Lefschetz fibrations (M, f ), (M , f ) are equivalent (i.e., there are two diffeomorphisms u : M → M , v : P1 → P1 such that f ◦ u = v ◦ f ) if and only if the two corresponding factorizations of the identity in the mapping class group are equivalent (under the equivalence relation generated by Hurwitz equivalence and by simultaneous conjugation). Remark 16. 1) A similar result holds for Lefschetz fibrations over the disc and we get a factorization τ1 ◦ τ2 ◦ · · · ◦ τm = φ of the monodromy φ of the fibration over the boundary of the disc D. 2) The fibration admits a symplectic structure if and only if each Dehn twist in the factorization is positively oriented. Assume that we are given two Lefschetz fibrations over P1C : then we can define the fiber sum of these two fibrations, which depends on a diffeomorphism chosen between two respective smooth fibers (cf. [GS99]). This operation translates (in view of the above quoted theorem of Kas) into the following definition of “conjugated composition” of factorization: Definition 22. Let τ1 ◦ τ2 ◦ · · · ◦ τm = φ and τ1 ◦ τ2 ◦ · · · ◦ τr = φ be two factorizations: then their by ψ conjugated composition is the factorization τ1 ◦ τ2 ◦ . . . τm ◦ (τ1 )ψ ◦ (τ2 )ψ ◦ · · · ◦ (τr )ψ = φ(φ )ψ . Remark 17. 1) If ψ and φ commute, we obtain a factorization of φφ . 2) A particular case is φ, φ = id and it corresponds to Lefschetz fibrations over P1 .

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5.3 Braid Monodromy and Chisini’ Problem Let B ⊂ P2C be a plane algebraic curve of degree d, and let P be a generic point not on B. Then the pencil of lines Lt passing through P determines a one parameter family of d-uples of points of C ∼ = Lt \{P }, i.e., Lt ∩ B. Therefore one gets a factorization of (∆2 )d in the braid group Bn , where (∆2 ) = (σd−1 σd−2 . . . σ1 )d is the generator of the center of the braid group. The equivalence class of the factorization does not depend on the point P (if it is chosen generic) and does not depend on B, if B varies in an equisingular family of curves. Chisini was mainly interested in the case of cuspidal curves (cf. e.g. [Chi44], [Chi55]), mainly because these are the branch curves of a generic projection f : S → P2C , for any smooth projective surface S ⊂ Pr . More precisely, a generic projection f : S → P2C is a covering whose branch curve has only nodes and cusps as singularities, and moreover is such that the local monodromy around a smooth point of the branch curve is a transposition. Maps with those properties are called generic coverings: for these the local monodromies are only Z/2 = S2 (at the smooth points of the branch curve B), S3 at the cusps, and Z/2 × Z/2 at the nodes. In such a case we have a cuspidal factorization, i.e. all factors are powers of a half twist, with respective exponent 1, 2, 3. Chisini posed the following Conjecture 3. (Chisini’s conjecture.) Given two generic coverings f : S → P2C , f : S → P2C , one of them of degree d ≥ 5, assume that they have the same branch curve B. Is it then true that f and f are equivalent? Observe that the condition on the degree is necessary, since counterexamples with d = 4 are furnished by the dual curve of a smooth plane cubic (as already known to Chisini, who gave a counterexample with d = 4, d = 3, while counterexamples with d = d = 4 were given in [Cat86b]). The conjecture has been proven under the hypothesis that the degree of each covering is at least 12, essentially by Kulikov (cf. [Kul99]). In fact, Kulikov proved the result under a more complicated assumption and shortly later Nemirovski [Nem01] noticed, just by using the Miyaoka-Yau inequality, that Kulikov’s assumption was implied by the simple assumption d ≥ 12. Later on generalizations of this result were obtained for singular (normal) surfaces [Kul03] or for curves with more complicated singularities [MP02]. A negative answer instead has the following problem of Chisini (due to work of B. Moishezon (cf. [Moi94]). Chisini’ s problem: (cf. [Chi55]). Given a cuspidal factorization, which is regenerable to the factorization of a smooth plane curve, is there a cuspidal curve which induces the given factorization?


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Regenerable means that there is a factorization (in the equivalence class) such that, after replacing each factor σ i (i = 2, 3) by the i corresponding factors (e.g. , σ 3 is replaced by σ◦σ◦σ) one obtains the factorization belonging to a non singular plane curve. Remark 18. 1) Moishezon proves that there exist infinitely many non equivalent cuspidal factorizations observing that π1 (P2C \B) is an invariant defined in terms of the factorization alone. On the other hand, the family of cuspidal curves of a fixed degree form an algebraic set, hence has a finite number of connected components. These two statements together give a negative answer to the above cited problem of Chisini. The examples of Moishezon have been recently reinterpreted in [ADK03], with a simpler treatment, in terms of symplectic surgeries. 2) In fact, as conjectured by Moishezon, a cuspidal factorization together with a generic monodromy with values in Sn induces a covering M → P2C , where M is a symplectic fourmanifold. Extending Donaldson’s techniques (for proving the existence of symplectic Lefschetz fibrations) Auroux and Katzarkov ([AK00]) proved that each symplectic 4-manifold is in a natural way ’asymptotically’ realized by such a generic covering. They propose to use an appropriate quotient of π1 (P2C \B) in order to produce invariants of symplectic structures, using the methods introduced by Moishezon and Teicher in a series of technically difficult papers ( see e.g. [MT92]). It seems however that, up to now, these groups π1 (P2C \B) allow only to detect homology invariants of the projected fourmanifold ([ADKY04]). 3) Suppose we have a surface S of general type and a pluricanonical embedding. Then a generic projection to P3C gives a surface with a double curve Γ . Now, project further to P2C and we do not only get the branch curve B, but also a curve Γ , image of Γ . Even if Chisini’s conjecture tells us that from the holomorphic point of view B determines the surface S and therefore the curve Γ , it does not follow that the fundamental group π1 (P2C \B) determines the group π1 (P2C \(B ∪ Γ )). It would be interesting to calculate this second fundamental group, even in special cases.

6 DEF, DIFF and Other Equivalence Relations As we said, one of the fundamental problems in the theory of complex algebraic surfaces is to understand the moduli spaces of surfaces of general type, and in particular their connected components, which parametrize the deformation equivalence classes of minimal surfaces of general type. Definition 23. Two minimal surfaces S and S are said to be def-equivalent (we also write: S ∼def S ) if and only if they are elements of the same connected component of the moduli space.

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By the classical theorem of Ehresmann, two def-equivalent algebraic surfaces are (orientedly) diffeomorphic. In the late eighties Friedman and Morgan (cf. [FM88]) conjectured that two algebraic surfaces are diffeomorphic if and only if they are def-equivalent. We will abbreviate this conjecture in the following by the acronym def = diff. The second author would like to point out here that he had made the opposite conjecture in the early eithties (cf. [Kat83]). Donaldson’s breaktrough results had made clear that diffeomorphism and homeomorphism differ drastically for algebraic surfaces (cf. [Don83]) and the success of gauge theory led Frieman and Morgan to “speculate” that the diffeomorphism type of algebraic surfaces determines the deformation class. After the first counterexamples of M. Manetti (cf. [Man01]) appeared, there were further counterexamples given by Catanese, KharlamovKulikov, Catanese-Wajnryb, Bauer-Catanese-Grunewald (cf. [Cat03], [KK02], [BCG05a], [CW04]). In the cited papers by Catanese, Kharlamov-Kulikov, Bauer-CataneseGrunewald, the counterexamples are given by pairs of surfaces, where one is the complex conjugate of the other. One could say that somehow these counterexamples are ’cheap’, and somehow in the air (cf. the definition of strong rigidity). The second author was very recently informed by R. Friedman that also he and Morgan were aware of such ’complex conjugate’ counterexamples, but for the case of elliptic surfaces. Since the beautiful examples of Manetti yield non simply connected surfaces, it made sense to weaken the conjecture def = diff in the following way. Question 7. Is the speculation def = diff true if one requires the diffeomorphism φ : S → S to send the first Chern class c1 (KS ) ∈ H 2 (S, Z) in c1 (KS ) and moreover one requires the surfaces to be simply connected? But even this weaker question turned out to have a negative answer, as it was shown by the second author and Wajnryb ([CW04]). Remark 19. If two surfaces are def-equivalent, then there exists a diffeomrophism sending the canonical class c1 (KS ) ∈ H 2 (S, Z) in the canonical class c1 (KS ). On the other hand, by the result of Seiberg-Witten theory we know that a diffeomorphism sends the canonical class of a minimal surface S to ±c1 (KS ). Therefore, if one gives at least three surfaces, which are pairwise diffeomorphic, one finds at least two surfaces with the property that there exists a diffeomorphism between them sending the canonical class of one to the canonical class of the other. Theorem 32. ([CW04]) For each natural number h there are simply connected surfaces S1 , . . . , Sh which are pairwise diffeomorphic, but are such that two of them are never def-equivalent.


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The above surfaces S1 , . . . , Sh belong to the class of the so-called (a, b, c)surfaces, obtained as minimal compactification of some affine surface described by the following two equations: z 2 = f (x, y), w2 = g(x, y), where f and g are suitable polynomials of respective bidegrees (2a, 2b), (2c, 2b). They can be compactified simply by bihomogenizing the polynomials f, g, thus obtaining Galois covers of P1 × P1 with Galois group Z/2Z. We remark that the above compactification is smooth if the two curves {f = 0} and {g = 0} in P1 × P1 are smooth and intersect transversally. We say that these surfaces are bidouble (i.e., Galois covers with Galois group (Z/2Z)2 ) covers of P1 ×P1 of type (2a, 2b), (2c, 2b) (cf. [Cat84], [Cat99]). The above theorem is implied by the two following results: Theorem 33. Let a, b, c, k be positive even numbers such that 1) a, b, c − k ≥ 4; 2) a ≥ 2c + 1; 3) b ≥ c + 2; and either 41 ) b ≥ 2a + 2k − 1 or 42 ) a ≥ b + 2. Furthermore, let S be an (a, b, c)-surface and S be an (a + k, b, c − k)surface. Then S is not def-equivalent to S . Theorem 34. Let S be an (a, b, c)-surface and S be an (a+1, b, c−1)-surface. Moreover, assume that a, b, c − 1 ≥ 2. Then S and S are diffeomorphic. Remark 20. Observe that the surfaces in question above are simply connected (cf. [Cat84], prop. 2.7.). The proof of the two theorems above are completely different in nature. The first theorem uses techniques which have been developped in a series of papers by the second author and by Manetti ([Cat84], [Cat87a], [Cat86a], [Man94], [Man97]). They use essentially the local deformation theory a` la Kuranishi, normal degenerations of smooth surfaces and a study of quotient singularities of rational double points and of their smoothings. One very elementary, but extremely important ingredient in the proof of the first theorem is the notion of natural deformations of a bidouble cover (introduced in [Cat84], p.494), which are parametrized by a quadruple of polynomials (f, g, φ, ψ) and given by the two equations z 2 = f (x, y) + wφ(x, y), w2 = g(x, y) + zψ(x, y),

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where f and g are polynomials of respective bidegrees (2a, 2b), (2c, 2b) as before and φ and ψ have respective bidegrees (2a − c, b), (2c − a, b). Under suitable hypotheses (rigid base, branch curve of sufficiently high degree), these natural deformations indeed give all small deformations. Moreover, since a ≥ 2c + 1, it follows that ψ ≡ 0, therefore every small deformation preserves the structure of an iterated double cover. The final point is to show that this structure also passes in a suitable way to the limit, so that we do not only have an open, but also a closed subset of the moduli space. We will now comment on the newer part, the proof of theorem 34. The key ideas are here the following: 1) Both surfaces S and S admit a holomorphic map to P1C given by the composition of the bidouble cover with the projection to the first coordinate x, and a small perturbation of this map realizes them as symplectic Lefschetz fibrations (cf. [Don99], [GS99]). 2) The respective fibrations are, by the accurate choice of the bidegrees of the curves, and especially because of the fact that the second degree is equal to 2b in both cases (thereby allowing, locally on the base, to ’rotate’ one branch curve to the other) fiber sums of the same pair of Lefschetz fibrations over the complex disc (the global effect of this local rotation is that the first curve {f = 0} loses a bidegree (2, 0), while the second {g = 0} gains a bidegree (2, 0)) 3) Once the first fiber sum is presented as composition of two factorizations and the second as the same composition of factorizations, just conjugated by the ’rotation’ Ψ , in order to prove that the two fiber sums are equivalent, it suffices, (thanks to Auroux’s lemma, [Aur02]) to show that the diffeomorphism Ψ is in the subgroup of the mapping class group generated by the Dehn twists which appear in the first factorization. 4)Figure 5 below shows the fibre C of the fibration in the case 2b = 6: it is a bidouble cover of P1 , which we can assume to be given by the equations z 2 = F (y), w2 = F (−y), where the roots of F are the integers 1, . . . , 2b. Moreover, one sees that the monodromy of the fibration at the boundary of the disc is trivial, and the map Ψ is the diffeomorphism of order 2 given by y → −y, z → w, w → z, which in our figure is given as a rotation of 180 degrees around an axis inclined in direction north-east. The figure shows a dihedral symmetry, where the automorphism of order 4 is given by y → −y, z → −w, w → z. Moreover, between the Dehn twists which appear in the factorization there are those which correspond to the inverse images of the segments between two consecutive integers (cf. figure 5). These circles can be organized on the curve C in six chains (not disjoint) and finally we have reduced ourselves to show that the isotopy class of Ψ is the same as the product of the six Coxeter elements associated to such chains.


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Fig. 5. The curve C with a dihedral symmetry

We recall that the Coxeter elements associated to a chain are products of the type ∆ = (Tα1 )(Tα2 Tα1 ) . . . (Tαn Tαn−1 . . . Tα1 ) of Dehn twists associated to the curves of the chain. In order to finally prove that such product (let us call it Ψ ) of Coxeter elements and Ψ are isotopic, one observes that if one removes the above cited chains of circles from the curve C, one obtains 4 connected components which are diffeomorphic to circles. By a result of Epstein it is then sufficient to verify that Ψ and Ψ send each such curve to a pair of isotopic curves: this last step

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needs a list of lengthy (though easy) verifications, for which it is necessary to have explicit drawings. For details we refer to the original paper [CW04]. It was observed by the second author (cf. [Cat02]) that a surface of general type has a canonical symplectic structure. In fact, he proves the following Theorem 35. A minimal surface of general type S has a canonical symplectic structure, unique up to symplectomorphism, such that the class of the symplectic form is the class of the canonical sheaf OS (KS ). We give the proof under the assumption that KS be ample, since the basic idea becomes clear in this simpler case. For the general case we refer to the original article. Proof. Let m be such that mKS is very ample (by Bombieri’s result it suffices any m ≥ 4), i.e., the pluricanonical map φm := φmKS : S → PPm −1 , where Pm := h0 (S, OS (mKS )) is the m-th plurigenus of S, is an embedding. We define ωm on S as follows: ωm :=

1 ∗ 1 φ ( ∂∂log|z|2 ), m m 2πi

i.e., we divide by m the pull back of the Fubini-Study form, whence ωm yields a symplectic form on S. It remains to show that the symplectomorphism class of (S, ωm ) is indeed independent of m. For this suppose that also φn gives an embedding of S: then the same holds for mn, whence it is sufficient to see that (S, ωm ) and (S, ωnm ) are symplectomorphic. Observe that the pull back of the Fubini-Study form under the n-th Veronese map vn is n times the Fubini-Study form and vn ◦ φm is a linear projection of φmn . Then by Moser’s theorem we are done. Therefore it seems natural to ask the following Question 8. Are the diffeomorphic (a, b, c)-surfaces of theorem 34, endowed with their canonical symplectic structure, indeed symplectomorphic? Remark 21. 1) In [Cat02] the second author shows that Manetti’s examples are indeed symplectomorphic. 2) A possible way of showing that the answer to the question above is yes (and therefore exhibiting symplectomorphic simply connected surfaces which are not def-equivalent) goes through the analysis of the braid monodromy of the branch curve of the “perturbed” (corresponding to the Lefschetz fibration) quadruple covering, and one would like to show that the involution ι on P1 , ι(y) = −y can be written as the product of braids which show up in the factorization. Anyhow, this approach turned out to be more difficult than the corresponding analysis which has been made in the mapping class group, because


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the braid monodromy contains very many ’tangency’ factors which do not come from local contributions to the regeneration of the branch curve from the union of the curves f = 0, g = 0 counted twice. In the rest of the paragraph we will discuss another equivalence relation the so-called Q.E.D. equivalence relation, which was introduced by the second author (cf. [Cat05]), and which seems worthwhile to examine for surfaces of general type. Observe that for any number g ≥ 2 there is a smooth curve of genus g, which is an étale covering of a curve of genus 2. Therefore all the smooth curves of Kodaira dimension 1 are equivalent by the equivalence relation generated by deformation and by étale maps. (Obviously, also all curves of Kodaira dimension 0, resp. −∞ are equivalent by this equivalence relation). Remark 22. More remarkable is what happens for algebraic surfaces of Kodaira dimension 0. Enriques surfaces admit an étale double cover which is a K3-surface, hyperelliptic surfaces have an étale cover which is a torus (in fact, the product of two elliptic curves). Therefore, in order to have some analogue to the curve case, one should “link” K3-surfaces and tori by étale maps and deformations. Obviously, this is not possible, since tori are K(π, 1)’s and K3-surfaces are simply connected. But the solution is simple: divide the torus by the involution x → −x, and obtain the (singular!) Kummer surface. A smoothing of this Kummer surface gives a K3-surface. The price we have to pay for going from curves to surfaces is that we have to allow morphisms which are not necessarily étale, but only étale in codimension 1. Moreover, we have to allow mild singularities: ordinary double points in this case, canonical singularities in a more general setting. This remark justifies the following Definition 24. We consider for complete algebraic varieties with canonical singularities defined over a fixed algebraically closed field the equivalence relation generated by 1) birational maps; 2) flat proper algebraic deformations π : X → B, with base B a connected algebraic variety, and all fibres having canonical singularities; 3) quasi étale morphisms f : X → Y , i.e., surjective morphisms which are étale in codimension 1 on X (i.e., there is Z ⊂ X of codimension ≥ 2 such that f |(X − Z) is étale). We will call this equivalence relation a.q.e.d.-relation, which means algebraic quasiétale-deformation relation and it will be denoted by X ∼a.q.e.d. X . It is rather clear that a completely analogous equivalence relation (called then C-q.e.d.-relation) can be defined also in the setting of compact complex spaces with canonical singularities. We refer to [Cat05] for more details.

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Remark 23. Trivially, the dimension of a variety is a q.e.d. invariant. By Siu’s recent result (cf. [Siu02]) also the Kodaira dimension is an invariant of a.q.e.d.-equivalence, if we restrict ourselves to projective varieties with canonical singularities (defined over the complex numbers). For surfaces of special type, i.e., of Kodaira dimension ≤ 1 the situation is as for curves. Theorem 36. Let S and S be smooth complex algebraic surfaces of the same Kodaira dimension ≤ 1. Then S and S are a.q.e.d.-equivalent. The ingredients of the proof of the above theorem are the Enriques classification of surfaces, the detailed knowledge of the deformation types of elliptic surfaces and the orbifold fundamental group of a fibration. The following question seems natural. Question 9. Is it possible to determine the q.e.d. equivalence classes inside the class of varieties with fixed dimension n, and with Kodaira dimension k? For curves and special algebraic surfaces over C there is only one a.q.e.d. class, but as shown in an appendix to [Cat05] by Fritz Grunewald, already for surfaces of general type the situation is completely different. Theorem 37. There are infinitely many q.e.d.-equivalence classes of algebraic surfaces of general type. The above surfaces are constructed from quaternion algebras (along general lines suggested by Shimura and explicitly described by Kuga and Shavel, cf. [Sha78]) are rigid, but the q.e.d.-equivalence class contains countably many distinct birational classes. The main points of the construction are the following: 1) the surfaces are quotients S = H×H/Γ of the two dimensional polydisk H × H via the free action of a discrete group Γ constructed from a quaternion algebra A over a totally real quadratic field k 2) since S is rigid, it suffices to show that if Γ is commensurable with Γ , then also Γ acts freely on H × H . 3) One sees by general theorems that Γ has as Q-linear span the same quaternion algebra A as Γ . 4) If Γ does not act freely, taking the tangent representation at a fixed point, we see by 3) that A contains a cyclotomic extension whose degree divides 4. 5) Using Hasse’s theorem, one chooses A such that the set of primes where it ramifies contains, one for each possible intermediate field K between the quadratic field k of A and one of the finitely many possible cyclotomic extensions above, a prime P such that K ⊗ kP is not an integral domain: this however contradicts 4) hence shows the desired assertion.


51

It remains open whether there are for instance varieties which are isolated in their q.e.d.-equivalence class (up to birational equivalence, of course). An interesting question is to determine, for surfaces of general type, the non standard a.q.e.d. classes (standard means: equivalent to a product of curves)

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[MT92] [Mum79] [Nai94] [Nem01]

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[Sch05]

[Ser64] [Sev32]

[Sha78] [Siu02]

[Tod81]

[Waj83] [Waj99] [Wit94] [Xia85a] [Xia85b] [Xia87] [Xia90] [Yau77] [Yau78]

[Zuc97] [Zuc03]

Schreyer, F.-O.: An experimental approach to numerical Godeaux surfaces. in Report n. 7/2005, Workshop “Komplexe Algebraische Geometrie”, Feb. 13–19, 2005. 434–436. M.F.O., Oberwolfach (2005) Serre, P.: Exemples de variétés projectives conjuguées non hom´ eomorphes. C. R. Acad. Sci. Paris 258, 4194–4196 (1964) Severi, F.: La serie canonica e la teoria delle serie principali di gruppi di punti sopra una superficie algebrica. Comment. Math. Helv. 4, 268–326 (1932) Shavel, I.H.: A class of algebraic surfaces of general type constructed from quaternion algebras. Pacific J. Math. 76, no. 1, 221–245 (1978) Siu, Y.T.: Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type. In: Bauer, I. (ed) et al. Complex geometry. Collection of papers dedicated to Hans Grauert on the occasion of his 70th birthday, 223–277. Springer, Berlin (2002) Todorov, A.N.: A construction of surfaces with pg = 1, q = 0 and 2 ≤ (K 2 ) ≤ 8. Counterexamples of the global Torelli theorem. Invent. Math. 63, no. 2, 287–304 (1981) Wajnryb, B.: A simple presentation for the mapping class group of an orientable surface. Israel J. Math. 45, no. 2–3, 157–174 (1983) Wajnryb, B.: An elementary approach to the mapping class group of a surface, Geom. Topol. 3, 405–466 (1999) Witten, E.: Monopoles and four-manifolds. Math. Res. Lett. 1, no. 6, 769–796 (1994) Xiao, G.: Finitude de l’application bicanonique des surfaces de type général. Bull. Soc. Math. France 113, no. 1, 23–51 (1985) Xiao, G.: Surfaces fibrées en courbes de genre deux. Lecture Notes in Math., 1137. Springer, Berlin (1985) Xiao, G.: Fibered algebraic surfaces with low slope. Math. Ann. 276, no. 3, 449–466 (1987) Xiao, G.: Degree of the bicanonical map of a surface of general type. Amer. J. Math. 112, no. 5, 713–736 (1990) Yau, S.T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. U.S.A. 74, no. 5, 1798–1799 (1977) Yau, S. T.: On the Ricci curvature of a compact K¨ ahler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math. 31, no. 3, 339–411 (1978) Zucconi, F.: A note on a theorem of Horikawa. Rev. Mat. Univ. Complut. Madrid 10, no. 2, 277–295 (1997) Zucconi, F.: Surfaces with pg = q = 2 and an irrational pencil. Canad. J. Math. 55, no. 3, 649–672 (2003)

Characteristic 0 and p Analogies, and some Motivic Cohomology Manuel Blickle, Hélène Esnault, Kay R¨ ulling Universit¨ at Duisburg-Essen, Mathematik, 45117 Essen, Germany [email protected] [email protected] [email protected]

Introduction The purpose of this survey is to explain some recent results about analogies between characteristic 0 and characteristic p > 0 geometry, and to discuss an infinitesimal variant of motivic cohomology. Homotopy invariance for motivic cohomology implies, in particular, that the Picard group of the affine line over a field k is trivial, i.e. Pic(A1k ) = 0. However, if instead of considering the Picard group, we consider the group of isomorphism classes of pairs (L, t) consisting of a line bundle L on A1k , and ∼ = an isomorphism t : OSpec(OA1 /mn ) −→ LSpec(OA1 /mn ) where m is the maximal ideal at the origin, then one obtains the group of 1 units in k[[t]]/(tn ), that × is {isom. classes (L, t)} ∼ = 1 + tk[[t]]/(1 + tn k[[t]]) . This group is indeed a ring, namely the ring of big Witt vectors Wn−1 (k) of length n − 1 over k. On the other hand, the theory of additive higher Chow groups, or Chow groups with modulus condition 2, developed in [10] and [11], allows to realize i of k as a group of 0-cycles (Theorems 2.1 and absolute differential forms Ωk/Z 2.3). In Sect. 2 these groups of 0-cycles and their higher modulus n generalization are presented. Theorem 2.7 – the main result of Sect. 2 – asserts that these groups of 0-cycles with higher modulus n compute big Witt differential forms Wn−1 Ωki . Cutting out the p-isotypical component in characteristic p > 0 by suitable correspondences, and extending Theorem 2.7 to smooth local rings over a perfect field would then describe crystalline cohomology of a smooth proper variety over a perfect field of characteristic p > 0 as the hypercohomology of a complex of sheaves of 0-cycles. If X is a smooth complex variety, the Riemann–Hilbert correspondence establishes an equivalence of categories between holonomic DX -modules (coherent) and constructible sheaves (topological). Emerton and Kisin developed

This work has been partly supported by the DFG Schwerpunkt “Komplexe Mannigfaltigkeiten” und by the DFG Leibniz Program

60

Manuel Blickle, Hélène Esnault, Kay R¨ ulling

in [22] for characteristic p > 0 a correspondence with properties analogous to the complex theory. The “topological like” side consists of sheaves of finite dimensional Fp -vector spaces which are locally constant in the étale topology, the “D-module like” side consists of locally finitely generated unit OX,F modules (see Theorem 3.1), which are modules on which the Frobenius acts in a certain way (unit). Even if the unit condition is quite restrictive – as will be explained shortly, it essentially corresponds to “slope zero” – the theory has a vaste range of applications and many objects naturally carry such a structure. For example OX itself is such a unit OX,F module. In Sect. 3 we investigate singularities of Y (at a point x ∈ Y ⊆ X, where X is smooth) from this viewpoint and obtain striking analogies for local invariants over the complex numbers and in characteristic p > 0. Postponing the definition of these invariants, which were introduced by Lyubeznik in [43], the main result (Theorem 3.1) of Sect. 3 asserts that these invariants can be expressed in terms of étale cohomology Hxi (Yét , Fp ) in characteristic p > 0 and respectively in terms of singular cohomology Hxi (Yan , C) over the complex numbers, by virtually the same expression. This extends earlier results in the isolated singular analytic case of [43] and [33]. This extension is made possible by the similarity between the two correspondences which allows to essentially treat both settings (char. p > 0 and char. 0) formally as one. In Sect. 1, we review recent results on congruences modulo q-powers for the number of Fq -rational points of algebraic varieties. They are all based on Deligne’s philosophy which predicts a deep analogy between the level of congruences for varieties defined over Fq and the Hodge level for varieties over the complex numbers. If a variety has Hodge level ≥ κ over the complex numbers, one expects that “over” Fq it will have the same number of rational points as Pn modulo q κ . Of course, the challenge is to make precise “over” as it can’t be the same variety. Divisibility of eigenvalues of the geometric Frobenius acting on -adic cohomology is one method to show the existence of congruences, slope computation in crystalline cohomology, or, in the singular case, in Berthelot’s rigid cohomology is another one. Of course, ideally one would wish to prove a motivic statement which would imply all those results at once. But it is often beyond reach. The main results are Theorem 1.5, which in particular gives a positive answer to the Lang-Manin conjecture asserting that Fano varieties over a finite field have a rational point, Theorem 1.7, asserting that the mod p reduction of a regular model of a smooth projective variety defined over a local field, the -adic cohomology of which is supported in codimension 1, carries one rational point modulo q, Theorem 1.11, asserting that two theta divisors on an abelian variety defined over a finite field carry the same number of points modulo q. Theorem 1.11 answers positively the finite consequence of a conjecture of Serre, and appears as a consequence of the slope Theorem 1.10 asserting that the slope < 1 piece of rigid cohomology is computed by Witt vector cohomology. Theorem 1.5 can be proven using either -adic cohomology or crystalline cohomology. Indeed, the geometric result behind is that Fano varieties are rationally connected and therefore

Characteristic 0 and p, and Motivic Cohomology

61

their Chow group of 0-cycles satisfies base change. Theorem 1.7 relies on a generalization to local fields of Deligne’s integrality theorem over finite fields (Theorem 1.8).

1 Hodge Type Over the Complex Numbers and Congruences for the Number of Rational Points over a Finite Field 1.1 Deligne’s Integrality Theorem over a Finite Field Deligne developed the theory of weights for complex varieties via the weight filtration in his mixed Hodge theory [17], and, for varieties defined over finite fields, via the absolute values with respect to any complex embedding of the eigenvalues of the geometric Frobenius operator acting on -adic cohomology [18]. The philosophy of motives, as conceived by him and Grothendieck, predicts a closed analogy between those two concepts of weights. It has led to many very fundamental results, the first of which being Deligne’s proof of the Weil conjecture for Hodge level one complete intersections and for K3 surfaces (Invent. math. 15 (1972)). On the other hand, Deligne shows the fundamental integrality theorem Theorem 1.1 (Deligne [19], Corollaire 5.5.3). Let X be a scheme of finite type defined over Fq . Then the eigenvalues of the geometric Frobenius acting on compactly supported -adic cohomology Hci (X ×Fq Fq , Q ) are algebraic integers. If X is defined over the complex numbers, its Hodge filtration F j satisfies grjF Hci (X) = 0 for j < 0 (see Hodge III [17]). In Deligne’s motivic philosophy, the integrality theorem 1.1 is analogous to the Hodge filtration starting in degrees ≥ 0 in Hodge theory. This analogy has been less studied in the past than the weight analogy. The purpose of this section is to demonstrate on some examples how the analogy works between the F -filtration in Hodge theory and the integrality over a finite field for -adic cohomology. It has led in very recent years to a series of results on congruences for the number of points on varieties defined over finite fields. Theorem 1.1 implies Theorem 1.2. Let X be a scheme of finite type defined over Fq . Then the eigenvalues of the geometric Frobenius acting on -adic cohomology H i (X ×Fq Fq , Q ) are algebraic integers. Strictly speaking, Deligne shows this for X smooth via duality, but applying de Jong’s alteration theorem, one easily reduces the theorem to the smooth case as in [20], Corollary 0.3.

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1.2 Divisibility and Rational Points The Grothendieck-Lefschetz trace formula [34] |X(Fq )| =

∞

(−1)i Trace Frobenius|Hci (X ×Fq Fq , Q )

(1)

i=0

together with Theorem 1.1 implies that if the eigenvalues of the geometric ¯ but also, for i ≥ 1, they Frobenius are not only algebraic integers, i.e. ∈ Z, ¯ are q-divisible as algebraic integers, i.e. ∈ q · Z, then one has |X(Fq )| ≡ dim Hc0 (X ×Fq Fq ) mod q.

(2)

X proper and geometrically connected ¯ with eigenvalues of geom. Frob. ∈ q · Z =⇒ |X(Fq )| ≡ 1 mod q.

(3)

So we conclude

Purity, for which the smoothness condition is definitely necessary, together with Theorem 1.2 implies Theorem 1.3. Let X be a smooth scheme of finite type defined over Fq . Then the eigenvalues of the geometric Frobenius acting on -adic cohomology i (X ×Fq Fq , Q ) with supports along A are algebraic integers divisible HA× F Fq

q

by q κ , if A ⊂ X is a closed subscheme of codimension ≥ κ.

(See [26], Lemma 2.1 for the analogous proof in crystalline cohomology). If X is no longer smooth, the conclusion of Theorem 1.3 is no longer true (see [20], Remark 0.5). But it remains true for generic hyperplanes for example, as purity is then true (see [27], Theorem 2.1). In Deligne’s philosophy, Theorem 1.3 is analogous to the Hodge level statement Theorem 1.4. Let X be a smooth scheme of finite type defined over C. Then the graded pieces for the Hodge filtration F on de Rham cohomology with i (X) = 0 for a < κ, if A ⊂ X is a closed subscheme of support fulfill graF HA codimension ≥ κ. The proof, based on purity, is the same as for Theorem 1.3. 1.3 Chow Group of 0-Cycles, Coniveau and Divisibility for Smooth Proper Varieties So if X is smooth proper geometrically connected over Fq , Theorem 1.3 implies that (3) holds if H i (X ×Fq Fq , Q ) is supported in codimension ≥ 1 for i ≥ 1. An equivalent terminology is to say that H i (X ×Fq Fq , Q ) has coniveau ≥ 1. According to Bloch’s decomposition of the diagonal ([8], Appendix to Lecture 1), this is the case if the Chow group of 0-cycles over a field containing the field of rational functions Fq (X) is trivial. We conclude


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Theorem 1.5 ([26], Corollary 1.2, Corollary 1.3). Let X be a smooth, proper, geometrically connected variety over Fq . Assume that CH0 (X ×Fq Fq (X)) = Q. Then |X(Fq )| ≡ 1 mod q. In particular, Fano varieties have a rational point, as conjectured by Lang [42] and Manin [45]. Originally, Bloch decomposed the diagonal in order to recover Mumford’s theorem (and its variants) asserting in its simplest form that if X is a smooth projective surface over C, and if gr0F H 2 (X) = 0, then it can’t be true that CH0 (X)deg=0 ∼ = Alb(X). Furthermore, that Fano varieties, that is smooth projective geometrically irreducible varieties X so that the inverse of the du−1 is ample, are rationally connected over any algebraically alizing sheaf ωX closed field, is a consequence of Mori’s break and bend theory, and has been proven independently by Koll´ ar-Miyaoka-Mori and Campana (see [41] and references there). 1.4 Singular Varieties Defined by Equations: Ax-Katz’ Theorem and Divisibility If X is no longer smooth, not only one can’t apply Theorem 1.3 to get divisibility of eigenvalues, but also Bloch’s decomposition of the diagonal does not work. Indeed, the diagonal has only a homology class, so it does not act on cohomology, while Grothendieck-Lefschetz trace formula (1) allowing to count points needs cohomology. Yet, Deligne’s philosophy on the analogy between Hodge type over C and eigenvalue divisibility over Fq , is still at disposal. There is an instance where one can directly generalize the Leitfaden sketched for the proof of Theorem 1.5, by refining the motivic cohomology used. Rather than considering CH0 (X) of the singular variety, one embedds X ⊂ Pn in a projective space, and considers relative motivic cohomology π → Pn is an alteration so that H 2n (P × U, Y × U, n) ([12]), Sect. 1), where P − Y = π −1 (X) is a normal crossings divisor and U = Pn \ X. Then the graph of π has a cycle class in this relative motivic group and one shows ([12], Theorem 1.2) that if X is a hypersurface in Pn of degree ≤ n, then this class decomposes in a suitable sense, generalizing Bloch’s decomposability notion in H 2dim(X) (X, dim(X)) ∼ = CH0 (X) when X is smooth. This implies immediately Hodge type ≥ 1 over C as well as eigenvalue q-divisibility over Fq , and yields a motivic proof of Ax’ theorem asserting the congruence (3) for hypersurfaces of degree ≤ n in Pn over Fq . Ax’ theorem generalizes the mod p congruence due to Chevalley-Warning. Another instance for which one can make Deligne’s philosophy work concerns closed subsets X of Pn defined by equations of degrees d1 ≥ . . . ≥ dr , without any other assumption. Those equations could be chosen in a highly non-optimal way. For example, one could take r times the same equation. r ]}, and Ax and Katz in [1], [40], assign to X a level κ := max{0, [ n−d2 −...−d d1 n κ show that |X(Fq )| ≡ |P (Fq )| mod q . On the other hand, one can compute

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([24] and [25]) that over C, graF Hci (Pn \ X) = 0 for a < κ. Thus one expects eigenvalue divisibility. Indeed one has Theorem 1.6 ( [27], Theorem 1.1, [28], Theorem 2.1). Let X ⊂ Pn be a closed subset defined by equations of degrees d1 ≥ . . . ≥ dr . Then the eigenvalues of the geometric Frobenius acting on Hci ((Pn \ X) ×Fq Fq , Q ) are ¯ in q κ · Z. According to (1), Theorem 1.6 implies Ax-Katz’ theorem. However, the proof of Theorem 1.6 is not good as it uses Ax-Katz’ theorem. One would like to understand a motivic proof in the spirit of [12]. We are very far from it, as, even if X is a smooth hypersurface of low degree, we do not know how to compute its Chow groups of higher dimensional cycles. 1.5 Singular Varieties in Families Singular varieties which are degenerations of smooth ones have more structure. Fakhruddin and Rajan ([31], Corollary 1.2) generalize the motivic method of Theorem 1.5 in a relative situation: if f : X → S is a proper dominant morphism of smooth irreducible varieties over a finite field k with CH0 (X ×S k(X)) = Q, then for any s ∈ S(k), one has |f −1 (s)| ≡ 1 mod |k|. Similarly on the Hodge side one proves ( [29], Theorem 1.1) that if f : X → S is a proper morphism with S a smooth connected curve and X smooth, then if gr0F H i (f −1 (s0 )) = 0 for some s0 in the smooth locus of f and all i ≥ 1, then gr0F H i (f −1 (s)) = 0 for all s and all i ≥ 1. Those two statements, the first one in equal characteristic p > 0 with its strong motivic assumption, the second one in equal characteristic 0 with its minimal Hodge type assumption, suggest, using Deligne’s philosophy, that the mod p reduction of a smooth projective variety in characteristic zero with gr0F H i (X) = 0 for all i ≥ 1 has eigenvalue q-divisibility, and therefore by (1), its number of rational points is congruent to one modulo q. One shows Theorem 1.7 ([30], Theorem 1.1, Section 4, Proof of Theorem 1.1). Let X be a smooth projective variety over a local field K with finite residue ¯ Q ) lives in codimension ≥ 1 for all i ≥ 1. field Fq . Assume H i (X ×K K, Then the eigenvalues of the geometric Frobenius acting on H i (Y ×Fq Fq , Q ) for i ≥ 1, where Y is the mod p reduction of a projective regular model, are ¯ In particular, |Y (Fq )| ≡ 1 mod q. lying in q · Z. If the local field K has equal characteristic p > 0, one expects Theorem 1.7 to be optimal. However, if the local field K has unequal characteristic, one would wish, following Deligne’s philosophy, to replace the assumption on the coniveau of -adic cohomology by gr0F H i (X) = 0 for all i ≥ 1. Due to the comparison of étale and de Rham cohomology, and due to the Hodge conjecture in codimension 1, those two assumptions are equivalent for surfaces. In general, Grothendieck’s generalized Hodge conjecture in codimension 1 predicts that if gr0F H i (X) = 0 then H i (X) lives in codimension ≥ 1. Thus


65

those two conditions are expected to be equivalent in general. So in unequal characteristic, Theorem 1.7 is optimal for surfaces, but in higher dimension, in absence of a proof of the generalized Hodge conjecture in codimension 1, one would wish to have another proof. On the other hand, a generalization of Bloch’s decomposition of the diagonal implies that the motivic assumption CH0 (X0 ⊗K0 Ω) = Q, where K0 ⊂ K is a subfield of finite type over the prime field over which X is defined, i.e. ¯ in unequal X = X0 ×K0 K, and Ω contains K0 (X0 ), so for example, Ω = K characteristic, implies the coniveau assumption of the theorem. So the motivic assumption implies the following direct corollary of Theorem 1.7: Let X be a smooth projective variety over a local field K with finite residue field Fq . Assume CH0 (X0 ×K0 Ω) = Q. Then the eigenvalues of the geometric Frobenius acting on H i (Y ×Fq Fq , Q ) for i ≥ 1, where Y is the mod p reduction of ¯ In particular, |Y (Fq )| ≡ 1 mod a projective regular model, are lying in q · Z. q ([30], Corollary 1.2). It is to be noted that for surfaces in characteristic zero, we are very far from knowing a positive answer to Bloch’s conjecture, asserting that gr0F H i (X) = 0 for i ≥ 1 implies the motivic condition. Thus the range of applicability of Theorem 1.7 is much larger than the one of its corollary. The proof of Theorem 1.7 relies on the specialization map, on Gabber’s purity theorem [32], Theorem 2.1.1, de Jong’s alteration [16], Theorem 6.5, and on the direct generalization of Deligne’s integrality theorem 1.1 to the local field case Theorem 1.8 ([20], Theorem 0.2, Corollary 0.3). Let X be a scheme of finite type defined over a local field K with finite residue field. Then the eigen¯ value of a lifting of the geometric Frobenius in the Galois group Gal(K/K) of i i ¯ ¯ the local field acting on Hc (X ×K K) and H (X ×K K) are algebraic integers for all i. For the proof of Theorem 1.7, one needs a form of the integrality theorem over local fields which is weaker than the one stated in Theorem 1.8, and which can be proven directly using alterations. Theorem 1.8 itself is a corollary of a more general integrality theorem for -adic sheaves. As a corollary, one has, as in Theorem 1.3 over finite field, the divisibility statement for smooth varieties Theorem 1.9 ([20], Corollary 0.4). Let X be a smooth scheme of finite type defined over a local field K with finite residue field Fq . Then the eigenvalues ¯ of a lifting of the geometric Frobenius in the Galois group Gal(K/K) of the i ¯ K, Q (X × ) are algebraic integers divisible by local field acting on HA× K ¯ KK q κ , if A ⊂ X is a closed subscheme of codimension ≥ κ. 1.6 Witt Vector Cohomology Intuitively, if instead of -adic cohomology, we consider crystalline or rigid cohomology, we expect a more direct link between Hodge type and slopes,

66


and consequently congruences for the number of points if the ground field is finite. The theorem of Bloch and Illusie asserts that if X is smooth proper over a perfect field k of characteristic p > 0, W = W (k), K = Frac(W ), then there is a functorial isomorphism ([7], III, 3.5 and [38], II, 3.5) ∼ =

H i (X/K) 0 so that a Gorenstein Fano variety over Fq would have one rational point modulo q. This would also require a generalization of Mori’s break and bend method to varieties with this type of mild singularities.


67

Next, let us remark that we did not discuss in this survey higher congruences, that is congruences modulo q κ , κ ≥ 2. Indeed, Theorem 1.3 implies that if X is projective smooth over Fq , then if H i (X ×Fq Fq , Q ) is supported in codimension ≥ κ up to the class of the j-th self-product of the polarization if i = 2j, then (1) yields |X(Fq )| ≡ |Pn (Fq )| mod q κ . However, following the Leitfaden explained in subsection 1.3, the coniveau κ condition is implied by triviality of CHi (X ×Fq Fq (X)), for all i < κ, and this is a condition we can basically never check. Finally, by Theorem 1.10, Witt vector cohomology computes the slope < 1 part of rigid cohomology. In particular, it is a topological cohomology theory. We do not know the relation between rigid and -adic cohomology if X is proper but not smooth. On the other hand, the slope = 0 part of crystalline cohomology, when X is smooth, is easier to understand as it is described by coherent cohomology. This viewpoint is developed in Sect. 3.

2 Additive Higher Chow Groups with Higher Modulus of Type (n,n) over a Field 2.1 Additive Higher Chow Groups Let X be a smooth variety over a field k. In [9] Bloch develops a theory of higher Chow groups, which are isomorphic to the motivic cohomology groups 2p−n CHp (X, n) ∼ = HM (X, Z(p)), p, n ≥ 0 (see [51]), by using the scheme ∆n = n 1 Spec k[t0 , . . . , tn ]/ ( i=0 ti − 1) or, in the cubical definition, the scheme (P \ n ti = 1 by ti = λ yields the {1}) . Replacing in the simplicial definition same groups as long as λ ∈ k× . The degenerate case λ = 0 is investigated in [10]. One obtains a theory of additive higher Chow groups, SHp (X, n), p ≥ 0, n ≥ 1. In analogy to the theorem of Nesterenko-Suslin and Totaro (see [47], [50]) CHn (k, n) ∼ = KnM (k), it is shown Theorem 2.1 ( [10], Theorem 5.3). Let k be field with char k = 2, then there is an isomorphism of groups n−1 SHn (k, n) ∼ . = Ωk/Z

The proof is in the spirit of the proofs in [47], [50]. We will sketch the proof for the corresponding statement with SHn (k, n) replaced by a cubical version of the higher additive Chow groups. This cubical version is defined in [11], so far only for a field and on the level of 0-cycles. The definition is as follows. n with coordinates (x, y1 , . . . , yn ) Consider the k-scheme Xn = A1 × (P1 \ {1}) n and denote the union of all faces by Yn = i=1 (yi = 0, ∞). Now denote by Z0 (k, n − 1) the free abelian group generated on all closed points of A1 \ {0} × (P1 \ {0, 1, ∞})n−1 . Let Z1 (k, n; 2) be the free abelian group generated on all irreducible curves C ⊂ Xn \ Yn satisfying the following properties

68


1. (Good position) ∂ij [C] = (yi = j).[C] ∈ Z0 (k, n − 1), for i = 1, . . . , n, j = 0, ∞. → P1 × (P1 )n be the normalization of the 2. (Modulus 2 condition) Let ν : C compactification of C, then in Z0 (C) 2 div(ν ∗ x) ≤

n

div(ν ∗ yi − 1).

(7)

i=1

Because of (i) one has a complex ∂=

n

(−1)i (∂i0 − ∂i∞ ) : Z1 (k, n; 2) → Z0 (k, n − 1) → 0.

i=1

Definition 2.2 ([11], Definition 6.2). The additive higher Chow groups of type (n, n) and modulus 2 of a field k are given by the homology of the above complex, i.e. Z0 (k, n − 1) . THn (k, n; 2) = ∂Z1 (k, n; 2) It is shown in [11, Theorem 6.4] (cf. Theorem 2.1 ) Theorem 2.3. Let k be a field with char k = 2, 3, then the map

n−1 THn (k, n; 2) −→ Ωk/Z ,

[P ] → Trk(P )/k

1 dy1 (P ) dyn−1 (P ) ··· x(P ) y1 (P ) yn−1 (P )

(8)

is an isomorphism of groups. Furthermore, the inclusion ι : (P1 \ {1})n−1 → Xn−1 , (y1 , . . . , yn−1 ) → (1, y1 , . . . , yn−1 ) induces a commutative diagram M Kn−1 (k) dlog

n−1 Ωk/Z

/ CHn−1 (k, n − 1) ι∗

/ THn (k, n; 2)

defined on elements by {a1 , . . . , an−1 } _ da1 a1

∧ ... ∧

dan−1 an−1

/ (a1 , . . . , an−1 ) _ / (1, a1 , . . . , an−1 ).


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The idea of the proof is the following. One first shows, that the map (8) is well defined, using the reciprocity law for rational differential forms on a nonsingular projective curve. Here one explicitly uses the modulus condition (7). Then one constructs an inverse map with the help of a representation of the absolute differential forms of k by generators and relations. Showing this map to be well defined is equivalent to finding 1-cycles in Z1 (k, n; 2), whose boundary yield the relations. This map is easily seen to be injective and the surjectivity follows from the fact, that the trace on the absolute differentials corresponds to the pushforward on the additive Chow groups. 2.2 Higher Modulus The cubical definition allows one to generalize the definition of the additive higher Chow groups (of type (n,n)) to higher modulus, i.e. replace the 2 in equation (7) by an integer m ≥ 2. The resulting groups are denoted by THn (k, n; m). (Notice that one has THn (k, n; 0) = CHn (A1 \ {0}, n − 1) = 0 = THn (k, n; 1).) Up to now it is not clear how to formulate this higher modulus condition in the simplicial setup. The attempt to generalize Theorem 2.3 leads to the following considerations. Let Pic(A1 , m{0}) be the relative Picard group of A1 with modulus m{0}. Then there is a natural surjective map Pic(A1 , m{0}) −→ TH1 (k, 1; m).

(9)

For this one observes that, if two 0-cycles divf and divg in the left hand side are equal, then the curve C ⊂ A1 × P1 \ {1} defined by f y = g satisfies the modulus condition ¯ m(x = 0).C¯ ≤ (y − 1).C, with C¯ ⊂ P1 × P1 the closure of C. Hence divf = divg also in the right hand side. By Theorem 2.3 this is an isomorphism for m = 2 and one can show that it is an isomorphism for all m ≥ 2. On the other hand we may identify Pic(A1 , m{0}) as a group with the additive group of the ring of big Witt vectors of length m − 1 of k via Pic(A1 , m{0}) ∼ =

1 + tk[t] 1 + tm+1 k[t]

×

∼ = Wm−1 (k).

(10)

This together with Theorem 2.3 leads to the prediction, that THn (k, n; m) is isomorphic to the group of generalized degree n − 1 Witt differential forms of length m − 1. These groups form the generalized de Rham-Witt complex of Hesselholt-Madsen generalizing the p-typical de Rham-Witt complex of BlochDeligne-Illusie (see [7], [38]). Before stating the generalization of Theorem 2.3 to the case of higher modulus, we describe the de Rham-Witt complex and some of his properties.

70


Definition 2.4 (see [36], [48], cf. [38]). Let A be a ring. A Witt complex over A is a projective system of differential graded Z-algebras ((Em )m∈N , R : Em+1 → Em ) together with families of homomorphisms of graded rings (Fn : Enm+n−1 → Em )m,n∈N and homomorphisms of graded groups (Vn : Em → Enm+n−1 )m,n∈N satisfying the following relations, for all n, r ∈ N (i) (ii) (iii) (iv)

RFn = Fn Rn , Rn Vn = Vn R, F1 = V1 = id, Fn Fr = Fnr , Vn Vr = Vnr . Fn Vn = n, and if (n, r) = 1, then Fr Vn = Vn Fr on Erm+r−1 . Vn (Fn (x)y) = xVn (y), for x ∈ Enm+n−1 , y ∈ Em . Fn dVn = d, with d the differential on Em and Enm+n−1 respectively.

Furthermore, there is a homomorphism of projective systems of rings 0 (λ : Wm (A) → Em )m∈N ,

under which the Frobenius and the Verschiebung maps on the big Witt vectors correspond to Fn and Vn , n ∈ N on E 0 and satisfies (v) Fn dλ([a]) = λ([a]n−1 )dλ([a]), for a ∈ A. A morphism of Witt complexes over A is a morphism of projective systems of dga’s compatible with all the structures. This yields a category of Witt complexes over A. Theorem 2.5 (see [36], [48], cf. [38] ). The category of Witt complexes has an initial object, called the de Rham-Witt complex of A and denoted by · )m∈N . (Wm ΩA Hesselholt-Madsen prove this using the Freyd adjoint functor theorem. In case · A is a Z(p) -algebra, p = 2 a prime, Wm ΩA may be constructed following Illusie · as a quotient of ΩWm (A) . It follows · · = ΩA/Z W1 ΩA

0 Wm ΩA = Wm (A).

Remark 2.6. Let A be a Z(p) -algebra, p = 2 a prime, and denote by · )n∈N0 the p-typical de Rham-Witt complex of Bloch-Deligne-Illusie(Wn ΩA Hesselholt-Madsen, then one has · ∼ · Wm ΩA Wn(j) ΩA , n(j) given by jpn(j) ≤ m < jpn(j)+1 . (11) = (j,p)=1

If X is a smooth variety over a perfect field, Bloch and Illusie also define · Wn ΩX , which is a complex of coherent sheaves on the scheme Wn (X). Its hypercohomology equals the crystalline cohomology. This is used for example to derive (4).


71

The generalization of Theorem 2.3 to the case of higher modulus is given by the Theorem 2.7 ([48]). Let k be a field with char = 2. Then the projective sys tem ( n≥1 THn (k, n; m + 1) → n≥1 THn (k, n; m))m≥2 can be equipped with the structure of a Witt complex over k and the natural map W∗ Ωk· → n≥0 THn+1 (k, n + 1; ∗ + 1) induced by the universality of the de Rham-Witt complex is an isomorphism. In particular Wm−1 Ωkn−1 ∼ = THn (k, n; m). In the following the Witt complex structure of THn+1 (k, n + 1; m + 1) Tm := n≥0

is described. The multiplication of a graded commutative ring on Tm is induced by the exterior product of cycles followed by a pushforward, which is induced by the multiplication map A1 × A1 → A1 . The differential is induced by pushing forward via the diagonal A1 → A1 × P1 and then restricting to A1 × P1 \ {1}. Fr (resp. Vr ) is induced by pushing forward (resp. pulling back) via 0 = TH1 (k, 1; m + 1) A1 → A1 , a → ar . And finally the map Wm (k) → Tm is given by the composition of (10) and (9). The relations (i)-(iii) and (v) in Definition 2.4 are already satisfied on the level of cycles. That Tm is a dga with (iv), follows from the result of Nesterenko-Suslin and Totaro, since one has a surjective map CH n−1 (k(P0 ), n − 1) → THn (k, n; m) P0 ∈A1 \{0}

induced by the inclusions {P0 }×(P1 \{1})n−1 → A1 \{0}×(P1 \{1})n−1 . Thus T is a Witt complex and this gives the natural map from Theorem 2.7. In [48] a trace map for arbitrary field extensions on the de Rham-Witt groups is constructed as well as a residue symbol for closed points of a smooth projective curve on the rational Witt differentials of this curve and a reciprocity law is proven, generalizing the corresponding notions and statements on K¨ ahler differentials (and the one obtained by Witt in [53]). Using this results one can generalize the proof of Theorem 2.3 to obtain, that the map THn (k, n; m) → Wm−1 Ωkn−1 1 d[y1 (P )] d[yn−1 (P )] ··· [P ] → Trk(P )/k [x(P )] [y1 (P )] [yn−1 (P )] gives the inverse map to the one obtained by the universality of the de Rhamuller lift. Witt complex. Here [−] : k(P ) → Wm−1 (k(P )) is the Teichm¨ Finally we explain how to describe the p-typical de Rham-Witt complex over a field k via the additive higher Chow groups. Denote by

72

Manuel Blickle, Hélène Esnault, Kay R¨ ulling i ∞ tp f = exp( − i ) ∈ 1 + tZ(p) [[t]] p i=0

the inverse of the Artin-Hasse eponential and by i fm ∈ (1 + ⊕m i=1 t Z(p) )

the truncation of f . Write m for the 0-cycle [divfm ] ∈ TH1 (k, 1; m + 1). Then it follows from the description of the additive higher Chow groups as the de Rham-Witt complex and from (11), that one has Wr Ωkn−1 ∼ = THn (k, n; pr + 1) ∗ pr , where we denote by ∗ the multiplication in the additive higher Chow groups explained above.

3 Riemann–Hilbert Type Correspondences and Applications to Local Cohomology Invariants This section starts with a very rough introduction to some aspects of the Riemann–Hilbert correspondence (over C) and, shortly thereafter, to a positive characteristic analog, developed recently by Emerton and Kisin [22]. We start at a basic level – merely motivating the correspondence with the help of a fundamental example – but progress quickly to a nontrivial construction which is central to our applications: the intermediate extension. The aim is to show how these correspondences can be put to good use in order to study singularities. Concretely we obtain a new characterization of invariants arising from local cohomology in terms of étale cohomology. One central interesting aspect of our approach is that the treatment is, up to the use of the respective correspondence, independent of the characteristic. The result we will discuss is a description of Lyubeznik’s local cohomology invariants in any characteristic [43] which are originally defined for a quotient A = R/I of a n-dimensional regular local ring (R, m) to be def

a (HIn−i (R))) λa,i = e(Hm

where e( ) denotes the D–module multiplicity (see Section 3.2). In [43] these are shown to be independent of the representation of A as the quotient of a regular local ring. In [4] and [5] these invariants were described in many cases in terms of étale cohomology; we discuss these results here as an application of the aforementioned correspondences: Theorem 3.1. Let k-be separably closed. Let A = OY,x for Y a closed k– n−i subvariety of a smooth variety X. If for i = d the modules H[Y ] (OX ) are supported at the point x then


73

1. For 2 ≤ a ≤ d one has λa,d (A) − δa,d = λ0,d−a+1 (A) 2.

and all other λa,i (A) vanish. d−a+1 dimFp H{x} (Yét , Fp ) λa,d (A) − δa,d = d−a+1 dimC H{x} (Yan , C)

if char k = p if k = C

where δa,d is the Kroneker delta function. The apparent analogy between the situation over C and over Fp suggested by this result is somewhat misleading. As we briefly discuss at the end of this section, étale cohomology with Fp –coefficients is only a very small part (the slope zero part) of, say, crystalline cohomology. For crystalline cohomology on the other hand there are comparison results to singular cohomology (via de Rham theory) hence Lyubeznik’s invariants really capture a different type of information in characteristic 0 than they do in characteristic p > 0. 3.1 Riemann–Hilbert and Emerton–Kisin Correspondence Let us first fix the following notation: Throughout this section X will be a smooth scheme over a field k of dimension n. Mostly k will be either C, the field of complex numbers, or Fq , the finite field with q = pe elements, as in Section 1. The Riemann–Hilbert Correspondence Now let k = C. In its simplest incarnation the Riemann–Hilbert correspondence asserts a one to one map between local systems of locally free coherent OX –modules ↔ with regular singular integrable connection C-vectorspaces This correspondence grew out of Hilbert’s 21st problem, motivated by work of Riemann, to find, given a monodromy action at some points, a Fuchsian (regular singular) differential equation with the prescribed monodromy action at the singular points. The correspondence is given via de Rham theory: A connection is a k– 1 linear map ∇ : M −→ ΩX ⊗OX M that satisfies the Leibniz rule ∇(rm) = r∇(m) + dr ⊗ m. One can extend ∇ in an (up to signs) obvious way to get a sequence of maps (each denoted also by ∇) ∇

∇

∇

1 2 n M −−→ ΩX ⊗ M −−→ ΩX ⊗ M −−→ . . . −→ ΩX ⊗M

and the connection is called integral if this sequence is a complex (i.e. ∇2 = 0), called the de Rham complex dR(M) associated to the connection ∇. Now, given such integrable connection its horizontal sections

74


M∇ = ker ∇ = H 0 (dR(M)) is a local system. The most trivial, but at the same time most important example, is M = d 1 OX with ∇ = d the universal differential OX −−→ ΩX . This is the OX –module ∂ , . . . , ∂x∂n ) · f = 0 where associated to the system of differential equations ( ∂x 1 x1 , . . . , xn are local coordinates at some point of X. The corresponding local system (the solutions to the differential equation) is of course the constant local system C. In fact more is true: By the Poincaré Lemma, the (analytic) de Rham complex is a resolution of the constant sheaf C and hence we can rephrase this by saying that C is quasi-isomorphic to the de Rham complex. In view of Grothendieck’s philosophy the above correspondence is flawed since the categories on either side are not closed under any reasonable functors. For example, the pushforward of a local system is generally not a local system i as the inclusion of a point {x} −−→ A1 readily illustrates (i∗ C is a skyscraper sheaf). However it is a constructible sheaf (that is one that is locally constant on each piece of a suitable stratification of X), and in fact on constructible sheaves all the functors one would like to have are defined. On the other side of the correspondence, modules with integrable connection are replaced by modules over the ring of differential operators DX . The conditions one has to impose are holonomicity (which is the crucial finiteness condition) and a further condition (namely that M is regular singualar) which will not be considered here. Without giving the precise definition, a holonomic DX –module is one of minimal possible dimension, and the category they form enjoys a strong finiteness condition: Proposition 3.2 ([15]). The category of holonomic DX –modules is abelian and closed under extensions. Every holonomic DX module has finite length. The correspondence should again be given via de Rham theory. However, it quickly becomes clear that one has to pass to the derived category. We give the following simple example as a small indication that the passage to the derived category cannot be avoided. ∂ Example 3.3. Let X = A1 = Spec C[x] such that DX = k[x, ∂x ]. Let M = 1 H{0} (OA1 ) denote the cokernel of the following injection of DX –modules

C[x] −→ C[x, x−1 ]. Then, as a C-vectorspace M has a basis consisting of de Rham complex associated to M is given

1 xi

for i > 0. The

∇

M −−→ ΩA1 1 ⊗ M where the map sends a basis element x1i −→ dx ⊗ x−i i+1 . Hence one immediately sees that ∇ is injective and that its cokernel is generated by dx ⊗ x1 , which is the skyscraper sheaf supported at 0. Hence we have an quasi-isomorphism


i∗ C{0}

1 dR(H{0} (OA1 ))[1].

75

(12)

Note the appearance of the shift [1] which is an indication that one cannot avoid to pass to the derived category. The ultimate generalization of the basic version above is the Riemann–Hilbert correspondence as proved by Mebkhout [46], Kashiwara [39], Beilinson and Bernstein [15]: Theorem 3.4. Let X be a smooth C–variety. On the level of bounded derived categories, there is an equivalence between constructible sheaves of holonomic DX –modules ↔ which are regular singular C-vectorspaces The correspondence is given by sending a complex M• of DX –modules to dR(M• ) = RHom(OX , M). This equivalence preserves the six standard functors f ∗ and f∗ , their duals f! and f ! and also ⊗ and Hom. Via duality on X the de Rham functor is related to the functor Sol( ) = RHom( , OX ). This functor yields therefore an anti-equivalence, and it is this anti-equivalence which can be obtained in positive characteristic. Emerton–Kisin Correspondence In positive characteristic a naive approach via de Rham theory does not work due to the failure of the Poincaré lemma. At least one should expect that the correspondence respects the relationship “Fp corresponds to OX ” in some way or another. But due to the fact that in characteristic p there are many functions with derivative zero (namely all pth powers), the de Rham theory is ill behaved. For the same reason, D–module theory in positive characteristic is also quite different from the one in characteristic zero. So what is the correct counterpart for constructible sheaves of Fp –vectorspaces? The solution arises from the Artin–Schreier sequence: x →xp

0 −→ Fp −→ OX −−−−−→ OX −→ 0. This is an exact sequence in the étale topology. Hence if one views the Riemann-Hilbert correspondence as a vast generalization of de Rham theory (in terms of the Poincaré lemma), then the correspondence of Emerton–Kisin – to be outlined shortly – is an analogous generalization of Artin–Schreier theory. Hence, the basic objects one studies in this correspondence is not DX – modules but rather quasi-coherent OX –modules M which are equipped with an action of the Frobenius F . That is, we have an OX –linear map FM : M −→ F∗ M. By adjunction, such a map is the same as a map θM : F ∗ M −→ M. Such objects have been studied in various forms for a long time, see for example [35] or [21, Exp. XXII].

76


Definition 3.5. An OX,F –module (M, θ) is a quasi-coherent OX –module M together with a OX –linear map θ : F ∗ M −→ M If θ is an isomorphism (M, θ) is called unit. If is called finitely generated if M is finitely generated when viewed as a module over the non-commutative ring OX,F . The following two are the essential examples of finitely generated unit OX,F – modules. Example 3.6. For simplicity assume that X = Spec R is affine. The natural identification F ∗ OX ∼ = OX gives OX the structure of a finitely generated unit OX,F –module. Let Rf be the localization of R at a single element f ∈ R. The natural map F ∗ Rf = R ⊗F Rf −→ Rf p

sending r ⊗ ab to r abp has a natural inverse given by sending ab −→ abp−1 ⊗ 1b . Hence Rf is naturally a unit module. In fact, Rf is even finitely generated as a unit R[F ]–module, generated by the element f1 , since F ( f1 ) = f1p . The main result of [44] about finitely genrated unit OX,F –modules, which makes them a suitable analog of holonomic D–modules, is Theorem 3.7. Let X be smooth. In the abelian category of (locally) finitely generated OX,F –modules, every object has finite length. Example 3.8. Considering a Cech resolution to compute coherent cohomology with support in some subscheme Z, the easy abelian category part of the preceding theorem and the preceding examples imply that HZi (OX ) is naturally a f.g. (finitely generated) unit OX,F –module. Now the correspondence that is proven by Emerton and Kisin in [22] can be summarized as follows: Theorem 3.9. For X smooth, there is an anti-equivalence on the level of derived categories locally finitely generated constructible sheaves of ↔ OX,F –modules Fp –vector-spaces on Xét The correspondence is given by sending a (complex of ) OX,F –modules M• to the constructible sheaf Sol(M• ) = RHom(M, OX ) and is roughly dual to the naive approach of taking the fixed points of the Frobenius (Artin-Schreier sequence) alluded to above. The correspondence preserves certain functors, namely f ! , f∗ (and ⊗) on the right hand side correspond to f ∗ , f! on the left hand side.


77

Intermediate Extensions From now on we treat the two situations – characteristic zero regular singular holonomic DX –modules with the Riemann–Hilbert correspondence on one hand and positive characteristic finitely generated unit OF,X –modules with the Emerton–Kisin correspondence on the other – formally as one. The crucial property they both share is the fact that all modules have finite length. This is key to the following construction. Due to the lack of duality in positive characteristic there is no analog of the functor j! available. However there is an adequate substitute, which still exists in our context [23, 6]. The substitute we have in mind is the intermediate extension j!∗ , usually constructed as the image of the “forget supports map” from j! −→ j∗ . This usual definition does not work since j! is not available. Nevertheless it turns out that all one needs to define j!∗ is the fact that the modules have finite length: Proposition 3.10 ([6], [23]). Let j : U ⊆ X be a locally closed immersion of smooth Fp –schemes (resp. C–schemes). Let M be a finitely generated unit OU,F –module. Then there is a unique submodule N of R0 j∗ M minimal with respect to the property that f ! N = M. This submodule N is called the intermediate extension and is denoted by j!∗ M. Proof. The key point is the fact that M has finite length which ensures the existence of minimal modules with the desired property (any decreasing chain is eventually constant). Let N1 and N2 be two modules with the desired property. Since f ! Ni = M their intersection cannot be zero. On the other hand the exact sequence 0 −→ N1 ∩ N2 −→ R0 j∗ M −→ R0 j∗ M/N1 ⊕ R0 j∗ M/N2 shows that their intersection is also a finitely generated unit OX,F –module (resp. r.s. holonomic DX –module) as it is the kernel of a map of such modules. By minimality one has N1 = N2 showing uniqueness. Under the correspondence the intermediate extensions behave well: In the situation as above we have Sol(j!∗ M) = Image(j! Sol(M) −→ R0 j∗ Sol(M)) such that they do in fact correspond to the intermediate extensions on the constructible side, where they can be identified as the image of the “forget supports” map. For the basic computations that follow we list some key properties both correspondences enjoy:

1. Sol(OX ) =

Fp [n] C[n]

if k = Fp if k = C

78


2. There are functors f ! and f∗ which behave under the correspondence in the expected way. 3. For Y ⊆ X a subvariety, local cohomology is defined in the categories and satisfies the triangle (a highbrow way of writing the long exact sequence for cohomology with supports) RΓ[Y ] M• −→ M• −→ Rj∗ j ∗ M• −−−→ +1

where j is the open inclusion of the complement of Y into X. 4. Again, let i : Y → X be the inclusion of a closed subset, then Sol ◦RΓ[Y ] ∼ = Ri! i∗ ◦ Sol . This follows via the preceding two items and the triangle Rj! j ! L −→ L −→ +1 Ri! i−1 L −−−→ for a (complex of) constructible sheaves L. 5. The preceding items allow us to compute (see also equation (12) on page 75) Sol(RΓY (OX )) = i! i∗ Fp [n] = i! Fp |Y [n] in positive characteristic and respectively i! C|Y [n] in characteristic zero. 3.2 Lyubeznik’s Local Cohomology invariants Let A = R/I for I an ideal in a regular (local) ring (R, m) of dimension n and containing a field k. The main results of [43, 37] state that the local a cohomology module Hm (HIn−i (R)) is injective and supported at m. Therefore a (HIn−i (R))) many copies of the injective it is a finite direct sum of e = e(Hm hull ER/m of the residue field of R. Lyubeznik shows in [43] that this number def

a λa,i (A) = e(Hm (HIn−i (R)))

does not depend on the auxiliary choice of R and I. At the same time, this number e(M) is the multiplicity of the holonomic DX –module M, respectively the finitely generated unit OX,F –module M. If A is a complete intersection, these invariants are essentially trivial (all are zero except λd,d = 1 where d = dim A). In general λa,i can only be nonzero in the range 0 ≤ a, i ≤ d. These invariants were first introduced by Lyubeznik in [43] and further studied by Walther in [52]. In [33] Garcia-L´ opez and Sabbah show Theorem 3.1 in the case of an isolated complex singularity. We now indicate briefly the proof of 3.1. As shown in [4] the condition imposed on the singularities in the Theorem 3.1 easily implies (via the spectral j a+j a H[Y sequence E2a,j = H[x] ] (OX ) ⇒ H[x] (OX )) part one. Part two is the point where the correspondences enter into the picture: The idea is of course to use that a a (HYn−d (OX ))) = dim Sol(H{x} (HYn−d (OX ))) e(Hm


79

by the correspondence, and then to compute the right hand side. As it is written here the right hand side is however not computable. The trick now is to replace HIn−d (R) by j!∗ HYn−d (OX )|X−{x} , which is easily checked to not affect our computation (long exact sequence for Γ{x} ). Now, the assumption on the singularity that for i = d the module HYn−i (OX ) is supported at x can simply be rephrased as HYn−d (OX )|X−{x} = RΓY −{x} (OX−{x} )[n − d]. Using that Sol commutes with j!∗ and the fact that Sol(RΓY −{x} (OX−{x} )) = i! (Fp )Y −{x} [n] one obtains that a e(Hm (HYn−d (OX ))) = dim(H −a j!∗ i! (Fp )Y −{x} [d]).

To compute the right hand side is now a feasible task that yields the desired result (feasible due to the fact that j is just the inclusion of the complement of a point, which makes it possible to effectively understand and calculate j!∗ ; see [4, 5] for details). 3.3 Comparison via Crystalline Cohomology We close this section with some remarks regarding the behaviour of these invariants under reduction to positive characteristic. There are by now classical examples that show that local cohomology does not behave well under reduction so one would expect that the invariants λa,i do not behave well either. On a superficial level, glancing at Theorem 3.1, one might however suspect a complete analogy between positive and zero characteristic. However, this is not true. The difference stems from the difference between the cohomology theories which describe λa,i . In positive characteristic, this is étale cohomology with coefficients in Fp , in characteristic zero however it is (topological) cohomology. Under reduction mod p the former only constitutes a very small part of the latter, namely the part of slope zero. For simplicity consider the situation of Y ⊆ Pn a smooth projective variety. Now. the local ring A of the cone of this projective embedding has an isolated singularity at its vertex and one can study the invariants λa,i (A). Theorem 3.1 shows that λa,d (A) are described (excision) by H d−a (Yét , Fp ) and H d−a (Yan , C) in positive and zero characteristic respectively. So far everything appears in complete analogy. But let us now consider reduction mod p and let Y be the special fiber of the smooth family Y −→ Spec W (k), where W (k) is the ring of Witt vectors over k and K denotes its field of fractions. Via the comparison results between topological cohomology, de Rham cohomology and Berthelot’s crystalline cohomology (see [3]) ∼ = d−a (YK ) ∼ H d−a (YC , C) −−→ HdR = H d−a (Y /K)

80


it turns out that H d−a (Yét , Fp ) is only a very small part of the crystalline cohomology H d−a (Y /K), namely the part on which the Frobenius acts with eigenvalue zero. This is an even smaller part then the part H d−a (Y /K) 0, Journal f¨ ur reine und angewandte Mathematik 491 (1997), 65–130. 45. Manin, Yu.: Notes on the arithmetic of Fano threefolds, Compos. math. 85 (1993), 37–55. 46. Mebkhout, Z.: Cohomologie locale des espaces analytiques complexes, Ph.D. thesis, Université Paris VII, 1979. 47. Nesterenko, Y. P., Suslin, A.: Homology of the general linear group over a local ring, and Milnor’s K-theory, Math. USSR-Izv. 34, No.1 (1990), 121–145. 48. R¨ ulling, K.: The generalized de Rham-Witt complex over a field is a complex of zero-cycles, preprint 2005. 49. Serre, J.-P.: Sur la topologie des variétés algébriques en caractéristique p, in Symposium internacional de topolog´ıa algebraica (International symposium on algebraic topology), 24–53, Universidad Nacional Autonoma de Mexico and UNESCO, Mexico City, 1958. 50. Totaro, B.: Milnor K-theory is the simplest part of algebraic K-theory, K-theory 6, No.2 (1992), 177–189. 51. Voevodsky, V.: Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not. 7 (2002), 7, 351–355. 52. Walther, U.: On the Lyubeznik numbers of a local ring, Proc. Amer. Math. Soc. 129 (2001), No. 6, 1631–1634 (electronic). 53. Witt, E.: Zyklische K¨ orper und Algebren der Charakteristik p vom Grad pn . Struktur diskret bewerteter perfekter K¨ orper mit vollkommenem Restklassenk¨ orper der Charakteristik p, J. Reine Angew. Math. 176 (1936), 126–140.

Vector Bundles and Torsion Free Sheaves on Degenerations of Elliptic Curves Lesya Bodnarchuk1 , Igor Burban2 , Yuriy Drozd3 , and Gert-Martin Greuel4 1 2 3 4

Technische Universit¨ at Kaiserslautern, Erwin-Schr¨ odinger-Straße, 67663 Kaiserslautern, Germany, [email protected] Institut f¨ ur Mathematik, Johannes Gutenberg-Universit¨ at Mainz, 55099 Mainz, [email protected] Kyiv Taras Shevchenko University, Department of Mechanics and Mathematics, Volodimirska 64, 01033 Kyiv, Ukraine, [email protected] Technische Universit¨ at Kaiserslautern, Erwin-Schr¨ odinger-Straße, 67663 Kaiserslautern, Germany, [email protected]

Summary. In this paper we give a survey about the classification of vector bundles and torsion free sheaves on degenerations of elliptic curves. Coherent sheaves on singular curves of arithmetic genus one can be studied using the technique of matrix problems or via Fourier-Mukai transforms , both methods are discussed here. Moreover, we include new proofs of some classical results about vector bundles on elliptic curves.

1 Overview The aim of this paper is to give a survey on the classification of vector bundles and torsion free sheaves on singular projective curves of arithmetic genus one. We include new proofs of some classical results on coherent sheaves on smooth elliptic curves, which use the technique of derived categories and FourierMukai transforms and are simpler than the original ones. Some results about singular curves are new or at least presented in a new framework. This research project had several sources of motivation and inspiration. Our study of vector bundles on degenerations of elliptic curves was originally motivated by the McKay correspondence for minimally elliptic surface singularities [Kah89]. Here we use as the main technical tool methods from the representation theory of associative algebras, in particular, a key tool in our approach to classification problems is played by the technique of “representations of bunches of chains ” or “Gelfand problems” [Bon92]. At last, but not least we want to mention that our research was strongly influenced by ideas and methods coming from the homological mirror symmetry [Kon95, PZ98, FMW99].

84

Lesya Bodnarchuk, Igor Burban, Yuriy Drozd, and Gert-Martin Greuel

Many different questions concerning properties of the category of vector bundles and coherent sheaves on degenerations of elliptic curves are encoded in the following general set-up: Problem 1. Let E −→ T be a flat family of projective curves of arithmetic genus one such that the fiber Et is smooth for generic t and singular for t = 0. What happens with the derived category Db (CohEt ), when t → 0?

In order to start working on this question one has to consider the absolute case first, where the base T is Spec(k). In particular, one has to describe indecomposable vector bundles and indecomposable objects of the derived category of coherent sheaves on degenerations of elliptic curves and develop a technique to calculate homomorphism and extension spaces between indecomposable torsion free sheaves as well as various operations on them, like tensor products and dualizing. For the first time we face this sort of problems when dealing with the McKay correspondence for minimally elliptic singularities. Namely, let S = Spec(R) be the spectrum of a complete (or analytical) two-dimensional −→ S its minimal resolution, and E minimally elliptic singularity, π : X the exceptional divisor. Due to a construction of Kahn [Kah89], the functor M → resE (π ∗ (M )∨∨ ) establishes a bijection between the reflexive R–modules (maximal Cohen-Macaulay modules) and the generically globally generated indecomposable vector bundles on E with vanishing first cohomology.5 A typical example of a minimally elliptic singularity is a Tpqr – singularity, given by the equation kx, y, z/(xp + y q + z r − λxyz), where p1 + 1q + 1r ≤ 1 and λ = 0. If p1 + 1q + 1r = 1, then this singularity is simple elliptic and the exceptional divisor E is a smooth elliptic curve. Thus, in this case a description 5

resE denotes the restriction to E

Vector Bundles on Degenerations of Elliptic Curves

85

of indecomposable maximal Cohen-Macaulay modules follows from Atiyah’s classification of vector bundles on elliptic curves [Ati57]. The main result of Atiyah’s paper essentially says: Theorem 1 (Atiyah). An indecomposable vector bundle E on an elliptic curve E is uniquely determined by its rank r, degree d and determinant det(E) ∈ Picd (E) ∼ = E. In Section 2.3 we give a new proof of this result. However, if p1 + 1q + 1r < 1, then S is a so-called cusp singularity and in this case E is a cycle of n projective lines En , where E1 denotes a rational curve with one node.

A complete classification of indecomposable vector bundles and torsion free sheaves on these curves in the case of an arbitrary base field k was obtained by Drozd and Greuel [DG01]. For algebraically closed fields there is the following description, which we prove in Section 3.2. Theorem 2. Let En denote a cycle of n projective lines and Ik be a chain of k projective lines, E an indecomposable torsion free sheaf on En . 1. If E is locally free, then there exists an étale covering π : Enr −→ En of degree r, a line bundle L ∈ Pic(Enr ) and a natural number m ∈ N such that E∼ = π∗ (L ⊗ Fm ), where Fm is an indecomposable vector bundle on Enr , recursively defined by the sequences 0 −→ Fm−1 −→ Fm −→ O −→ 0,

m ≥ 2,

F1 = O.

2. If E is not locally free then there exists a finite map p : Ik −→ En and a line bundle L ∈ Pic(Ik ) (where k, p and L are determined by E) such that E∼ = p∗ (L). This classification is completely analogous to Oda’s description of vector bundles on smooth elliptic curves [Oda71] and provides quite simple rules for the computation of the decomposition of the tensor product of any two vector bundles into a direct sum of indecomposable ones. It allows to describe the

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dual sheaf of an indecomposable torsion free sheaf as well as the dimensions of homomorphism and extension spaces between indecomposable vector bundles (and in particular, their cohomology), see [Bur03,BDG01]. We carry out these computations in Section 3.2. However, the way we prove this theorem, essentially uses ideas from representation theory and the technique of matrix problems [Bon92]. Using a similar approach, Theorem 2 was generalized by Burban and Drozd [BD04] to classify indecomposable complexes of the bounded (from the right) derived category of coherent sheaves D− (Coh(E)) on a cycle of projective lines E = En , see also [BD05] for the case of associative algebras. The situation turns out to be quite different for other singular projective curves of arithmetic genus one. For example, in the case of a cuspidal rational curve zy 2 = x3 even a classification of indecomposable semi-stable vector bundles of a given slope is a representation-wild problem.6 However, if we restrict our attention only on stable vector bundles, then this problem turns out to be tame again.7 Moreover, the combinatorics of the answer is essentially the same as for smooth and nodal Weierstraß curves8 : Theorem 3 (see [BD03, BK3]). Let E be a cuspidal cubic curve over an algebraically closed field k then a stable vector bundle E is completely determined by its rank r, its degree d, that should be coprime, and its determinant det(E) ∈ Picd (E) ∼ = k. The technique of matrix problems is a very convenient tool for the study of vector bundles on a given singular projective rational curve of arithmetic genus one. However, to investigate the behavior of the category of coherent sheaves on genus one curves in families one needs other methods. One possible approach is provided by the technique of derived categories and Fourier-Mukai transforms [Muk81,ST01], see Section 3.4. The key idea of this method is that we can transform a sky-scraper sheaf into a torsion free sheaf by applying an auto-equivalence of the derived category. In a relative setting of elliptic fibrations with a section one can use relative Fourier-Mukai transforms to construct examples of relatively semi-stable torsion free sheaves, see for example [BK2]. Theorem 4 (see [BK1]). Let E be an irreducible projective curve of arithmetic genus one over an algebraically closed field k. Then 6

7 8

An exact k–linear category A over an algebraically closed field k is called wild if it contains as a full subcategory the category of finite-dimensional representations of any associative algebra. For a formal definition of tameness we refer to [DG01], where a wild-tame dichotomy for vector bundles and torsion free sheaves on reduced curves was proven. In this paper we call a plane cubic curve Weierstraß curve. If k is algebraically closed and char(k) = 2, 3 then it can be written in the form zy 2 = 4x3 − g2 xz 2 − g3 z 3 , where g2 , g3 ∈ k. It is singular if and only if g23 = 27g32 and unless g2 = g3 = 0 the singularity is a node, whereas in the case g2 = g3 = 0 the singularity is a cusp.


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1. The group of exact auto-equivalences of the derived category Db (Coh(E)) transforms stable sheaves into stable ones and semi-stable sheaves into semi-stable ones. 2. For any rational number ν the abelian category Cohν (E) of semi-stable coherent sheaves of slope ν is equivalent to the category Coh∞ (E) of coherent torsion sheaves and this equivalence is induced by an exact autoequivalence of Db (Coh(E)). 3. For any coherent sheaf F on E such that End(F ) = k there exists a point x ∈ E and Φ ∈ Aut(Db (Coh(E))) such that F ∼ = Φ(k(x)). This theorem shows a fundamental difference between a nodal and a cuspidal Weierstraß curve. Namely, let E be a singular Weierstraß curve and s its singular point. Then the category of finite-dimensional modules over the complete local ring O E,s has different representation types in the nodal and cuspidal cases. For a nodal curve, the category of finite dimensional representations of kx, y/xy is tame due to a result of Gelfand and Ponomarev [GP68]. In the second case, the category of finite length modules over the ring kx, y/(y 2 − x3 ) is representation wild, see for example [Dro72]. The correspondence between sky-scraper sheaves and semi-stable vector bundles on irreducible Weierstraß curves was first discovered by Friedman, Morgan and Witten [FMW99] (see also [Teo00]) and afterwards widely used in the physical literature under the name “spectral cover construction”. Theorem 5 (see [FMW99]). Let E be an irreducible Weierstraß curve, p0 ∈ E a smooth point and E a semi-stable torsion free sheaf of degree zero. Then the sequence ev

0 −→ H 0 (E(p0 )) ⊗ O −→ E(p0 ) −→ coker(ev) −→ 0 is exact. Moreover, the functor Φ : E → coker(ev) establish an equivalence between the category Coh0 (E) of semi-stable torsion-free sheaves of degree zero and the category of coherent torsion sheaves Coh∞ (E). This correspondence between torsion sheaves and semi-stable coherent sheaves can be generalized to a relative setting of an elliptic fibration E −→ T. In [FMW99] it was used to construct vector bundles on E which are semi-stable of degree zero on each fiber, see also [BK2]. As was shown in [BK1], the functor Φ is the trace of a certain exact auto-equivalence of the derived category Db (Coh(E)). Using this equivalence of categories and a concrete description of kx, y/xy–modules in terms of their projective resolutions, one can get a description of semi-stable torsion free sheaves of degree zero on a nodal Weierstraß curve in terms of étale coverings [FM, BK1]. In Section 3.4 we give a short overview of some related results without going into details. Acknowledgement. Work on this article has been supported by the DFGSchwerpunkt ”Globale Methoden in der komplexen Geometrie”. The second-

88


named author would like to thank Bernd Kreußler for numerous discussions of the results of this paper.

2 Vector Bundles on Smooth Projective Curves In this section we review some classical results about vector bundles on smooth curves. However, we provide non-classical proofs which, as we think, are simpler and fit well in our approach to coherent sheaves over singular curves. The behavior of the category of vector bundles on a smooth projective curve X is controlled by its genus g(X). If g(X) = 0 then X is a projective line P1 and any locally free sheaf on it splits into a direct sum of line bundles OP1 (n), n ∈ Z. This result, usually attributed to Grothendieck [Gro57], was already known in an equivalent form to Birkhoff [Bir13]. We found it quite instructive to include Birkhoff’s algorithmic proof in our survey. A classification of vector bundles in the case of smooth elliptic curves, i.e. for g(X) = 1 was obtained by Atiyah [Ati57]. He has shown that the category of vector bundles on an elliptic curve X is tame and an indecomposable vector bundle E is uniquely determined by its rank r, its degree d and a point of the curve x ∈ X. A modern, and in our opinion more conceptual way to prove Atiah’s result uses the language of derived categories and is due to Lenzing and Meltzer [LM93]. In the case of an algebraically closed field of characteristics zero an alternative description of indecomposable vector bundles via étale coverings was found by Oda [Oda71]. This classification was a cornerstone in the proof of Polishchuk and Zaslow [PZ98] of Kontsevich’s homological mirror symmetry conjecture for elliptic curves, see also [Kre01]. The case of curves of genus bigger than one is not considered in this survey. In this situation even the category of semi-stable vector bundles of slope one is representation wild and the main attention is drawn to the study of various moduli problems and properties of stable vector bundles, see for example [LeP97]. Throughout this section we do not require any assumptions about the base field k. 2.1 Vector Bundles on the Projective Line We are going to prove the following classical theorem. Theorem 6 (Birkhoff-Grothendieck). Any vector bundle E on the projective line P1 splits into a direct sum of line bundles: OP1 (n)rn . E∼ = n∈Z


89

A proof of this result based on Serre duality and vanishing theorems can be found in a book of Le Potier [LeP97]. However, it is quite interesting to give another, completely elementary proof, based on a lemma proven by Birkhoff in 1913. A projective line P1 is a union of two affine lines A1i (i = 0, 1). If (x0 : x1 ) are homogeneous coordinates in P1 then A1i = {(x0 : x1 )|xi = 0}. The affine coordinate on A10 is z = x1 /x0 and on A11 it is z −1 = x0 /x1 . Thus we can identify A10 with Spec(k[z]) and A11 with Spec(k[z −1 ]), their intersection is then Spec(k[z, z −1 ]). Certainly, any projective module over k[z] is free, i.e. all vector bundles over an affine line are trivial. Therefore to define a vector bundle over P1 one only has to prescribe its rank r and a gluing matrix M ∈ GL(r, k[z, z −1 ]). Changing bases in free modules over k[z] and k[z −1 ] corresponds to the transformations M → T −1 M S, where S and T are invertible matrices of the same size, over k[z] and k[z −1 ] respectively. Proposition 1 (Birkhoff [Bir13]). For any matrix M ∈ GL(r, k[z, z −1 ]) there are matrices S ∈ GL(r, k[z]) and T ∈ GL(r, k[z −1 ]) such that T −1 M S is a diagonal matrix diag(z d1 , . . . , z dr ). Proof. One can diagonalize the matrix M in three steps. Step 1. Reduce the matrix M = (aij ) to a lower triangular form with diagonal entries aii = z mi , where mi ∈ Z and m1 ≤ m2 ≤ · · · ≤ mr . Indeed, since k[z] is a discrete valuation ring, using invertible transformations of columns over k[z] we can reduce the first row (a11 , a12 , . . . , a1r ) of M to the form (a1 , 0, . . . , 0), where a1 is the greatest common divisor of a11 , a12 , . . . , a1r . Let M be the (r − 1) × (r − 1) matrix formed by the entries aij , i, j ≥ 2. Since det(M ) = a1 · det(M ) and det(M ) is a unit in k[z, z −1 ], it implies that a1 = z m1 and det(M ) is a unit in k[z, z −1 ] too. Then we proceed with the matrix M inductively. Note, that the diagonal entries can be always reordered to satisfy m1 ≤ m2 ≤ · · · ≤ mr . Step 2. Consider the case of a lower-triangular (2 × 2)-matrix zm 0 M= p(z, z −1 ) z n with m ≤ n. We show by induction on the difference n − m that M can be diagonalized performing invertible transformations of rows over k[z −1 ] and invertible transformations of columns over k[z]. If m = n then we can simply kill the entry p = p(z, z −1 ). Assume now that m < n and p = 0. Without loss of generality we may suppose that p ∈ z m+1 , . . . , z n−1 . Therefore there exist two mutually prime polynomials n−d az a and b in k[z] such that ap + bz n = z d and m < d < n. Then b p/z d belongs to GL(2, k[z]) and

90


zm 0 p zn

a z n−d b p/z d

=

z m a z n+m−d zd 0

.

In order to conclude the induction step it remains to note that |n + m − 2d| < |n − m|. Step 3. Let M be a lower-diagonal matrix with the diagonal elements r z m1 , . . . , z mr with m1 ≤ m2 ≤ · · · ≤ mr . We show by induction on |mi − mj | that M can be diagonalized. This statement is obvious for 0. Assume that

r

r

i,j=1

|mi −mj | =

i,j=1

|mi − mj | = N > 0. Introduce an ordering on the set

i,j=1

{(i, j)|1 ≤ j ≤ i ≤ r}: (2, 1) < (2, 3) < · · · < (r − 1, r) < (3, 1) < · · · < (r − 2, r) < · · · < (r, 1). Let (i0 , j0 ) be the smallest pair such that ai0 j0 = 0. Then we can apply the algorithm from the Step 2 to the (2 × 2) matrix formed by the entries r |mi − mj |. This (j0 , j0 ), (j0 , i0 ), (i0 , j0 ) and (i0 , i0 ) to diminish the sum i,j=1

completes the proof of Birkhoff’s lemma. Now it remains to note that 1 × 1 matrix (td ) defines the line bundle OP1 (−d). This implies the statement about the splitting of a vector bundle on a projective line into a direct sum of line bundles. 2.2 Projective Curves of Arithmetic Genus Bigger Then One Are Vector Bundle Wild In this subsection we are going to prove the following Theorem 7 (see [DG01]). Let X be an irreducible projective curve of arithmetic genus g(X) > 1 over an algebraically closed field k. Then the abelian category of semi-stable vector bundles of slope9 one is representation wild. In order to show the wildness of a category A one frequently uses the following lemma: Lemma 1. Let A be an abelian category, M, N ∈ Ob(A) with HomA (M, N ) = 0 and ξ, ξ ∈ Ext1A (N, M ) two extensions α

β

α

β

ξ : 0 −→ M −→ K −→ N −→ 0, ξ : 0 −→ M −→ K −→ N −→ 0. ∼ K if and only if there exist two isomorphisms f : M −→ M and Then K = g : N −→ N such that f ξ = ξ g. 9

The slope of a coherent sheaf F is µ(F ) =

deg(F) . rk(F)


91

Proof. The statement is clear in one direction: if f ξ = ξ g in Ext1A (N, M ), then K∼ = K by the 5-Lemma. Now suppose K ∼ = K and let h : K −→ K be an isomorphism. Then im(hα) is a subobject of im(α ). Indeed, otherwise the map β hα : M −→ N would be non-zero, a contradiction. Therefore we get the following commutative diagram: 0

/M f

α

/K h

β

/N

/0

g

/ 0. / M α / K β / N 0 −1 In the same way we proceed with h . Hence f is an isomorphism, what proves the lemma.

Let us come back to the proof of the theorem. Suppose now that X is an irreducible projective curve of arithmetic genus g > 1, O := OX . Then for any two points x = y from X we have Hom(O(x), O(y)) ∼ = H 0 (X, O(y − x)) = 0 and the Riemann-Roch theorem implies that Ext1 (O(x), O(y)) ∼ = H 1 (X, O(y− g−1 . Fix 5 different points x , . . . , x of the curve X, choose nonx)) ∼ k = 1 5 zero elements ξij ∈ Ext1 (O(xj ), O(xi )) for i = j and consider vector bundles F (A, B), where A, B ∈ M at(n × n, k) and F (A, B) is given as an extension 0 −→ (O(x1 ) ⊕ O(x2 ))n −→ F(A, B) −→ (O(x3 ) ⊕ O(x4 ) ⊕ O(x5 ))n −→ 0 B

A

corresponding to the element ξ(A, B) of Ext1 (A, B) presented by the matrix ξ13 I ξ14 I ξ15 I , ξ23 I ξ24 A ξ25 B where I denotes the unit n × n matrix. If (A , B ) is another pair of matrices, and F (A, B) → F (A , B ) any morphism, then the previous lemma implies that there are morphisms φ : A → A and ψ : B → B such that ψξ(A, B) = ξ(A , B )φ. Now one can easily deduce that Φ = diag(S, S, S) and Ψ = diag(S, S) for some matrix S ∈ M at(n × n, k) such that SA = A S and SB = B S. If we consider a pair of matrices (A, B) as a representation of the free algebra kx, y in 2 generators, the correspondence (A, B) → F (A, B) becomes a full, faithful and exact functor kx, y − mod −→ VB(X). In particular, it maps non-isomorphic modules to non-isomorphic vector bundles and indecomposable modules to indecomposable vector bundles. Using the terminology of representation theory of algebras, we say in this situation that the curve X is vector bundle wild. For a precise definition of wildness we refer to [DG01]. Recall that the algebra kx, y here can be replaced by any finitely generated algebra Λ = ka1 , . . . , an . Indeed, any Λ–module M such that dimk (M ) = m is given by a set of matrices A1 , . . . , An of size m × m. One

92


gets a full, faithful and exact functor Λ − mod −→ kx, y − mod mapping the module M to the kx, y–module of dimension m · n defined by the pair of matrices ⎛ ⎞ ⎞ ⎛ A1 I . . . 0 λ1 I 0 . . . 0 ⎜ 0 A2 . . . 0 ⎟ ⎜ 0 λ2 I . . . 0 ⎟ ⎜ ⎟ ⎟ ⎜ Y = X=⎜ . ⎜ .. .. . . .. ⎟ , ⎟ . . .. . . . .. ⎠ ⎝ . . ⎝ .. . . ⎠ 0

0 . . . λn I

0 0 . . . An

where λ1 , . . . , λn are different elements of the field k. Thus a classification of vector bundles over X would imply a classifications of all representations of all finitely generated algebras, a goal that perhaps nobody considers as achievable (whence the name “wild”). 2.3 Vector Bundles on Elliptic Curves In this subsection we shall discuss a classification of indecomposable coherent sheaves over smooth elliptic curves. Modulo some facts about derived categories we give a self-contained proof of Atiyah’s classification of indecomposable vector bundles which is probably simpler than the original one. Definition 1. An elliptic curve E over a field k is a smooth projective curve of genus one having a k–rational point p0 . The category Coh(E) of coherent sheaves on an elliptic curve E has the following properties, sometimes called “the dimension one Calabi-Yau property”: • It is abelian, k–linear, Hom-finite, noetherian and of global dimension one. • Serre Duality: for any two coherent sheaves F and G on E there is an isomorphism Hom(F , G) ∼ = Ext1 (G, F )∗ , functorial in both arguments. It is interesting to note that these properties almost characterize the category of coherent sheaves on an elliptic curve: Theorem 8 (Reiten – van den Bergh [RV02]). Let k be an algebraically closed field and A an indecomposable abelian Calabi-Yau category of dimension one. Then A is equivalent either to the category of finite-dimensional kt– modules or to the category of coherent sheaves on an elliptic curve E. This theorem characterizes Calabi-Yau abelian categories of global dimension one. We shall need one more formula to proceed with a classification of indecomposable coherent sheaves.


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Theorem 9 (Riemann–Roch formula). For any two coherent sheaves F and G on an elliptic curve E there is an integral bilinear Euler form deg(G) deg(F ) 1 . F, G := dimk Hom(F , G) − dimk Ext (F , G) = rk(G) rk(F ) In particular, , is anti-symmetric: F, G = −G, F . Now we are ready to start with the classification of indecomposable coherent sheaves. Theorem 10 (Atiyah). Let E be an elliptic curve over a field k. Then 1. Any indecomposable coherent sheaf F on E is semi-stable. 2. If F is semi-stable and indecomposable then all its Jordan-H¨ older factors are isomorphic. 3. A coherent sheaf F is stable if and only if End(F ) = K, where k ⊂ K is some finite field extension. Proof. It is well-known that any coherent sheaf F ∈ Coh(E) has a HarderNarasimhan filtration 0 ⊂ Fn ⊂ . . . ⊂ F1 ⊂ F0 = F whose factors Aν := Fν /Fν+1 are semi-stable with decreasing slopes µ(An ) > µ(An−1 ) > . . . > µ(A0 ). Using the definition of semi-stability, this implies Hom(Aν+i , Aν ) = 0 for all ν ≥ 0 and i > 0. Therefore, Ext1 (A0 , F1 ) ∼ = Hom(F1 , A0 )∗ = 0, and the exact sequence 0 → F1 → F → A0 → 0 must split. In particular, if F is indecomposable, we have F1 = 0 and F ∼ = A0 and F is semi-stable. The full sub-category of Coh(E) whose objects are the semi-stable sheaves of a fixed slope is an abelian category in which any object has a Jordan-H¨ older filtration with stable factors. If F and G are non-isomorphic stable sheaves which have the same slope then Ext1 (F , G) = 0. Based on this fact we deduce that an indecomposable semi-stable sheaf has all its Jordan-Hölder factors isomorphic to each other. It is well-known that any non-zero automorphism of a stable coherent sheaf F is invertible, i.e. End(F ) is a field K. Since E is projective, the field extension k ⊂ K is finite. On a smooth elliptic curve, the converse is true as well, which equips us with a useful homological characterization of stability. To see this, suppose that all endomorphism of F are invertible but F is not stable. This implies the existence of an epimorphism F → G with G stable and µ(F ) ≥ µ(G). Serre duality implies dimk Ext1 (G, F ) = dimk Hom(F , G) > 0, hence, G, F = dimk Hom(G, F ) −dimk Ext1 (G,F ) < dimk Hom(G, F ). By Riemann-Roch formula G, F = µ(F ) − µ(G) rk(F ) rk(G) > 0, thus Hom(G, F ) = 0. But this produces a non-zero composition F → G → F which is not an isomorphism, in contradiction to the assumption that End(F ) is a field.

94


Remark 1. Usually one speaks about stability of vector bundles on projective varieties in the case of an algebraically closed field of characteristics zero. However, due to a result of Rudakov [Rud97] one can introduce a stability notion for fairly general abelian categories. The following classical fact was, probably first, proven by Dold [Dol60]: Proposition 2. Let A be an abelian category of global dimension one and F an object of the derived category Db (A). Then there is an isomorphism H i (F )[−i], i.e. any object of Db (A) splits into a direct sum of its F ∼ = i∈Z

homologies. This proposition in particular means that the derived category Db (A) of a hereditary abelian category A and the abelian category A itself have the same representation type. However, it turns out that the derived category has a richer structure and more symmetry then the corresponding abelian category. First of all note that the group Aut(Db (Coh(E))) acts on the K–group K(E) of Coh(E) preserving the Euler form , . Hence, it leaves invariant the radical of the Euler form rad , = {F ∈ K(E)|F, = 0} and induces an acZ tion on K(E)/rad , . Since by Riemann-Roch theorem Z : K(E)/rad , −→ Z2 is an isomorphism, where Z(F ) := (rk(F ), deg(F )) ∈ Z2 , we get a group homomorphism Aut(Db (Coh(E))) −→ SL(2, Z). We call the pair Z(F ) ∈ Z2 the charge of F . Theorem 11 (Mukai, [Muk81]). Let E be an elliptic curve. Then the group homomorphism Aut(Db (Coh(E))) −→ SL(2, Z) is surjective. Proof. By the definition of an elliptic curve there is a k–rational point p0 on E inducing an exact equivalence O(p0 )⊗−. Let P = OE×E ∆−(p0 ×E)−(E×p0 ) then the Fourier-Mukai transform ΦP : Db (Coh(E)) → Db (Coh(E)),

ΦP (F ) = Rπ2∗ (P ⊗ π1∗ F )

is an exact auto-equivalence of the derived category, see [Muk81]. The actions of O(p0 ) ⊗ − and ΦP on K(E)/rad , in the basis {[O], [k(p0 )]} are given by 0 1 which are known to generate SL(2, Z). This the matrices ( 11 01 ) and −1 0 shows the claim. The technique of derived categories makes it easy to give a classification of indecomposable coherent sheaves on an elliptic curve. Theorem 12. Let F be an indecomposable coherent sheaf on an elliptic curve E. Then there exits a torsion sheaf T and an exact auto-equivalence Φ ∈ Aut(Db (Coh(E))) such that F ∼ = Φ(T ). Proof. Let F be an indecomposable coherent sheaf on E with the charge Z(F ) = (r, d), r > 0. Let h = g.c.d.(r, d) be the greatest common divisor, then


95

there exists a matrix F ∈ SL(2, Z) such that F ( dr ) = ( h0 ). We can lift the matrix F to an auto-equivalence Φ ∈ Aut(Db (Coh(E)), then Z(Φ(F )) = ( h0 ). Since Aut(Db (Coh(E))) maps indecomposable objects of the derived category to indecomposable ones and since the only indecomposable objects in the derived category are shifts of indecomposable coherent sheaves, we can conclude that Φ(F ) is isomorphic to a shift of some indecomposable sheaf of rank zero, what proves the theorem. Let ME (r, d) denote the set of indecomposable vector bundles on E of rank r and degree d. Theorem 13 (Atiyah). Let E be an elliptic curve. Then for any integer h > 0 there exists a unique indecomposable vector bundle Fh ∈ ME (h, 0) such that H 0 (Fh ) = 0. The vector bundles Fh are called unipotent. Moreover, the following properties hold: 1. H 0 (Fh ) = H 1 (Fh ) = k for all h ≥ 1. 2. If char(k) = 0 then Fh ∼ = Symh−1 (F2 ). Moreover F e ⊗ Ff ∼ =

f −1

Fe+f −2i−1

i=0

Sketch of the proof. Since Fh is indecomposable of degree zero, it has a unique Jordan-H¨ older factor L ∈ Pic0 (E). From the assumption Hom(O, Fh ) = 0 we conclude that L ∼ = O, so each bundle Fh can be obtained by recursive self-extensions of the structure sheaf. Since by Theorem 12 the category of semi-stable vector bundles of degree zero is equivalent to the category of torsion sheaves, we conclude that the category of semi-stable sheaves with the Jordan-H¨ older factor O is equivalent to the category of finite-dimensional kt-modules. The exact sequence 0 −→ Fh−1 −→ Fh −→ O −→ 0. corresponds via Fourier-Mukai transform ΦP to 0 −→ kt/th−1 −→ kt/th −→ k −→ 0. In the same way we conclude that Hom(O, Fh ) = Homkt (k, kt/th ) = k. Moreover, one can show that ΦP (kt/te ⊗k kt/tf ) ∼ = Fe ⊗ Ff , hence we have the same rules for the decomposition of the tensor product of unipotent vector bundles and of nilpotent Jordan cells, see [Ati57,Oda71,Muk81,PH05]. Remark 2. Atiyah’s original proof from 1957 was written at the time when the formalism of derived and triangulated categories was not developed yet. However, his construction of a bijection between ME (r, d) and ME (h, 0) corresponds exactly to the action of the group of exact auto-equivalences of the derived category of coherent sheaves on the set of indecomposable objects. This was probably for the first time observed by Lenzing and Meltzer in [LM93]. For further elaborations, see [Pol03, PH05, BK3].

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Actually, Atiyah’s description of indecomposable vector bundles on an elliptic curve E is more precise. Theorem 14 (Atiyah). Let E be an elliptic curve over an algebraically closed field k. For any pair of coprime integers (r, d) with r > 0 pick up some E(r, d) ∈ ME (r, d). Then 1. 2. 3. 4.

ME (r, d) = {E(r, d) ⊗ L| L ∈ Pic0 (E)}. E(r, d) ⊗ L ∼ = O. = E(r, d) if and only if Lr ∼ The map det : ME (r, d) −→ ME (1, d) is a bijection. If char(k) = 0, then Fh ⊗ − : ME (r, d) −→ ME (rh, dh) is a bijection.

Remark 3. If k is not algebraically closed and E is a smooth projective curve of genus one over k, without k–rational points, then we miss the generator O(p0 ) in the group of exact auto-equivalences Db (Coh(X)) and the method used for elliptic curves can not be immediately applied. This problem was solved in a paper of Pumpl¨ un [Pum04]. We may sum up the discussed properties of indecomposable coherent sheaves on elliptic curves: Proposition 3. Let E be an elliptic curve over a field k. Then 1. Any indecomposable coherent sheaf F on E is semi-stable with a unique stable Jordan-H¨ older factor. 2. An indecomposable vector bundle is determined by its charge (r, d) ∈ Z2 and a closed point x of the curve E. 3. Let Cohν (E) be the category of semi-stable sheaves of slope ν. Then for any µ, ν ∈ Q ∪ {∞} the abelian categories Cohν (E) and Cohµ (E) are equivalent and this equivalence is induced by an auto-equivalence of Db (Coh(E)). 4. In particular, each category Cohµ (E) is equivalent to the category of coherent torsion sheaves. 5. If F ∈ Cohν (E), G ∈ Cohµ (E) and ν < µ then Ext1 (F , G) = 0 and dimk Hom(F , G) = deg(G)rk(F ) − deg(F )rk(G). The case ν > µ is dual by Serre duality. This gives a pretty complete description of the category of coherent sheaves on an elliptic curve and of its derived category. However, in applications one needs another description of indecomposable vector bundles, see [Pol02,PZ98]. The following form of Atiyah’s classification is due to Oda [Oda71]. It was used by Polishchuk and Zaslow in their proof of the homological mirror symmetry conjecture for elliptic curves, see [PZ98, Kre01]. Theorem 15 (Oda). Let k = C and E = Eτ = C/1, τ be an elliptic curve, E ∈ ME (rh, dh) an indecomposable vector bundle, where g.c.d.(r, d) = 1. Then there exists a unique line bundle L of degree d on Erτ such that


97

E∼ = π∗ (L) ⊗ Fh ∼ = π∗ (L ⊗ Fh ), where π : Erτ −→ Eτ is an étale covering of degree r. Proof. Let L be a line bundle on Erτ of degree d. Since the morphism π is étale, π∗ (L) is a vector bundle on Eτ of rank r. The Todd class of an elliptic curve is trivial, hence by Grothendieck-Riemann-Roch theorem we obtain deg(π∗ (L)) = deg(L) = d. Now let us show that π∗ (L) is indecomposable. To do this it suffices to prove that End(π∗ L) = C. Consider the fiber product diagram E ⏐ ⏐ p2

p1

−−−−→ Erτ ⏐ ⏐π1 π

Erτ −−−2−→ Eτ is a union of r copies of the elliptic curve Erτ : One can easily check that E r ! (i) = Eirτ , where each p1 : Eirτ −→ Erτ , i = 1, . . . , r can be chosen to be E i=1 (i)

the identity map and p2 (z) = z + ri τ . Since all morphisms πi , pi , i = 1, 2 are affine and flat, the functors πi∗ , πi∗ , pi∗ , p∗i are exact. Moreover, p!i = p∗i , since the canonical sheaf of an elliptic curve is trivial.10 Using the base change isomorphism and Grothendieck duality we have HomEτ π1∗ L, π2∗ L ∼ = HomErτ π2∗ π1∗ L, L ∼ = HomE p2∗ p∗ L, L ∼ = Hom p∗ L, p∗ L . 1

rτ

E

1

2

(i)∗ (i)∗ It remains to note that HomE p1 L, p2 L = 0 for i = 0. If E is an indecomposable vector bundle on Eτ of rank rh and degree dh, then by Theorem 14 there exists M ∈ Pic0 (Eτ ) such that E ∼ = π∗ (L)⊗M⊗Fh . By the projection formula E ∼ = π∗ (L ⊗ π ∗ M) ⊗ Fh . Moreover, passing to an étale covering kills the ambiguity in the choice of M. It remains to show that π ∗ (Fh ) ∼ = Fh . To do this it suffices see that ∗ π (F2 ) ∼ = F2 , since Fh ∼ = Symh−1 (F2 ) and the inverse image commutes with all tensor operations. The only property we have to check is that π ∗ (F2 ) does not split. It is equivalent to say that the map π ∗ : H 1 (OEτ ) −→ H 1 (OErτ ) is non-zero. This follows from the commutativity of the diagram:

10

Recall that if f : X → Y is a finite morphism of Gorenstein projective schemes then HomX (F , f ! G) ∼ = HomE (f∗ F , G) for any coherent sheaf F on X and a coherent sheaf G on Y.

98


ZO 2 ( 10 0r ) Z2

H1 (Eτ , Z) O

∼ =

/ H 1 (Eτ , Z)

∼ =

/ H 1 (Erτ , Z)

π∗

π∗

H1 (Erτ , Z)

/ H 1 (Eτ , 0)

π∗

/ H 1 (Erτ , 0).

3 Vector Bundles and Torsion Free Sheaves on Singular Curves of Arithmetic Genus One In this paper we discuss two approaches for the study of the category of coherent sheaves on a singular projective curve of arithmetic genus one. The first uses the technique of derived categories and Fourier-Mukai transforms . Its key point is that any semi-stable torsion free sheaf on an irreducible Weierstraß curve can be obtained from a torsion sheaf by applying an autoequivalence of the derived category. This technique can be generalized to the case of elliptic fibrations: we can transform a family of torsion sheaves to a family of sheaves, which are semi-stable on each fiber. However, the approach via Fourier-Mukai transforms allows to describe only semi-stable sheaves. In order to get a description of all indecomposable torsion free sheaves, another technique turns out to be useful. Namely, we relate vector bundles on a singular rational curve X and on its normalizap can map −→ X. The inverse image functor p∗ : VB(X) −→ VB(X) tion X non-isomorphic bundles into isomorphic ones. The full information about the fibers of this map is encoded in a certain matrix problem. In the case of an algebraically closed field this approach leads to a very concrete description of indecomposable vector bundles on cycles of projective lines via étale coverings (no assumption on char(k) is needed). Combining both methods, we get a quite complete description of the category of torsion free sheaves on a nodal Weierstraß curve. 3.1 Vector Bundles on Singular Curves via Matrix Problems Let X be a reduced projective curve over a field k. Introduce the following notation: −→ X the normalization of X; • p:X = p∗ O ; • O = OX and O X • J = AnnO (O/O) the conductor of O in O; . • A = O/J and A = O/J Note that A and A are skyscraper sheaves supported at the singular locus of X. Since the morphism p is affine, p∗ identifies the category of coherent and the category Coh of coherent modules on the ringed sheaves Coh(X) O space (X, O). Let S be the subscheme of X defined by the conductor J , S its


99

and I = I its ideal sheaf on X. Then p∗ scheme-theoretic pull-back on X S also induces an equivalence between the category of OX /I–modules and the category of A–modules. A, A} on the topological space X, denote For a sheaf of algebras Λ ∈ {O, O, by TFΛ the category of torsion free coherent Λ–modules and by VBΛ its full subcategory of locally free sheaves. The usual way to deal with vector bundles on a singular curve is to lift them to the normalization, and then work on a smooth curve, see for example [Ses82]. Passing to the normalization we loose information about the isomorphism classes of objects of VBO since nonisomorphic vector bundles can have isomorphic inverse images. In order to describe the fibers of the map VBO −→ VBO and to be able to deal with arbitrary torsion free sheaves we introduce the following definition: Definition 2. The category of triples TX is defined as follows: 1. Its objects are triples (F, M, ˜i), where F is a locally free O–module, M is a coherent A–module and ˜i : M ⊗O A −→ F ⊗O A is an epimorphism of A–modules, which induces a monomorphism of A–modules i : M −→ ˜i M ⊗O A −→ F ⊗ A. O

(F,f ) 2. A morphism (F1 , M1 , ˜i1 ) −→ (F2 , M1 , ˜i2 ) is given by a pair (F, f ), where f F and M1 −→ M2 is a morphism F1 −→ F2 is a morphism of O-modules of A-modules, such that the following diagram

M1 ⊗O A

˜i1

f¯

/ F1 ⊗ A O F¯

˜i2 / F ⊗ M2 ⊗O A 2 O A is commutative in CohA , where F¯ = F ⊗ id and f¯ = ϕ ⊗ id. The main reason to introduce the formalism of triples is the following theorem: Ψ

Theorem 16. The functor TFO −→ TX mapping a torsion free sheaf F to the M = F ⊗O A and ˜i : F ⊗O ⊗O O), triple (F, M, ˜i), where F = F ⊗O O/tor(F A −→ F ⊗O A, is an equivalence of categories. Moreover, the category of vector bundles VBO is equivalent to the full subcategory of TX consisting of those triples (F, M, ˜i), for which M is a free A–module and ˜i is an isomorphism. Ψ

Sketch of the proof. We construct the quasi-inverse functor TX −→ TFX as follows. Let (F, M, ˜i) be some triple. Consider the pull-back diagram / J F /M /0 /F 0

0

id

/ J F

i

/ F

/ F ⊗ A O

/0

100


in the category of O–modules. Since the pull-back is functorial, we get a Ψ functor TX −→ Coh(X). Since the map i is injective, F −→ F is injective as well, so F is torsion free. It remains to show that the functors Ψ and Ψ are quasi-inverse to each other. We refer to [DG01] for the details of the proof. Remark 4. There is a geometric way to interpret the above construction of the p −→ X its normalization, s : category of triples. Let X be a singular curve, X S −→ X the inclusion of the closed subscheme defined by the conductor ideal its pull-back on the normalization. Consider the Cartesian and s˜ : S −→ X diagram s˜ / X S p˜

p

s / S X. Theorem 16 says that a torsion free sheaf F on a singular curve X can be ∗ reconstructed from its “normalization” p∗ (F )/tor(p s∗ F on F ), its pull-back ∗ ∗ ∗ ∗ ∗ ∗ ∗ S and the “gluing map” p˜ s F −→ s˜ p F −→ s˜ p F /tor(p F ) . Now let us see how this construction can be used to classify torsion free sheaves on degenerations of elliptic curves. Let char(k) = 2 and E be a nodal Weierstraß curve, given by the equation zy 2 − x3 − zx2 = 0, s = (0 : 0 : 1) its p −→ singular point, P1 = E E the normalization map. Choose coordinates on P1 in such a way that the preimages of s are 0 = (0 : 1) and ∞ = (1 : 0). The previous theorem says that a torsion free sheaf F on the curve E is uniquely determined by the corresponding triple Ψ (F ) = (F, M, ˜i). Here F is a locally free O–module, or as we have seen, a locally free OP1 -module. Using the notation O(n) = p∗ (OP1 (n)), due to the theorem of Birkhoff rn Grothendieck, F ∼ O(n) . = n∈Z

Since A = O/J = k(s) and A = O/J = (k × k)(s), the sheaf M can be identified with its stalk at s and the map ˜i : M⊗A A −→ F ⊗O A can be viewed as a pair (i(0), i(∞)) of linear maps of k–vector spaces. In order to write ˜i in terms of matrices we identify O(n) ⊗O A with p∗ (OP1 (n) ⊗OP1 OP1 /I). 1 The choice of coordinates on P fixes two canonical sections z0 and z1 of H 0 (OP1 (1)) and we use the trivializations OP1 (n) ⊗ I −→ k(0) × k(∞) given by ζ ⊗ 1 → (ζ/z1n (0), ζ/z0n (∞)). Note, that this isomorphism only depends on the choice of coordinates of P1 . In such a way we supply the A–module F ⊗ A = F(0) ⊕ F(∞) with a basis and get isomorphisms rn ∼ ∼ F (0) = k(0) and F (∞) = k(∞)rn . With respect to all choices the n∈Z

n∈Z

morphism ˜i is given by two matrices i(0) and i(∞), divided into horizontal blocks:

Vector Bundles on Degenerations of Elliptic Curves .. .

101

.. .

n−1

n−1

}rn

n

n

n+1

n+1

.. .

.. .

i(0)

}rn

i(∞)

From the definition of the category of triples it follows that the matrices i(0) and i(∞) have to be of full row rank and the transposed matrix (i(0)|i(∞))t has to be monomorphic. Vector bundles on E correspond to invertible square matrices i(0) and i(∞). rn O(n) and M = kN (s), two different pairs Of course, for a fixed F = n∈Z

of matrices (i(0)|i(∞)) and (i (0)|i (∞)) can define isomorphic torsion free sheaves on E. However, since the functor Ψ : TFO −→ TX preserves isomorphism classes of indecomposable objects, two triples (F, M, ˜i) and (F, M, i ) define isomorphic torsion free sheaves if and only if there are automorphisms ¯˜i = i f¯. F : F −→ F and f : M −→ M such thatrF n O(n) can be written in a matrix form: An endomorphism F of F = n∈Z

F = (Fkl ), where Fkl is a rl × rk –matrix with coefficients in the vector space ∼ Hom(O(k), O(l)) = k[z0 , z1 ]l−k . In particular, the matrix F is lower triangular and the diagonal rn × rn blocks Fnn are just matrices over k. The morphism F is an isomorphism if and only if all Fnn are invertible. Let r = rank(F). With respect to the chosen trivialization of OP1 (n) at 0 and ∞ the map F¯ : kr (0)⊕kr (∞) −→ kr (0)⊕kr (∞) is given by the pair of matrices (F (0), F (∞)) and we have the following transformation rules for the pair (i(0), i(∞)): i(0), i(∞) → F (0)−1 i(0)S, F (∞)−1 i(∞)S , where F is an automorphism of

rn O(n) and S an automorphism of kN . n∈Z

Note, that the matrices Fkl (0) and Fkl (∞), k, l ∈ Z, k > l can be arbitrary and Fnn (0) = Fnn (∞) can be arbitrary invertible for n ∈ Z. As a result we get the following matrix problem. Matrix problem for a nodal Weierstraß curve. We have two matrices i(0) and i(∞) of the same size and both of full row rank. Each of them is divided into horizontal blocks labeled by integers (they are called sometimes weights). Blocks of i(0) and i(∞), labeled by the same integer, have the same size. We are allowed to perform only the following transformations: 1. We can simultaneously do any elementary transformations of columns of i(0) and i(∞).

102


2. We can simultaneously do any invertible elementary transformations of rows inside of any two conjugated horizontal blocks. 3. We can in each of the matrices i(0) and i(∞) independently add a scalar multiple of any row with lower weight to any row with higher weight. The main idea is that we can transform the matrix i into a canonical form which is quite analogous to the Jordan normal form. These types of matrix problems are well-known in representation theory. First they appeared in a work of Nazarova and Roiter [NR69] about the classification of kx, y/(xy)–modules. They are called, sometimes, “Gelfand problems” or “representations of bunches of chains”. Example 1. Let E be a nodal Weierstraß curve. • The following triple (F, M, ˜i) defines an indecomposable vector bundle of ⊕ O(n), rank 2 on E: the normalization F = O n = 0, M = k2 (s) and matrices: i(0) =

1 0

0

and i(∞) =

0 1

0

λ ∈ k∗ .

0 1 n λ 0 n • The triple O(−1), k2 , ˜i = 1 0 0 1 describes the unique torsion free sheaf that is not locally free of degree zero, which compactifies the Jacobian Pic0 (E). A Gelfand matrix problem is determined by a certain partially ordered set together with an equivalence relation on it. Such a poset with an equivalence relation is called a bunch of chains. Before giving a general definition, we give an example describing the matrix problem which corresponds to a nodal Weierstraß curve. There are two infinite sets E0 = {E0 (k)|k ∈ Z} and E∞ = {E∞ (k)|k ∈ Z} with the ordering · · · < E∗ (−1) < E∗ (0) < E∗ (1) < . . . , ∗ ∈ {0, ∞} and two one-point sets {F0 } and {F∞ }. On the union E F = (E0 ∪ E∞ ) (F0 ∪ F∞ ) we introduce an equivalence relation: E0 (k) ∼ E∞ (k), where k ∈ Z and F0 ∼ F∞ . F0 F∞ ppppp ! !

E0 (k) E0 (k+1)

◦ ◦ ◦

ppppp ppppp ppppp

◦ E∞ (k) ◦ E∞ (k+1) ◦


103

This picture contains complete information about the corresponding matrix problem. The circles denote the elements of E, the diamonds denote elements F0 and F∞ , dotted lines connect equivalent elements and vertical arrows describe the partial order in E0 and E∞ . The sets E0 ∪F0 and E∞ ∪F∞ correspond to matrices i(0) and i(∞) respectively, elements E0 (k) and E∞ (k) label their horizontal stripes, k ∈ Z. We also say that a row from the horizontal block E∗ (k) has weight k, where ∗ is either 0 or ∞. The equivalence relation E0 (k) ∼ E∞ (k) means that horizontal blocks of weight k have the same number of rows and F0 ∼ F∞ tells that i(0) and i(∞) have the same number of columns. Moreover, elementary transformations inside of two conjugated blocks have to be done simultaneously. The total order on E0 and E∞ means that we can add any scalar multiple of any row of smaller weight to any row with a bigger weight. Such transformations can be performed in the matrices E0 and E∞ independently. Definition 3. Let k be an arbitrary field. A cycle of n projective lines is a projective curve En over k with n irreducible components, each of them is isomorphic to P1 . We additionally assume that all components intersect n and the completion of the transversally with intersection matrix of type A local rings of any singular point of En is isomorphic to ku, v/uv. In a similar way, a chain of k projective lines Ik is a configuration of projective k lines with intersection matrix of type Ak−1 . Remark 5. Let k = R and E be the cubic curve zy 2 = x3 − zx2 . Then s = (0 : 2 2 0 : 1) is the singular point of E and O E,s = Ru, v/(u + v ). Then E is not a cycle of projective lines in the sense of Definition 3 and the combinatorics of the indecomposable vector bundles on E will be considered elsewhere. its normalization. Let E be either a chain or a cycle of projective lines and E The matrix problem we get is given by the following partially ordered set. Consider the set of pairs {(L, a)}, where L is an irreducible component of E and a ∈ L a preimage of a singular point. To each such pair we attach a totally ordered set E(L,a) = {E(L,a) (i)|i ∈ Z}, where · · · < E(L,a) (−1) < E(L,a) (0) < E(L,a) (1) < . . . and a one-point set F(L,a) . On the union E

F=

(E(L,a) ∪ F(L,a) ),

(L,a)

we introduce an equivalence relation: 1. F(L ,a ) ∼ F(L ,a ) , where a and a are preimages of the same singular point a ∈ E. 2. E(L,a ) (k) ∼ F(L,a ) (k) for k ∈ Z and a , a ∈ L. Such a partially ordered set with an equivalence relation is called a bunch of chains [Bon92]. A representation of such a bunch of chains is given by a set

104


of matrices M (L, a), for each element (L, a). Every matrix M (L, a) is divided into horizontal blocks labelled by the elements of E(L,a) . Of course, all but finitely many labels corresponds to empty blocks. The principle of conjugation of blocks is the same as for a rational curve with one node. The category of representations of a bunch of chains is additive and has two types of indecomposable representations: bands and strings [Bon92]. Hence, the technique of representations of bunches of chains allows to describe indecomposable torsion free sheaves on chains and cycles of projective lines. Let E = En be a cycle of n projective lines, {a1 , a2 , . . . , an } the set of n p = ! Li , where each −→ E the normalization of E, E singular points of E, E i=1

Li is isomorphic to a projective line and {ai , ai } = p−1 (ai ). Assume that ai , ai+1 ∈ Li , where an+1 = a1 . Fix coordinates on each projective line Li in such a way, that ai = (0 : 1) and ai+1 = (1 : 0). Definition 4. A band B(d, m, p(t)) is an indecomposable vector bundle of rank rmk. It is determined by the following parameters: 1. d = (d1 , d2 , . . . , dn , dn+1 , dn+2 , . . . , d2n , . . . , drn−n+1 , drn−n+2 , . . . , drn ) ∈ This sequence Zrn is a sequence of degrees on the normalized curve E. . should be non-periodic, i.e. not of the form es = ee . . e, where e = s times

e1 , e2 , . . . , eqn is another sequence and q = rs . 2. p(t) = tk + a1 tk−1 + · · · + ak ∈ k[t] is an irreducible polynomial of degree k, p(t) = t. 3. m ∈ Z+ is a positive integer. In particular, one can recover from the sequence d the pull-back of B(d, m, p(t)) it is on the l-th irreducible component of E: p∗l (B(d, m, p(t))) ∼ =

r

OLl (dl+in )mk .

i=1

A string S(d, f ) is a torsion free sheaf which depends only on two discrete parameters f ∈ {1, 2, . . . , n} and d = (d1 , d2 , . . . , dt ), t > 1. Now we are going to explain the way of construction of gluing matrices of triples corresponding to bands B(d, m, p(t)) and strings S(e, f ). Algorithm 1. Bands. Let d = (d1 , d2 , . . . , drn ) ∈ Zrn be a non-periodic sequence, m ∈ Z+ and p(t) ∈ k[t] an irreducible polynomial of degree k. We have 2n matrices M (Li , ai ) and M (Li , ai+1 ), i = 1, . . . , n occurring in the triple, corresponding to B(d, m, p(t)). Each of them has size mrk × mrk. Divide these matrices into mk × mk square blocks. Consider the sequences d(i) = di di+n . . . di+(r−1)n and label the horizontal strips of M (Li , ai ) and


105

M (Li , ai+1 ) by integers occurring in each d(i). If some integer d appears l times in d(i) then the horizontal strip corresponding to the label d consists of l substrips having mk rows each. Recall now an algorithm of writing the components of the matrix ˜i in a normal form [Bon92]: 1

1

1

1

1. Start with the sequence (L1 , a1 ) −→ (L1 , a2 ) −→ (L2 , a2 ) −→ (L2 , a3 ) −→ 1 2 2 r r 1 · · · −→ (Ln , a1 ) −→ (L1 , a1 ) −→ · · · −→ (Ln , an ) −→ (Ln , a1 ) −→ . It is convenient to imagine this sequence as a cyclic word broken at the place (L1 , a1 ). 2. Unroll the sequence d. This means that we write over each (Li , a) the corresponding term of the subsequence d(i) together with the number of its previous occurrences in d(i) including the current one: (L1 , a1 )(d1 ,1) −→ (L2 , a2 )(d2 ,1) −→ (L2 , a3 )(d2 ,1) −→ 1

1

1

· · · −→ (Ln , a1 )drn ,∗ −→ (Ln , a1 )(drn ,1) −→ . 1

r

r

3. Now we can fill the entries of the matrices M (L, a). Consider each arrow l

(L, a)(d,i) −→ . Then insert the matrix Imk in the block ((d, i), l) of the matrix M (L, a), which is defined as the intersection of the i-th substrip of the horizontal strip labeled by d and the l-th vertical strip. 4. Put at the ((drn , 1), r)-th place of M (Ln , a1 ) the Frobenius block Jm (p(t)). Strings. Let e = (e1 , e2 , . . . , es ) ∈ Zs and f ∈ {1, 2, . . . , n}. The algorithm to write the matrices for the torsion free sheaf S(d, f ) is essentially the same as for bands. The parameter f denotes the number of the component Lf of which the sequence (Lf , a ) −→ (Lf , a ) −→ . . . −→ (Lf +s , a ) −→ E f f +1 f +s (Lf +s , af +s+1 ) starts with. Then we unroll e using the same algorithm as for bands. The only difference is that we insert instead of Imk the unit (1 × 1)– matrix. Note, that some of the matrices M (Li , ai ) and M (Li , ai ) can be nonsquare, but they are automatically of full row rank. Example 2. Let E = E2 be a cycle of two projective lines, d = (0, 1, 1, 3, 1, −2) and p(t) ∈ kt an irreducible polynomial of degree k. Then B(d, m, p(t)) is a vector bundle of degree 3m with the normalization mk OL1 ⊕ OL1 (1)2mk ⊕ OL2 (−2)mk ⊕ OL2 (1)mk ⊕ OL2 (3)mk and gluing matrices

Imk

M (L1 , a1 ) =

0 Imk Imk

1 Imk

= M (L1 , a2 )

Imk Imk

106

Lesya Bodnarchuk, Igor Burban, Yuriy Drozd, and Gert-Martin Greuel −2

Jm

M (L2 , a1 ) =

Imk

= M (L2 , a2 ),

1 Imk

Imk Imk

3

Imk

where Jm is the Frobenius block corresponding to the k[t]-module k[t]/p(t)m . The corresponding unrolled sequence looks as follows: 1 1 1 2 (L1 , a1 )(0,1) −→ (L1 , a2 )(0,1) −→ (L2 , a2 )(1,1) −→ (L2 , a1 )(1,1) −→ 2 2 2 3 (L1 , a1 )(1,1) −→ (L1 , a2 )(1,1) −→ (L2 , a2 )(3,1) −→ (L2 , a1 )(3,1) −→ 3 3 3 1 (L1 , a1 )(1,2) −→ (L1 , a2 )(1,2) −→ (L2 , a2 )(−2,1) −→ (L2 , a1 )(−2,1) −→ . Let f = 2 and d = (−1, 0, 1, −1, 1). Then the corresponding torsion free sheaf S(d, f ) has normalization F = OL1 (−1) ⊕ OL1 ⊕ OL2 (−1) ⊕ OL2 (1)2 and gluing matrices

M (L1 , a1 ) =

M (L2 , a1 )

=

1

1

−1

1

0

1

−1

1

1

1

1

= M (L1 , a2 )

1 = M (L2 , a2 )

1 1

The corresponding unrolled sequence is 1 1 1 1 (L2 , a2 )(−1,1) −→ (L2 , a1 )(−1,1) −→ (L1 , a1 )(0,1) −→ (L1 , a2 )(0,1) −→ 2 2 2 2 (L2 , a2 )(1,1) −→ (L2 , a1 )(1,1) −→ (L1 , a1 )(−1,1) −→ (L1 , a2 )(−1,1) −→ 3 3 (L2 , a2 )(1,2) −→ (L2 , a1 )(1,2) −→ . Summing everything up, we get the following theorem. Theorem 17 ( [DG01]). Let E = En be a cycle of n projective lines over a filed k. Then • any indecomposable vector bundle on E is isomorphic to some B(d, m, p(t)), where d = (d1 , d2 , . . . , drn ) ∈ Zrn is a non-periodic sequence, m ∈ Z+ and p(t) = tk + a1 tk−1 + · · · + ak ∈ k[t] is an irreducible polynomial, p(t) = t. • Any torsion free but not locally free coherent sheaf is isomorphic to some S(d, t), where t ∈ {1, 2, . . . , n} and d ∈ Zrn . The only isomorphisms between indecomposable vector bundles are generated by


• B(d, m, p(t)) ∼ = B(d◦ , m, q(t)), d◦ = drn , drn−1 , . . . , d1 and q(t) = • B(d, m, p(t)) ∼ = B(d , m, p(t)), with

107

k

t ak p(1/t)

d = dn+1 , dn+2 , . . . , d2n , d2n+1 , . . . , d1 , d2 , . . . , dn . ∼ S(e◦ , f ◦ ), where e◦ is The only isomorphisms between strings are S(e, f ) = ◦ the opposite sequence e = es , es−1 , . . . , e1 . If s = nk + s with 0 ≤ s < n, then f ◦ = s + f taken modulo n. Remark 6. If E = E1 is a Weierstraß nodal curve, then there is no choice for the parameter f in the definition of a string and one simply uses the notation S(d). If the field k is algebraically closed, we write B(d, m, λ) instead of B(d, m, t − λ). As a direct corollary of the combinatorics of strings and bands we obtain the following theorem Theorem 18 ( [DG01]). Let X = In be a chain of n projective lines, then any vector bundle E on X splits into a direct sum of lines bundles. Moreover, Pic(In ) ∼ = Zn and a line bundle is determined by its restrictions on each irreducible component. A description of torsion free sheaves is similar: any indecomposable torsion free sheaf is isomorphic to the direct image of a line bundle on a subchain of projective lines. 3.2 Properties of Torsion Free Sheaves on Cycles of Projective Lines Throughout this subsection, let k be an algebraically closed field and E = En a cycle of n projective lines over k. As we have seen in the previous subsection, indecomposable vector bundles on E are bands B(d, m, λ) and indecomposable torsion free but not locally sheaves are strings S(d, f ). They were described in terms of a certain problem of linear algebra. However, in the case of an algebraically closed field there is a geometric way to present the classification of indecomposable torsion free sheaves on E without appealing to the formalism of bunches of chains. This description, the proof of which we give here for the first time, is completely parallel to Oda’s one for vector bundles on elliptic curves [Oda71]. We start with a lemma describing unipotent vector bundles on E. Lemma 2. For any integer m ≥ 1 there exists a unique indecomposable vector bundle Fm on En appearing in the exact sequence 0 −→ Fm−1 −→ Fm −→ O −→ 0,

F1 = O.

In our notation we have Fm ∼ = B(0, m, 1), where 0 = (0, 0, . . . , 0). n times

108


Sketch of the proof. Since the dualizing sheaf ωE ∼ = O is trivial, we have Ext1 (O, O) = k and there is a unique non-split extension 0 −→ O −→ F2 −→ O −→ 0. Then using the same arguments as in [Ati57], we can inductively construct indecomposable vector bundles Fm , m ≥ 1 such that H 0 (Fm ) ∼ = H 1 (Fm ) = k together with exact sequences 0 −→ Fm−1 −→ Fm −→ O −→ 0,

F1 = O.

On the other hand, B(0, m, 1) is the unique indecomposable vector bundle on E of rank m and normalization OEm with a non-zero section. Hence Fm ∼ = B(0, m, 1). The proof of the following proposition is straightforward: Proposition 4. Let Ψ : VB(E) −→ TE be the functor establishing an equivalence between the category of vector bundles on E and the category of triples. Then Ψ preserves tensor products: Ψ (E ⊗ F) ∼ = Ψ (E) ⊗ Ψ (F ), where M, ˜i) ⊗ (F, N , (E, j) = (E ⊗O F, M ⊗A N , ˜i ⊗ j). In particular, • We have an isomorphism B((d), m, λ) ∼ = B((d), 1, λ) ⊗ Fm . • There is the following rule for a decomposition of the tensor product of two unipotent vector bundles: Fhi , F f ⊗ Fg ∼ = i

where integers hi are the same as in the decomposition k[t]/tf ⊗k k[t]/tg ∼ = k[t]/thi in the category of k[t]–modules. i∈Z

• In particular, if k is of characteristics zero, we have F f ⊗ Fg ∼ =

g

Ff −g−1+2j .

j=1

Now we formulate a geometric description of indecomposable torsion free sheaves on a cycle of projective lines in the case of an algebraically closed field. Theorem 19. Let E = En be a cycle of n projective lines and Ik be a chain of k projective lines, E an indecomposable torsion free sheaf on En . 1. If E is locally free, then there is an étale covering πr : Enr −→ En , a line bundle L ∈ Pic(Enr ) and a natural number m ∈ N such that E∼ = πr∗ (L ⊗ Fm ).


109

Moreover, if char(k) = 0, then integers r and m are uniquely determined and the line bundle L is unique up to the action of Aut(Enr /Er ). Other way around, for given r, m ∈ Z+ and L ∈ Pic(Enr ) the vector bundle πr∗ (L ⊗ Fm ) is indecomposable if and only if L does not belong to the image of the map πt∗ : Pic(Enr/t ) → Pic(Enr ) for any proper divisor t of r. 2. If E is not locally free then there exists a map pk : Ik −→ En and a line bundle L ∈ Pic(Ik ) such that E ∼ = pk∗ (L).

Other way around, the torsion free sheaf pk∗ (L) is indecomposable for any line bundle L ∈ Pic(Ik ). Proof. Let E be a cycle of projective lines and πE : E −→ E an étale covering of degree r, F a torsion free sheaf on E . In the notation of Remark 4 we have the commutative diagram πS

S

S ~? p ~~ ~ i ~~ ~~ πS

i ?~ E ~~ ~~p ~ ~ E

/ S

/S @ p˜ i

˜i

πE

πE

/E @ p /E

in which all squares are pull-back diagrams. In order to prove the theorem we have to compute the triple describing the torsion free sheaf π∗ (F ). Note that each map I −→ E from a chain of projective lines to a cycle of projective lines factors through an étale covering E −→ E. So, in order to prove the second

110


part of the theorem about the characterization of strings we may consider an étale covering of E as well. Note the following simple fact about pull-back diagrams: Lemma 3. Let Y

g

f

X

/Y f

g

/X

be a pull-back diagram, where all maps f, g, f , g are affine. Then for any coherent sheaf F on Y it holds g ∗ f∗ F ∼ = f∗ g ∗ F . ∗ ∗ The morphism p∗ πE∗ (F )/tor(p∗ πE∗ (F )) −→ πE∗ p (F )/tor(p (F )) is an isomorphism. Indeed, we have a surjection ∼ =

∗ ∗ ∗ p∗ πE∗ (F ) −→ πE∗ p F −→ πE∗ (p (F )/tor(p (F )))

which induces a surjective map ∗ ∗ p∗ πE∗ (F )/tor(p∗ πE∗ (F )) −→ πE∗ p (F )/tor(p (F )) of torsion free sheaves. Since both sheaves have the same rank on each ir we conclude that this map is also injective and reducible component of E, therefore an isomorphism. We need one more simple statement about étale coverings. Lemma 4. Let π : Y −→ X be an étale map of reduced schemes and F a coherent sheaf on Y. Then there is a canonical isomorphism π∗ (F /tor(F )) −→ π∗ (F )/tor(π∗ (F )). Proof. The canonical map F −→ F/tor(F ) induces the morphism π∗ (F ) −→ π∗ (F /tor(F )). Since π is étale, the sheaf π∗ (F /tor(F )) is torsion free and the induced morphism π∗ (F /tor(F )) −→ π∗ (F )/tor(π∗ (F )) is an isomorphism since it is an isomorphism on the stalks. ∗

Let ε : p˜∗ i∗ (F ) −→ i (p∗ (F )/tor(p∗ (F ))) be the gluing map describing the torsion free sheaf F in the corresponding triple. From the commutativity of the diagram


111

/ ˜i∗ p∗ πE∗ (F )/tor(p∗ πE∗ (F ))

p˜∗ i∗ πE∗ (F )

˜i∗ π p∗ (F )/tor(π p∗ (F )) E∗ E∗ ˜i∗ π (p∗ (F )/tor(p∗ (F )) E∗

p˜∗ πS∗ i∗n (F ) ∗ πS∗ p i∗ (F )

πS∗ ( ε)

/ π i ∗ p∗ (F )/tor(p∗ (F )) S∗

we conclude that the direct image sheaf πE∗ F is described by the gluing maε), which are exactly the matrices constructed in the Algorithm 1. trices πS∗ ( This completes the proof. Remark 7. For a given cycle of projective lines E over an arbitrary field k there is always an étale covering π : E −→ E of a given degree r. For example, let E = E1 be a rational curve with one node, Xi = P1 (i = 1, 2), fi : Spec(k × k) −→ Xi be two closed embeddings with the image 0 and ∞ and gi : Xi −→ E two normalization maps mapping the points 0 and ∞ on Xi to the singular point of E. Then the push-out of X1 and X2 over Spec(k × k) (in the category of all schemes and affine maps) is a cycle of two projective lines E2 and the induced map g : E2 −→ E1 is an étale covering of degree two. The general case can be considered in a similar way. Note that this is quite different to the case of elliptic curves, where the existence of an étale covering of a given degree strongly depends on the arithmetics of the curve. Similarly to the proof of Theorem 19 we have the following proposition. Proposition 5. Let πr : Enr −→ En be an étale covering of degree r, n nr nr = ! L be the normalizations of En and Enr . Let n = ! Li and E E j i=1

j=1

{a1 , a2 , . . . , an } be the set of singular points of En and {b1 , b2 , . . . , bn , bn+1 , . . . , bnr } the singular points of Enr and πr−1 (ai ) = {bi , bi+n , . . . , bi+(r−1) }. Al , ˜i), where Assume E is a vector bundle of rank l, given by the triple (E, E ∼ = E1 ⊕ E2 ⊕ · · · ⊕ En and ˜i is given by matrices M (L1 , a1 ), M (L1 , a2 ), . . . , M (Ln , a1 ). Then the pull-back πr∗ (E) corresponds to the triple (E , Al , ˜i ), where E |Li+nj ∼ = Ei , 0 ≤ j ≤ r − 1 and i is given by matrices M (L , b ) = M (L, a ) and M (L , b ) = M (L, a ) if πr (L ) = L and πr (b) = a. From this proposition follows the following corollary: Corollary 1. Let Let E = En be a cycle of n projective lines and πr : Erd −→ En an étale covering of degree r. Then

112


1. πr∗ B(d, 1, λ) ∼ = B(dr , 1, λr ). 2. If char(k) = 0, then πr∗ (Fm ) ∼ = Fm . In particular, we have an isomorphism B(d, m, λ) ∼ = B(d, 1, λ) ⊗ Fm . Proof. The proof of the first part is straightforward. To prove the second, m , Am , ˜i), where ˜i note that πr∗ (Fm ) is given by a triple, isomorphic to (O is given by matrices M (L1 , a1 ) = Im , M (L1 , a2 ) = Im , . . . , M (Ln , an ) = Im , M (Ln , a1 ) = Jm (1)r , where Jm (1) is the Jordan (m × m)–block with the eigenvalue 1. If char(k) = 0 then Jm (1)r ∼ Jm (1) and we get the claim. Note, that in the case char(k) = p we have Jp (1)p = Ip that implies πp∗ (Fp ) ∼ = Op . To complete the proof of the second claim note, that B(d, m, λ) ∼ = πr∗ (L(d, λ) ⊗ Fm ) ∼ = πr∗ (L(d, λ) ⊗ πr∗ Fm ) ∼ = πr∗ (L(d, λ)) ⊗ Fm ∼ = B(d, 1, λ) ⊗ Fm .

Remark 8. As we have already seen, the technique of étale coverings requires special care in the case of positive characteristics. For example, let E = E1 be a Weierstraß nodal curve and π2 : E2 −→ E1 an étale covering of degree 2. 2 , k2 (s), ˜i), where Then the vector bundle π2∗ (O) corresponds to the triple (O ˜i is given by matrices i(0) =

1 0

and i(∞) =

0 1

0 1 1 0

Then for char(k) = 2 we have 0

1

1

0

∼

1 0 0 −1

and π2∗ O ∼ = O ⊕ B(0, 1, −1). However, for char(k) = 2 0

1

1

0

∼

1 1 0 1

and π2∗ O ∼ = F2 . From Theorem 19 one can derive formulas for the cohomology groups of indecomposable torsion free sheaves, a formula for the dual of an indecomposable torsion free sheaf and rules for the computation of the direct sum decomposition of two indecomposable vector bundles. This is what we are going to describe now.


113

Lemma 5 ( [BDG01,BK1]). If d = (d1 , . . . , drn ) ∈ Zrn , e = (e1 , e2 , . . . , ek ), λ ∈ k∗ , m ≥ 1, 1 ≤ f ≤ n, we have: (i) B(d, m, λ)∨ ∼ = B(−d, m, λ−1 )

(−1, 0, . . . , 0, −1) (ii)S(e, f )∨ ∼ = S(κ − e, f ) with κ = −2

if k ≥ 2 if k = 1.

Proof. If f : X → E is a finite morphism, F a coherent sheaf on X and G a locally free sheaf on E then there is a natural isomorphism of f∗ OX –modules f∗ HomX (F , f ! G) ∼ = HomE (f∗ F , G). Recall that f ! ωE is a dualizing sheaf on X if ωE is one on E. In our situation ωE ∼ = OE and we obtain an isomorphism f∗ HomX (F , ωX ) ∼ = HomE (f∗ F , OE ) ∼ = (f∗ F )∨ . To show (i), we consider X = En and f = πn . The claim follows now from ∨ ∼ ωEn ∼ = OEn , Fm = Fm and L(d, λ)∨ ∼ = L(−d, λ−1 ) on En . For the proof of (ii) we let X = Ik and f = pk . Now ωIk ∼ = L(κ) and the result follows from L(d)∨ ∼ = L(−d) on Ik . Using the description of indecomposable vector bundles via étale coverings it is not difficult to compute their cohomology. Lemma 6 ( [DGK03]). There is the following formula for the cohomology of indecomposable vector bundles dimk H 0 (B(d, m, λ)) = m

rn

(di + 1)+ − θ(d) + δ(d, λ)

i=1

and dimk H 1 (B(d, m, λ)) = rm − dimk H 0 (B(d, m, λ), where δ(d, λ) = 1 if d = (0, . . . , 0), λ = 1 and 0 otherwise; k+ = k if k > 0 and zero otherwise. The number θ(d) is defined as follows: call a subsequence p = (dk+1 , . . . , dk+l ), where 0 ≤ k < rn and 1 ≤ l ≤ rn a positive part of d if all dk+j ≥ 0 and either l = rn or both dk < 0 and dk+l+1 < 0. For such a positive part put θ(p) = l if either l = rs or p = (0, . . . , 0) and θ(p) = l + 1 otherwise. Then θ(d) = θ(p), where we take a sum over all positive subparts of d. In order to compute the tensor product of two indecomposable vector bundles we shall need the following lemma. Lemma 7. Let E be a cycle of projective lines, πi : Ei −→ E two étale coverings i = 1, 2 and Ei a vector bundle on Ei . Let E be the fiber product of E1 and E2 over E:

114

Lesya Bodnarchuk, Igor Burban, Yuriy Drozd, and Gert-Martin Greuel p1

E −−−−→ E1 ⏐ ⏐ ⏐p2 ⏐π1 π

E2 −−−2−→ E. Denote by π : E −→ E the composition π1 p1 , then π1∗ (E1 ) ⊗ π2∗ (E2 ) ∼ ∗ p∗1 (E1 ) ⊗ p∗2 (E2 ) . =π Proof. By the base change and projection formula π ∗ p∗1 (E1 ) ⊗ p∗2 (E2 ) ∼ = ∗ ∗ ∼ ∼ p π π π p (E ) ⊗ E π (E ) ⊗ E π2∗ p2∗ p∗1 (E1 ) ⊗ p∗2 (E2 ) ∼ = 2∗ 2 1∗ 1 = = 2∗ 2∗ 1 1 2 2 π1∗ (E1 ) ⊗ π2∗ (E2 ). The following proposition describes the fiber product of two étale coverings of a given cycle of projective lines. Proposition 6 ( [Bur03]). Let En be a cycle of n projective lines and πi : Edi n −→ En two étale covering of degree di , i = 1, 2. Choose a labeling of the irreducible components of each cycle Ed1 n and Ed2 n by consecutive nonnegative integers 0, 1, 2, . . . , such that πi maps the zero component into a zero component, i = 1, 2. Let d = g.c.d.(d1 , d2 ) be the greatest common divisor and D = [d1 , d2 ] the smallest common multiple of d1 and d2 . Then the fiber product d = ! E(i) and p1 : E(i) −→ Ed n is the étale covering determined by the is E 1 Dn Dn i=1

(i)

assumption that it maps the i-th component of EDn to the 0-th component of (i) Ed1 n . The second morphism p2 : EDn −→ Ed2 n is the étale covering, mapping the zero component to the zero component. These properties allow to describe a decomposition of the tensor product of any two indecomposable vector bundles into a direct sum of indecomposable ones. In particular, in the case of a nodal Weierstraß curve we get the following concrete algorithm, obtained for the first time in [Yud01]. Theorem 20 ( [Yud01, Bur03]). Let E be a Weierstraß nodal curve over an algebraically closed field k of characteristics zero, B(d, 1, λ) and B(e, 1, µ) two vector bundles on E of rank k and l respectively, d = d1 d2 . . . dk and e = e1 e2 . . . el . Let D be the smallest common multiple and d the greatest common divisor of k and l. Consider d sequences f 1 = d1 + e1 , d2 + e2 , . . . , dk + el , f 2 = d1 + e2 , d2 + e3 , . . . , dk + e1 , .. . f d = d1 + ed , d2 + ed+1 , . . . , dk + ed−1 , of length D. Then the following decomposition holds:


B(d, 1, λ) ⊗ B(e, 1, µ) ∼ =

d

l

115

k

B(fi , 1, λ d µ d ).

i=1

If some fi is periodic, then we use the isomorphism B(g , 1, λ) = l

l

√ l B(g, 1, ξ i λ),

i=1

where g l = gg . . . g and ξ a primitive l–th root of 1. l

Even possessing a complete classification of indecomposable torsion free sheaves on a Weierstraß nodal curve, an exact description of stable vector bundles is a non-trivial problem. It can be shown by many methods that for a pair of coprime integers (r, d) ∈ Z2 , r > 0 the moduli space of stable vector bundles of rank r and degree d is k∗ , see for example [BK3]. However, for applications it is important to have a description of stable vector bundles via étale coverings. In order to get such a classification note the following useful fact. Lemma 8 ( [Bur03]). Let E be an irreducible Weierstraß curve. Then a coherent sheaf F on E is stable if and only if it is simple i.e. End(F ) = k. In general, for irreducible curves stability implies simplicity, but in the case of irreducible curves of arithmetic genus one both conditions are equivalent. Then one can prove the following theorem: Theorem 21 ( [Bur03]). Let E be a nodal Weierstraß curve and E a stable vector bundle on E of rank r and degree d, 0 < d < r. Then g.c.d.(r, d) = 1, E∼ = B(d, 1, λ) and d can be obtained by the following algorithm. 1. Let y = min(d, r−d), x = max(d, r−d). If x = y, then d = (0, 1). Assume now x > y. Consider the triple (x, y, x + y) and write x + y = (k + 1)y + s, where 0 < s < y and k ≥ 1. If s > y − s then replace (x, y, x + y) by (s, y − s, y) and say say that (x, y, x + y) is obtained from (s, y − s, y) by the blow-up of type (A, k). If s < y − s then replace (x, y, x + y) by (y − s, s, y) and say that (x, y, x + y) is obtained from (y − s, s, y) by the blow-up of type (B, k). 2. Repeat this algorithm until we get the triple (p, 1, p + 1). Consider the (C1 ,k1 )

sequence of reductions (x, y, x + y) = (x0 , y0 , x0 + y0 ) −→ (x1 , y1 , x1 + (Cn ,kn )

y1 ) −→ · · · −→ (xn , yn , xn + yn ) = (p, 1, p + 1), where Ci ∈ {A, B} and ki ≥ 1 for 1 ≤ i ≤ n. Now we can recover the vector d :

116


1. Start with sequence α, α, . . . , α, β, which corresponds to the triple (xn , yn , p times

xn + yn ) = (p, 1, p + 1). If Cn = A then replace each letter α by the block α, α, . . . , α and each β by the block α, α, . . . , α. Between these new kn + 1

kn

blocks insert the letter β. If Cn = B then replace each letter α by the block α, α, . . . , α and each letter β by the block α, α, . . . , α. Between these new kn

kn + 1

blocks insert the letter β again. We have got a new sequence of letters α and β of total length xn−1 + yn−1 with xn−1 letters α and yn−1 letters β. 2. Proceed inductively until we get a sequence of length r with max(d, r − d) letters α and min(d, r − d) letters β. 3. If d > r − d then replace each letter α by 1 and each letter β by 0. In case d ≤ r − d replace α by 0 and β by 1. The resulted sequence is the vector d we are looking for. Example 3. Let rank r = 19 and degree 11. The sequence of reductions is (B,1)

(A,1)

(11, 8, 19) −→ (5, 3, 8) −→ (2, 1, 3). Using the algorithm we get the sequence of blowing-ups (A,1)

α, α, β −→ α, α, β, α, α, β, α, β (B,1)

α, α, β, α, α, β, α, β −→ α, β, α, β, α, α, β, α, β, α, β, α, α, β, α, β, α, α, β and hence d = (1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0). This result was generalized by Mozgovoy [Moz] to get a recursive description of semi-stable torsion free sheaves of arbitrary slope. Note the following important difference between smooth and singular curves of arithmetic genus one. In the smooth case any indecomposable coherent sheaf is either locally free or torsion free and is automatically semi-stable. This is no longer true for singular curves, in particular, in that case there are indecomposable coherent sheaves which are neither torsion nor torsion free. Example 4. Let E be a nodal Weierstraß curve, s its singular point, n : P1 −→ E the normalization map. Then Ext1 (n∗ (OP1 ), k(s)) = H 0 (Ext1 (n∗ (OP1 ), k(s))) = k2 . Let w ∈ Ext1 (n∗ (OP1 ), k(s)) be a non-zero element and i

p

0 −→ k(s) −→ F −→ n∗ (OP1 ) −→ 0


117

the corresponding extension. Then F is an indecomposable coherent sheaf which is neither torsion nor torsion free. To see that F is indecomposable assume F ∼ = F ⊕ F . Then one of its direct summands, say F is a torsion sheaf. Since Hom(F , n∗ (OP1 )) = 0, F belongs to the kernel ker(p) and hence is isomorphic to k(s). Therefore the map i has a left inverse, hence w = 0, and that is a contradiction. Proposition 7 ( [BK3]). Let E be a singular Weierstraß curve and F ∈ Coh(E) an indecomposable coherent sheaf which is not semi-stable. Then, all Harder-Narasimhan factors of F are direct sums of semi-stable sheaves of infinite homological dimension. Proof. Let 0 ⊂ Fn ⊂ . . . ⊂ F1 ⊂ F0 = F be the Harder-Narasimhan filtration of F with semi-stable factors Aν := Fν /Fν+1 of decreasing slopes µ(An ) > µ(An−1 ) > . . . > µ(A0 ). Assume Aν ∼ = Aν ⊕ Aν and Aν has finite global dimension. Since Fν+1 is filtered by semi-stable sheaves Fµ for µ > ν and Hom(Aµ , Aν ) = 0, we get Ext1 (Aν , Fν+1 ) ∼ = Hom(Fν+1 , Aν )∗ = 0. Therefore Fν contains Aν as a direct summand: Fν ∼ = Fν ⊕ Aν . From the exact sequence 0 −→ Fν ⊕ Aν −→ Fν−1 −→ Aν−1 −→ 0 and the isomorphism Ext1 (Aν−1 , Aν ) ∼ = Hom(Aν , Aν−1 )∗ = 0 we conclude that Fν−1 contains Aν as a direct summand as well. Proceeding inductively we obtain that F itself contains Aν as a direct summand, a contradiction. We see that a difference between the combinatorics of indecomposable coherent sheaves on smooth and singular Weierstraß curves is due to the existence of semi-stable sheaves of infinite global dimension together with the failure of the Serre duality on singular curves. In order to classify indecomposable coherent sheaves it is convenient to consider a more general problem: the description of indecomposable objects of the derived category D− (Coh(E)). It turns out that the last problem is again tame and can be solved using the technique of representations of bunches of chains, see [BD04] for the details. 3.3 Vector Bundles on a Cuspidal Cubic Curve As we have mentioned in the introduction, the category of vector bundles on a curve of arithmetic genus one, different from a cycle of projective lines, is vector bundle wild, see also Corollary 2. Nevertheless, if we restrict ourselves to the subcategory of simple vector bundles VBs , or even to the subcategory of simple torsion free sheaves TFs , then the classification problem becomes tame again and, moreover, the combinatorics of the answer resembles the case of smooth and nodal Weierstraß curves (see Theorem 3 and Theorem 21). Theorem 22. Let E be a cuspidal cubic curve over an algebraically closed field k. Then

118


1. the rank r and the degree d of a simple torsion free sheaf F over E are coprime; 2. for every pair (r, d) of coprime integers with positive r, the isomorphism classes of simple vector bundles E ∈ VBs (r, d) are parametrized by A1 and there is a unique simple torsion free but not locally free sheaf F of rank r and degree d. Note that A1 ∼ = Ereg is isomorphic to the Picard group Pic◦ (E). It can be shown that for a pair of coprime integers r > 0 and d the moduli space of TFs (r, d) is isomorph to E, moreover, vector bundles E correspond to nonsingular points of E and the unique torsion free but not locally free sheaf F corresponds to the singular point s. Sketch of proof. Let E be a cuspidal cubic curve given by the equation x3 − p ∼ y 2 z = 0. Choose coordinates (z0 : z1 ) on the normalization E = P1 −→ E such that the preimage of the singular point s = (0 : 0 : 1) of E is (0 : 1). Let U = {(z0 : z1 )|z1 = 0} be an affine neighborhood of (0 : 1) and z = z0 /z1 . In the notations of Section 3.1 we have: A ∼ = k(s) and A ∼ = k[ε]/ε2 (s). M, ˜i) be the correLet F be a torsion free sheaf of rank r on E and (F, sponding triple. Then, as in the case of a nodal rational curve, we have ∼ O(n)rn , with • a splitting F = n∈Z rn = r; n∈Z

• an isomorphism M ∼ = At , for some t ≥ r, and t = r if and only if F is a vector bundle; which is an " ⊗A A −→ F ⊗ A, ˜i : M • an epimorphism of A–modules O isomorphism if and only if F is a vector bundle. In order to write ˜i in matrix form remember that we identify F ⊗O A with 2 p∗ p∗ (F ) ⊗OP1 OP1 /I , where I = I(0:1) is the ideal sheaf of the schemetheoretic preimage of s. We choose a basis of M ∼ = k(s)r and fix the trivializations OP1 (n) ⊗ OP1 /I −→ k[ε]/ε2 (s) given by the map ζ ⊗ 1 → pr( zζn ) for a local section ζ of O(n) on an open 1 set V containing (0 : 1), where pr : k[V ] −→ k[ε]/ε2 be the map induced by k[z] −→ k[ε]/ε2 , z → ε. Using these choices we may write ˜i = i(0) + εiε (0), where both i(0) and iε (0) are square r × r matrices. Since by Theorem 16 the isomorphism classes of triples are in bijection with the isomorphism classes of torsion free sheaves, we have to study the action of automorphisms of (F, M, ˜i) on the matrices i(0) and iε (0). The condition for ˜i to be surjective is equivalent to the surjectivity of i(0). Similarly, for vector bundles we have that ˜i is invertible if and only if i(0) is invertible. If we have a morphism O(n) −→ O(m) given by a homogeneous form Q(z0 , z1 ) of degree m − n then the induced map O(n) ⊗ O/I −→ O(m) ⊗ O/I dQ (0 : 1). is given by the map pr(Q(z0 , z1 )/z1m−n ) = Q(0 : 1) + ε dz 0


119

Moreover, for any endomorphism (F, f ) of the triple (F, M, ˜i) the induced dF (0 : 1). map F¯ : F ⊗ O/I −→ F ⊗ O/I has the form F¯ = F (0 : 1) + ε dz 0 2 ¯ If (F, f ) is an automorphism then F ∈ GLr (k[ε]/ε ) and the transformation rule F¯˜i = i˜ f in matrix form reads i (0) = F (0 : 1)i(0)f −1 dF iε (0) = dz (0 : 1)i(0)f −1 + F (0 : 1)iε (0)f −1 . 0 As a result, the matrix problem is as follows: we have two matrices i(0) and iε (0) with r rows and t columns, and rank(i(0)) = r. In the case of a vector bundle i(0) and iε (0) are square matrices and i(0) is invertible. The matrices i(0) and iε (0) are divided into horizontal blocks labelled by integers called weights. Any two blocks of i(0) and iε (0) marked by the same label are called conjugated and have the same number of rows.

.. .

.. .

n−1 n

n−1

}rn

n

n+1

n+1

.. .

.. .

i(0)

}rn

iε (0)

The permitted transformations are listed below: 1. We can simultaneously do any elementary transformations of columns of i(0) and of iε (0). 2. We can simultaneously perform any invertible elementary transformations of rows of i(0) and iε (0) inside of any two conjugated horizontal blocks. 3. We can add a scalar multiple of any row with lower weight to any row with higher weight simultaneously in i(0) and iε (0). 4. We can add a row of i(0) with a lower weight to any row of iε (0) with a higher weight. This matrix problem turns to be wild, see corollary 2. However, the simplicity condition of a triple (F, M, ˜i) implies additional restrictions, which make the problem tame. First note that if F contains O(c) ⊕ O(d) with d > c + 1 as a direct summand, then the pair (F, f ) := (z0d−c , 0) defines a non-scalar endomorphism of the triple (F, M, ˜i), which, therefore, can not be simple. r2 r1 ⊕ O(c+1) Thus, for a simple torsion free sheaf F we may assume F ∼ = O(c) for some c ∈ Z and the matrix ˜i consists of two horizontal blocks. We consider the case of vector bundles first. Although the case of torsion free but not locally free sheaves is similar, it should be considered separately in order to make the presentation clearer.

120


As was mentioned above, if F is a vector bundle then ˜i is an isomorphism and by transformations 1 and 2 the matrix i(0) can be reduced to the identity matrix. Moreover, by applying transformation 4 we can make the left lower block of iε (0) zero, as indicated below: i(0) =

Ir1

and

iε (0) =

Ir2

B1 B12

.

(∗)

B2

Here In denotes the identity matrix of size n, an empty space stands for a zero block and B1 , B12 , B2 denote nonreduced blocks. Thus we can assume that i(0) is the identity matrix and concentrate on the matrix iε (0), taking into account only those transformations, which leave i(0) unchanged. Then we obtain the category of block matrices BM = BM(r1 , r2 ). (r1 ,r2 )

Objects of BM(r1 , r2 ) are matrices of the form iε (0) in formula (*), i.e. upper triangular block matrices B consisting of the blocks (B1 , B12 , B2 ), where (B1 , B2 ) are square matrices of sizes r1 and r2 respectively. Morphisms C : B → B are given by lower triangular block matrices: C=

C1 C21 C2

with block sizes (r1 , r2 ) and satisfying equations CB = B C. In term of blocks this equation can be written as: C21 , C1 B1 = B1 C1 + B12 C1 B12 = B12 C2 , C2 B2 + C21 B12 = B2 C2 .

(∗∗)

Two matrices B and B are called equivalent (i.e. correspond to isomorphic vector bundles) if there is a non-degenerate morphism C : B → B , i.e. if B = CBC −1 . In terms of transformations this means: we can add a row k with lower weight to a row j with higher weight and simultaneously add the column j to the column k. A matrix B ∈ BM(r1 , r2 ) is called simple if any endomorphism C : B → B is scalar. Obviously, simplicity is a property defined on equivalence classes. The full subcategory BM(r1 , r2 ) consisting of simple objects B is denoted by BMs (r1 , r2 ). Note that, if a block B12 has a zero-row k and a zero-column j, then by adding column j to column k and row k to row j we construct a nonscalar endomorphism, hence B is not simple. In particular, if r1 = r2 then B12 is a square matrix and can be reduced to the identity matrix I. Having B12 = I we can reduce one of matrices B1 and B2 , let us say B1 , to zero and the other


121

one B2 to its Jordan normal form. If r2 = 1 then B2 = λ , λ ∈ k, in this case B is simple, but for r2 > 1 the Jordan normal form has an endomorphism, which can be extended to an endomorphism of B. Therefore, if B is simple then B12 can be reduced to one of the following forms ⎧ 0 ⎪ ⎪ ⎪ ⎪ Ir2 if r1 > r2 , ⎪ ⎪ ⎨ B12 = Ir1 0 if r2 > r1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 if r1 = r2 = 1, From the system of equations (**) we get that in case r1 > r2 block B2 can be reduced to the zero matrix and block B1 to the upper triangular block-matrix formed by three nonzero subblocks (B1.1 , B1.12 , B1.2 ). Long but straightforward calculations show that the transformations of B which preserve already reduced blocks are uniquely determined by the automorphisms of B1 in the category BMs . Moreover, EndBMs (B1 ) = EndBMs (B). In the same way the matrix B can be reduced in case r2 > r1 . Thus the problem BMs (r1 , r2 ) is self-reproducing, that means we get a bijection between BMs (r1 , r2 ) and BMs (r1 − r2 , r2 ) if r1 > r2 , between BMs (r1 , r2 ) and BMs (r1 , r2 − r1 ) if r2 > r1 , and if r1 = r2 > 1 then BMs (r1 , r1 ) is empty. In this reduction one can easily recognize the Euclidian algorithm. Moreover, the reduction terminates after finitely many steps when we achieve r1 = r2 = 1. Without loss of generality we may assume that the matrix B ∈ BMs (1, 1) has the form B=

0

1

.

λ λ

(∗ ∗ ∗)

1

.) (Note that this matrix is equivalent to 0 Objects of BMs (1, 1) are parametrized by a continuous parameter λ ∈ k, thus the same holds for BMs (r1 , r2 ) with coprime r1 and r2 . Let E be a vector bundle of rank r and degree d with normalization E = + 1)r2 . Taking into account that by Riemann-Roch theorem O(c)r1 ⊕ O(c (Theorem 9) r2 = d mod r and r1 + r2 = r, we obtain the statements about vector bundles of Theorem 22. Moreover, if coprime integers r > 0 and d are given then for λ ∈ k one can construct the matrix B(λ) ∈ BMs (r1 , r2 ), and hence, the unique vector bundle E(r, d, λ), by reversing the reduction procedure described above: Algorithm 2. Let (r, d) ∈ Z2 be coprime with positive r, and λ ∈ k. • First, by the Euclidean algorithm we find integers c, r1 and r2 , 0 < r1 ≤ r, 0 ≤ r2 < r such that cr + r2 = d and r1 + r2 = r. Thus we recover the + 1)r2 . r1 ⊕ O(c normalization sheaf F = O(c)

122


• If r1 = r2 = 1 the matrix B(λ) has form (***). Using this input data we construct the matrix B(λ) ∈ BMs (r1 , r2 ) inductively: • Let r1 + r2 > 2 and r1 > r2 . Assume we have the matrix B1 (λ) ∈ BMs (r1 − r2 , r2 ), then B(λ) ∈ BMs (r1 , r2 ) has form B(λ) =

B1 (λ)

0 Ir2 . 0

• Let r1 + r2 > 2 and r1 < r2 . Assume that we have the matrix B2 (λ) ∈ BMs (r1 , r2 − r1 ), then B(λ) ∈ BMs (r1 , r2 ) has form 0 Ir1 0 B(λ) =

. B2 (λ)

• Finally, we get the matrix ˜i = i(0) + εiε (0) = Ir + εB(λ). Let us illustrate this with a small example: Example 5. Let E ∈ VBs (7, 12) be an indecomposable vector bundle of rank 7 and degree 12. To obtain the matrices i(0) and iε (0) we calculate the normal 2 ⊕ O(2) 5 . Thus, in our notations r1 = 2 and ization sheaf E first: E = O(1) r2 = 5. The Euclidian algorithm applied to the pair (2, 5) gives: (2, 5) → (2, 3) → (2, 1) → (1, 1). Reversing this sequence, by the above reduction procedure, we obtain a sequence of bijections: ∼

∼

∼

BMs (1, 1) −→ BMs (2, 1) −→ BMs (2, 3) −→ BMs (2, 5), and finally for the matrices we get:

0

1 λ

0 →

1

0 λ

0 1 0

→

0

0

1

0

0

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

0 .

0 λ 1

0

0

0

1

0

0

0

0

0 λ 1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

1 0

0 0

→

0

0

The reduction for torsion free but not locally free sheaves can be done in a similar way. The only difference is that the matrices i(0) and iε (0) are no longer square:


i(0) =

Ir1

and iε (0) =

Ir2

B1 B12 B13

123

.

B2 B23

The matrix iε (0) has two additional blocks B13 and B23 with a new column size r3 > 0. Investigating such matrices inductively for simple matrices, we get r3 = 1. Moreover, if r1 + 1 and r2 + 1 are coprime then there is a unique simple matrix ˜i, and there is no simple matrices otherwise. This unique simple matrix ˜i corresponds to the unique torsion free but not locally free sheaf F , which can be considered as a compactifying object of the family VBs (r, d). Let us illustrate this for small ranks. as the normalization Example 6. Vector bundles E of VBs (1, 0) have E = O sheaf and the corresponding matrices ˜i are 1 + ε λ , λ ∈ k. For the unique torsion free but not locally free sheaf F of rank 1 and degree 0, one computes = deg(F ) − 1 = −1, thus F = O(−1) that deg(F) and the corresponding ˜ matrix i is 1 0 + ε · 0 1 . Example 7. Vector bundles E from VBs (2, 1) have as normalization sheaf E = ⊕ O(1) O thus the corresponding matrices are ˜i = 1 0

0

0

1

1

+ε·

0 1

0

0 λ

1

,

where λ ∈ k. The normalization sheaf F of the torsion free but not locally 2 free sheaf F ∈ TFs (2, 1) has degree deg(F) = deg(F ) − 1 = 0 thus F = O and the corresponding matrix is ˜i = 1 0 0 1

0 0

0

+ε·

0 1

0

0 0

1

0.

3.4 Coherent Sheaves on Degenerations of Elliptic Curves and Fourier-Mukai Transforms The technique of Fourier-Mukai transforms on elliptic curves led to a classification of indecomposable coherent sheaves. This can be generalized to the case of singular Weierstraß cubic curves. Theorem 23 ( [BK1]). Let E be an irreducible projective curve of arithmetic genus one over an algebraically closed field k, p0 ∈ E a smooth point and P = I∆ ⊗ π1∗ (O(p0 )) ⊗ π2∗ (O(p0 )), where I∆ is the ideal sheaf of the diagonal ∆ ⊂ E × E. We have the following properties of the Fourier-Mukai transform Φ = ΦP :

124


1. Φ is an exact equivalence and Φ ◦ Φ ∼ = i∗ [−1], where i : E −→ E is an involution of E. 2. Φ transforms semi-stable sheaves to semi-stable ones and stable sheaves to stable ones. 3. In particular, Φ induces an equivalence between the abelian categories 1 Cohν (E) and Coh− ν (E), where ν ∈ Q ∪ {∞}. 4. Let F be a semi-stable sheaf of degree zero. Then the sequence ev

0 −→ H 0 (E(p0 )) ⊗ O −→ E(p0 ) −→ coker(ev) −→ 0 ∼ coker(ev) establishes an equivais exact. Moreover, the functor Φ(E) = lence between the category Coh0 (E) of semi-stable torsion-free sheaves of degree zero and the category of torsion sheaves Coh∞ (E). From this theorem follows that for any pair of coprime integers (r, d) ∈ Z2 , r > 0 the moduli space ME (r, d) of stable sheaves of rank r and degree d is isomorphic to E. The unique singular point of ME (r, d) corresponds to the stable sheaf which is not locally free. Let T be an indecomposable torsion sheaf on E. If T has support at a smooth point p ∈ E than T ∼ = OE,p /mnp for some n > 0. The structure of torsion sheaves supported at the singular point s ∈ E is much more complicated. First of all note that the categories of finite dimensional modules over OE,s and O E,s are equivalent. So, in order to understand semi-stable sheaves on singular Weierstraß curves we have to analyze the structure of finite-dimensional representations of kx, y/(xy) and kx, y/(y 2 − x3 ) first. Let R = kx, y/(xy), then it is easy to show that all indecomposable finite length R-modules generated by one element are M((n, m), 1, λ) = R/(xn + λy m ) for n, m ≥ 1, λ ∈ k∗ and N (0, (n, m), 0) = R/(xn+1 , y m+1 ) for n, m ≥ 0. A classification of all indecomposable R–modules was obtained by Gelfand and Ponomarev [GP68] and independently by Nazarova and Roiter [NR69], see also [BD04] for a description via derived categories. We identify an indecomposable torsion module T supported at s with the corresponding kx, y/(xy)–module. Theorem 24 ( [BK1]). The Fourier-Mukai transform ΦP maps the torsion module M((n, m), 1, λ) to the degree zero semi-stable vector bundle m

n

B((1, 0, . . . , 0, −1, 0, . . . , 0), 1, (−1)(n+m) λ) and N (0, (n, m), 0) to the semi-stable torsion free sheaf m

n

S(0, . . . , 0, −1, 0, . . . , 0).


125

In [BK1] a complete correspondence between torsion sheaves and semistable sheaves of degree zero was described. Using the technique of relative Fourier-Mukai transforms one gets a powerful tool to construct interesting examples of relatively stable and semi-stable sheaves on elliptically fibered varieties, see [FMW99, BK2]. In a similar way, if R = kx, y/(y 2 − x3 ) = kt2 , t3 , then a finite length R–module is given by a finite dimensional vector space V over k and two nilpotent commuting endomorphisms X, Y : V → V which satisfy Y 2 − X 3 = 0. It is again very easy to classify all R–modules of the form R/I, where I is an ideal in R: there are one-parameter families of modules of projective dimension one R/(tn +λtn+1 ), n ≥ 2 and λ ∈ k, and discrete series of modules of infinite projective dimension, R/(tn , tn+1 ), where n ≥ 2. However, there is an essential difference between the rings kx, y/(xy) and R = kx, y/(y 2 − x3 ): the first has tame representation type [GP68, NR69] whereas the second is wild. Proposition 8 ( [Dro72]). The category of finite length modules over the ring kx, y/(y 2 − x3 ) is wild. Proof. We have a fully-faithful exact functor kz1 , z2 −→ RepΓ , where Γ is b / • . This functor maps a kz1 , z2 –module (Z1 , Z2 , kn ) the quiver a :• to the representation of Γ given by Z1 Z2 , B = 0 In . A= In 0 Moreover, we have another fully-faithful exact functor RepΓ −→ kx, y/(y 2 − x3 ), mapping a representation (A, B, kn , km ) to the kx, y/(y 2 −x3 )–module given by the matrices ⎞ ⎛ ⎞ ⎛ X 1 0 X2 03n 0 I Y = ⎝ 0 0m 0 ⎠ , X = ⎝ 0 0 X3 ⎠ , 0 0 X1 0 0 03n where

⎞ 0n 0 I X1 = ⎝ 0 0n 0 ⎠ , 0 0 0n ⎛

⎞ 0n 0 0 X2 = ⎝ I 0n 0 ⎠ , 0 A 0n ⎛

X3 = 0m×n Bm×n 0m×n .

Taking the composition of these two functors we see that the category of finite dimensional kx, y/(y 2 − x3 )–modules is wild. Since via an appropriate Fourier-Mukai transform the category of torsion modules over the ring O E,s is equivalent to the category of semi-stable torsion free sheaves of a given slope ν with non-locally free Jordan-H¨ older quotient, we obtain the following corollary.

126


Corollary 2. Let E be a cuspidal Weierstraß curve and ν ∈ Q ∪ {∞}, then the category Cohν (E) of semi-stable torsion free sheaves of slope ν on E has wild representation type.

References [Ati57]

Atiyah, M.: Vector bundles over an elliptic curve. Proc. London Math. Soc., 7, 414–452 (1957) [Bir13] Birkhoff, G.: A theorem on matrices of analytic functions. Math. Ann., 74, no. 1, 122–133 (1913) [Bon92] Bondarenko, V. M.: Representations of bundles of semi-chains and their applications. St. Petersburg Math. J., 3, 973-996 (1992) [BD03] Bodnarchuk, L., Drozd, Yu.: Stable vector bundles over cuspidal cubics. Cent. Eur. J. Math., 1 no. 4, 650–660 (2003) [Bur03] Burban, I.: Stable vector bundles on a rational curve with one node. Ukra¨ın. Mat. Zh., 55, no. 7, 867–874 (2003) [BD04] Burban, I., Drozd, Yu.: Coherent sheaves on rational curves with simple double points and transversal intersections. Duke Math. J. 121, no. 2, 189–229 (2004) [BD05] Burban, I., Drozd, Yu.: Indecomposables of the derived categories of certain associative algebras. Proceedings of the ICRA-X Conference, Toronto, fields institute communications, 45, 109 – 127 (2005); arxiv: math.RT/0307062 [BDG01] Burban, I., Drozd, Yu., Greuel, G.-M.: Vector bundles on singular projective curves. Applications of algebraic geometry to coding theory, physics and computation 1–15 (Eilat 2001), NATO Sci. Ser. II Math. Phys. Chem., 36, Kluwer Acad. Publ., Dordrecht (2001) [BK1] Burban, I., Kreußler, B.: Fourier-Mukai transforms and semi-stable sheaves on nodal Weierstraß cubics. J. Reine Angew. Math., 584, 45– 82 (2005); arxiv: math.AG/0401437 [BK2] Burban, I., Kreußler, B.: On a relative Fourier-Mukai transform on genus one fibrations. Arxiv: math.AG/0410349 [BK3] Burban, I., Kreußler, B.: Derived categories of irreducible projective curves of arithmetic genus one. To apear in Compositio Mathematica; arxiv: math.AG/0503496 [Dol60] Dold, A.: Albrecht Zur Homotopietheorie der Kettenkomplexe. Math. Ann., 140, 278–298 (1960) [Dro72] Drozd, Yu.: Representations of commutative algebras. Funct. Anal. Appl., 6, 286–288 (1972); translation from Funkts. Anal. Prilozh., 6, 41–43 (1972) [DG01] Drozd, Yu., Greuel, G.-M.: Tame and wild projective curves and classification of vector bundles. J. Algebra, 246, 1–54 (2001) [DGK03] Drozd, Yu., Greuel, G.-M., Kashuba, I.: On Cohen-Macaulay modules on surface singularities. Mosc. Math. J., 3 no. 2, 397–418 (2003) [FM] Friedman, R., Morgan, J.: Holomorphic Principal Bundles Over Elliptic Curves III. Singular Curves and Fibrations. Arxiv: math.AG/0108104 [FMW99] Friedman, R., Morgan, J., Witten, E.: Vector bundles over elliptic fibrations. J. Algebr. Geom.8, 279–401 (1999)

Vector Bundles on Degenerations of Elliptic Curves [GP68] [Gro57] [PH05]

[Kah89] [Kon95]

[Kre01]

[LM93]

[LeP97] [Moz] [Muk81] [NR69]

[Oda71] [Pol03]

[Pol02] [PZ98] [Pum04] [RV02] [Rud97] [Ses82] [Teo00]

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Gelfand, I., Ponomarev, V.: Indecomposable representations of the Lorentz group. Uspehi Mat. Nauk 23, no. 2 140, 3–60 (1968) Grothendieck, A.: Sur la classification des fibrés holomorphes sur la sphére de Riemann. Amer. J. Math. 79, 121–138 (1957) Hein, G., Ploog, D.: Fourier-Mukai transforms and stable bundles on elliptic curves. Beitr¨ age zur Algebra und Geometrie, 46, no. 2, 423–434 (2005) Kahn, C.: Reflexive modules on minimally elliptic singularities. Math. Ann. 285, no. 1, 141–160 (1989) Kontsevich, M.: Homological algebra of mirror symmetry. Proceedings of the international congress of mathematicians, ICM’94, Vol. I. Basel: Birkh¨ auser, 120–139 (1995) Kreußler, B.: Homological mirror symmetry in dimension one. Advances in algebraic geometry motivated by physics. Proceedings of the AMS special session on enumerative geometry in physics, Contemp. Math. 276, 179-198 (2001) Lenzing, H., Meltzer, H.: Sheaves on a weighted projective line of genus one, and representations of a tubular algebra. Representations of algebras (Ottawa, ON, 1992), 313–337, CMS Conf. Proc. 14, AMS Providence, RI (1993) Le Potier, J.: Lectures on vector bundles. Cambridge Studies in Advanced Mathematics, 54 (1997) Mozgovoy, S.: Classification of semi-stable sheaves on a rational curve with one node. Arxiv: math.AG/0410190 ˆ with its application to Picard Mukai, S.: Duality between D(X) and D(X) sheaves. Nagoya Math. J., 81, 153–175 (1981) Nazarova, L.A., Roiter, A.V.: Finitely generated modules over a dyad of two local Dedekind rings, and finite groups with an Abelian normal divisor of index p. Math. USSR Izv., 3, 65–86 (1969); translation from Izv. Akad. Nauk SSSR, Ser. Mat., 33, 65–89 (1969) Oda, T.: Vector bundles on an elliptic curve. Nagoya Math. J., 43, 41–72 (1971) Polishchuk, A.: Abelian varieties, theta functions and the Fourier transform. Cambridge Tracts in Mathematics, 153, Cambridge University Press, (2003) Polishchuk, A.: Classical Yang-Baxter equation and the A∞ -constraint. Adv. Math. 168, no. 1, 56–95 (2002) Polishchuk, A., Zaslow, E.: Categorical mirror symmetry: The elliptic curve. Adv. Theor. Math. Phys., 2, no. 2, 443–470 (1998) Pumpl¨ un, S.: Vector bundles and symmetric bilinear forms over curves of genus one and arbitrary index. Math. Z., 246, no. 3, 563–602 (2004) Reiten, I., Van den Bergh, M.: Noetherian hereditary abelian categories satisfying Serre duality. J. Amer. Math. Soc. 15, no. 2, 295–366 (2002) Rudakov, A.: Stability for an Abelian Category, J. Algebra, 197, 231–245 (1997) Seshadri, C.S.: Fibrés vectoriels sur les courbes algébriques. Astérisque, 96, Société Mathématique de France, Paris, (1982) Teodorescu, T.: Vector bundles on curves of arithmetic genus one. PhD Thesis, Columbia University (2000)

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Lesya Bodnarchuk, Igor Burban, Yuriy Drozd, and Gert-Martin Greuel Seidel, P., Thomas, R.P.: Braid group actions on derived categories of coherent sheaves. Duke Math. J., 108, no. 1, 37–108 (2001) Yudin, I.: Tensor product of vector bundles on curves of arithmetic genus one. Diploma thesis, Kaiserslautern (2001)

Indices of Vector Fields and 1-Forms on Singular Varieties W. Ebeling1 and S. M. Gusein-Zade2 1 2

Universit¨ at Hannover, Institut f¨ ur Algebraische Geometrie, Postfach 6009, D-30060 Hannover, Germany [email protected] Moscow State University, Faculty of Mechanics and Mathematics, Moscow, 119992, Russia [email protected]

Summary. We discuss different generalizations of the classical notion of the index of a singular point of a vector field to the case of vector fields or 1-forms on singular varieties, describe relations between them and formulae for their computation.

Introduction An isolated singular point (zero) of a (continuous) vector field on a smooth manifold has a well known invariant — the index. A neighbourhood of a point P in a manifold M n of dimension n can be identified with a neighbourhood of the origin in the affine (coordinate) space Rn . A germ X of a vector field n ∂ Xi (x) ∂x . Let Bεn (0) be the ball of on (Rn , 0) can be written as X(x) = i i=1

radius ε centred at the origin in Rn such that the vector field X is defined in a neighbourhood of this ball and has no zeros in it except at the origin. Let Sεn−1 (0) be the (n − 1)-dimensional sphere ∂Bεn (0). The vector field X defines a map X : Sεn−1 (0) → S1n−1 . "X" The index indP X of the vector field X at the point P is defined as the degree X . One can see that it is independent of the chosen coordinates. of the map X If the vector field and its singular point P is non-degenerate, ' X is smooth ( ∂Xi i.e. if JX,P := det ∂xj (0) = 0, then indP X = sgn JX,P , i.e. it is equal to 1 if JX,P > 0 and is equal to −1 if JX,P < 0. The index of an arbitrary isolated singular point P of a smooth vector field X is equal to the number of non-degenerate singular points which split from the point P under a generic of the vector field X in a neighbourhood of the point P counted deformation X with the corresponding signs (sgn JX, ). This follows, in particular, from the P fact that the index of a vector field satisfies the law of conservation of number (see below).

130

W. Ebeling and S. M. Gusein-Zade

One of the most important properties of the index of a vector field is the Poincaré–Hopf theorem. Suppose that the manifold M is closed, i.e. compact without boundary, and that the vector field X has finitely many singular points on it. Theorem 1 (Poincar´ e-Hopf ). The sum indP X P ∈Sing X

of indices of singular points of the vector field X is equal to the Euler characteristic χ(M ) of the manifold M . If M is an n-dimensional complex analytic (compact) manifold, then its Euler characteristic χ(M ) is equal to the characteristic number cn (T M ), [M ], where cn (M ) is the top Chern class of the manifold M . If a vector field X on a complex manifold M is holomorphic and a singular point P of it is nondegenerate, then the index indP X is equal to +1. The index of an isolated singular point P of a holomorphic vector field X is positive. It is equal to the number of non-degenerate singular points which split from the point P under a generic deformation of the vector field X in a neighbourhood of the point P. There exists an algebraic formula for the index indP X of an isolated singular point of a holomorphic vector field. In local coordinates centred at the n ∂ Xi (x) ∂x . Let OCn ,0 be the ring point P , let the vector field X be equal to i i=1

of germs of holomorphic functions of n variables. Theorem 2. indP X = dimC OCn ,0 /(X1 , . . . , Xn ) , where (X1 , . . . , Xn ) is the ideal generated by the germs X1 , . . . , Xn . This statement was known for a long time, however, it seems that the first complete proof appeared in [Pa67]. An algebraic formula for the index indP X of an isolated (in fact for an algebraically isolated: see the definition below) singular point of a smooth (say, C ∞ ) vector field was given in [EL77, Kh77]. In local coordinates centred n ∂ Xi (x) ∂x . Let ERn ,0 be the ring of germs of smooth at the point P , let X = i i=1

(C ∞ ) functions of n variables. Definition 1. The singular point P of the vector field v is algebraically isolated if the factor ring ERn ,0 /(X1 , . . . , Xn ) has a finite dimension as a vector space. If a singular point of a vector field is algebraically isolated, then it is isolated. If, in the local coordinates x1 , . . . , xn , the vector field X is (real)

Indices of Vector Fields

131

analytic (i.e., if the components Xi are analytic functions of x1 , . . . , xn ), one can consider its complexification XC which is a holomorphic vector field in a neighbourhood of the origin in Cn . In this case the singular point of the vector field X is algebraically isolated if and only if the singular point of the vector field XC (the origin) is isolated. Let ∂Xi JX = JX (x) := det ∂xj be the Jacobian of the vector field X. One can prove that JX = 0 in the local ring RX = ERn ,0 /(X1 , . . . , Xn ). Moreover, the ring RX has a one dimensional minimal ideal and this ideal is generated by the Jacobian JX . Let : RX → R be a linear function on the ring RX (considered as a vector space) such that (JX ) > 0. Consider the quadratic form Q on RX defined by Q(ϕ, ψ) = (ϕ · ψ) . Theorem 3 (Eisenbud-Levine-Khimshiashvili). The index indP X of the singular point P of the vector field X is equal to the signature sgn Q of the quadratic form Q. For a proof of this theorem see also [AGV85]. In what follows we shall, in particular, discuss generalizations of the notion of the index of a vector field to real or complex analytic variety with singularities and problems of their computation.

1 1-Forms Versus Vector Fields Traditionally the definition of the index and the corresponding results (say, the Poincaré–Hopf and the Eisenbud–Levine–Khimshiashvili theorems) are formulated for vector fields. However, instead of vector fields one can consider 1-forms. Using a Riemannian metric one can identify vector fields and 1-forms on a smooth (C ∞ ) manifold. Therefore on a smooth manifold all notions and statements concerning vector fields can be formulated for 1-forms as well. In particular, the index indP ω of a 1-form ω on a smooth n-dimensional manifold M at an isolated singular point (zero) P is defined (in local coordinates) as ω : Sεn−1 → S1n−1 , where S1n−1 is the unit sphere the degree of the map ω in the dual space. In the local complex analytic situation one can identify germs of complex analytic vector fields and germs of complex analytic 1∂ corresponds to the 1forms using local coordinates: the vector field Xi ∂z i form Xi dzi . In the Poincaré–Hopf theorem for 1-forms on an n-dimensional complex manifold M the sum of indices of singular points of a 1-form is equal to cn (T ∗ M ), [M ] = (−1)n χ(M ). The only essential difference between vector fields and 1-forms in the smooth complex analytic case is that non-trivial complex analytic (global) vector fields and 1-forms exist, generally speaking,

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on different complex analytic manifolds: in the odd-dimensional case vector fields (respectively 1-forms) can exist only on manifolds with non-negative (respectively non-positive) Euler characteristic. However, for the case of singular varieties the situation becomes quite different. In a series of papers instead of considering vector fields on a variety we started to consider 1-forms. For 1-forms on germs of singular varieties (real or complex analytic) one can define several notions of indices usually corresponding to appropriate analogues for vector fields. However, the properties of the indices for vector fields and for 1-forms are different. For example, N ∂ Xi ∂x on a real variety just as above, to an analytic vector field X = i i=1

(V, 0) ⊂ (RN , 0) (or on a complex analytic variety (V, 0) ⊂ (CN , 0)) one can N associate the 1-form ω = Xi dxi (dependent on the choice of coordinates i=1

x1 , . . . , xN in (RN , 0) or in (CN , 0)). If the vector field X has an isolated singular point on (V, 0), then, for a generic (!) choice of the coordinates x1 , . . . , xN , the corresponding 1-form ω has an isolated singular point as well. This correspondence does not work in the other direction. Moreover, whereas for the radial index on a real analytic variety (see below) the index of a vector field coincides with that of the corresponding 1-form, this does not hold for other indices, say, for the so called GSV index (on an isolated complete intersection singularity: ICIS). In some cases one can say that the notion of an index of a 1-form (say, of a holomorphic one) is somewhat more natural than that of a vector field. For example, the notion of the GSV index of a 1-form (see below) is “more complex analytic” (does not use the complex conjugation for the definition) and “more geometric” (uses only objects of the same tensor type). Moreover the GSV index of an isolated singular point of a holomorphic 1-form on a complex ICIS can be described as the dimension of an appropriate algebra. Finally, on a real ICIS the real index of a 1-form which is the differential of a germ of a function with an algebraically isolated singular point plus-minus the Euler characteristic of a (real) smoothing of the ICIS can be expressed in terms of the signature of a quadratic form on a space (certain space of thimbles), the dimension of which is equal to the (complex) index of the corresponding complexification. The condition for a vector field to be tangent to a germ of a singular variety is a very restrictive one. For example, holomorphic vector fields with isolated zeros exist on a germ of a complex analytic variety with an isolated singularity [BG94], but not in general. Example 1. Consider the surface X in C3 given by the equation xy(x − y)(x + zy) = 0. It has singularities on the line {x = y = 0}. It can be considered as a family of four lines in the (x, y)-plane with different cross ratios. Then any holomorphic vector field tangent to X vanishes on the line {x = y = 0}, because the


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translation along such a vector field has to preserve the cross ratio of the lines. However, 1-forms with isolated singular points always exist. The idea to consider indices of 1-forms instead of indices of vector fields (in some situations) was first formulated by V.I.Arnold ([Ar79], see also [Ar04]).

2 Radial Index One can say that the notion of the radial index of a vector field or of a 1-form on a singular variety is a straightforward generalization of the usual index inspired by the Poincaré–Hopf theorem. Let us start with the setting of (real: C ∞ ) manifolds with isolated singularities. A manifold with isolated singularities is a topological space M which has the structure of a smooth (say, C ∞ –) manifold outside of a discrete set S (the set of singular points of M ). A diffeomorphism between two such manifolds is a homeomorphism which sends the set of singular points onto the set of singular points and is a diffeomorphism outside of them. We say that M has a conelike singularity at a (singular) point P ∈ S if there exists a neighbourhood of the point P diffeomorphic to the cone over a smooth manifold WP (WP is called the link of the point P ). In what follows we assume all manifolds to have only cone-like singularities. A (smooth or continuous) vector field or 1form on a manifold M with isolated singularities is a (smooth or continuous) vector field or 1-form respectively on the set M \ S of regular points of M . The set of singular points SX of a vector field X or the set of singular points Sω of a 1-form ω on a (singular) manifold M is the union of the set of usual singular points of X or of ω respectively on M \ S (i.e., points at which X or ω respectively tends to zero) and of the set S of singular points of M itself. For an isolated usual singular point P of a vector field X or of a 1-form ω there is defined its index indP X or indP ω respectively. If the manifold M is closed and has no singularities (S = ∅) and the vector field X or the 1-form ω on M has only isolated singularities, then the sum of these indices over all singular points is equal to the Euler characteristic χ(M ) of the manifold M . Let (M, P ) be a cone-like singularity (i.e., a germ of a manifold with such a singular point). Let X be a vector field defined on an open neighbourhood U of the point P . Suppose that X has no singular points on U \ {P }. Let V be a closed cone–like neighbourhood of the point P in U (V ∼ = CWP , V ⊂ U ). On the cone CWP = (WP × I)/(WP × {0}) (I = [0, 1]), there is defined a natural vector field ∂/∂t (t is the coordinate on I). Let Xrad be the corresponding be a (continuous) vector field on U which coincides vector field on V . Let X with X near the boundary ∂U of the neighbourhood U and with Xrad on V and has only isolated singular points.

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Definition 2. The radial index indrad (X; M, P ) (or simply indrad X) of the vector field X at the point P is equal to 1+ indP X ∈S " \{P } P X

except P itself ). (the sum is over all singular points P of the vector field X Analogously, we define the radial index of a 1-form ω defined on an open neighbourhood U of the point P which has no singular points on U \ {P }. Let V again be a closed cone–like neighbourhood of P in U (V ∼ = CWP , V ⊂ U ). On the cone CWP = (WP × I)/(WP × {0}) (I = [0, 1]) there is defined a natural 1-form dt. Let ωrad be the corresponding 1-form on V . Let ω be a (continuous) 1-form on U which coincides with ω near the boundary ∂U of the neighbourhood U and with ωrad on V and has only isolated singular points. Definition 3. The radial index indrad (ω; M, P ) (or simply indrad ω) of the 1-form ω at the point P is equal to 1+ indP ω ∈Sω P \{P }

(the sum is over all singular points P of the 1-form ω except P itself ). For a cone-like singularity at a point P ∈ S, the link WP and thus the cone structure of a neighbourhood are, generally speaking, not well-defined (cones over different manifolds may be locally diffeomorphic). However it is not difficult to show that the indices indrad X and indrad ω do not depend on the choice of a cone structure on a neighbourhood and on the choice of the and the 1-form ω vector field X respectively. Example 2. The index of the radial vector field Xrad and of the radial 1-form ωrad is equal to 1. The index of the vector field (−Xrad ) and of the 1-form (−ωrad ) is equal to 1 − χ(WP ) where WP is the link of the singular point P . Proposition 1. For a vector field X or a 1-form ω with isolated singular points on a closed manifold M with isolated cone-like singularities, the statement of Theorem 1 holds. Now let (V, 0) ⊂ (RN , 0) be the germ of a real analytic variety,generally q speaking, with a non-isolated singular point at the origin. Let V = i=1 Vi be a Whitney stratification of the germ (V, 0). Let X be a continuous vector field on (V, 0) (i.e., the restriction of a continuous vector field on (RN , 0) tangent to V at each point) which has an isolated zero at the origin (on V ). (Tangency to the stratified set V means that for each point Q ∈ V the vector X(Q) is tangent to the stratum Vi which contains the point Q.) Let Vi be a stratum


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of dimension k and let Q be a point of Vi . A neighbourhood of the point Q in V is diffeomorphic to the direct product of a linear space Rk and the cone CWQ over a compact singular analytic variety WQ . (Here a diffeomorphism between two stratified spaces is a homeomorphism which is a diffeomorphism on each stratum.) Let ε > 0 be small enough so that in the closed ball Bε of radius ε centred at the origin in RN the vector field X has no zeros on V \{0}. on It is not difficult to show that there exists a (continuous) vector field X (V, 0) such that: is defined on V ∩ Bε . 1. The vector field X 2. X coincides with the vector field X in a neighbourhood of V ∩Sε in V ∩Bε (Sε is the sphere ∂Bε ). has only a finite number of zeros. 3. The vector field X 4. Each point Q ∈ U (0) with X(Q) = 0 has a neighbourhood diffeomorphic k to (R , 0) × CWQ in which X(y, z) (y ∈ Rk , z ∈ CWQ ) is of the form Y (y) + Xrad (z), where Y is a germ of a vector field on (Rk , 0) with an isolated singular point at the origin, Xrad is the radial vector field on the cone CWQ . on V ∩ Bε . For a point Let SX be the set of zeros of the vector field X k " Q ∈ SX , let ind(Q) := ind (Y ; R , 0), where Y is the vector field on (Rk , 0) described above. If k = 0 (this happens at the origin if it is a stratum of the " := 1. stratification {Vi }), we set ind(Q) Definition 4. The radial index indrad (X; V, 0) of the vector field X on the variety V at the origin is the number " ind(Q). Q∈SX "

One can say that the idea of this definition goes back to M.-H. Schwartz who defined this index for so called radial vector fields [Schw65, Schw86a, Schw86b]; cf. also [KT99]. It was used to define characteristic classes for singular varieties, see e.g. the surveys [Br00, Se02]. Let ω be an (arbitrary continuous) 1-form on a neighbourhood of the origin in RN with an isolated singular point on (V, 0) at the origin. (A point Q ∈ V is a singular point of the 1-form ω on V if it is singular for the restriction of the 1-form ω to the stratum Vi which contains the point Q; if the stratum Vi is zero-dimensional, the point Q is always singular.) Let ε > 0 be small enough so that in the closed ball Bε of radius ε centred at the origin in RN the 1-form ω has no singular points on V \ {0}. We say that a 1-form ω on a germ (W, 0) is radial if for any analytic curve γ : (R, 0) → (W, 0) different from the trivial one (γ(t) ≡ 0) the value ω(γ(t)) ˙ of the 1-form ω on the tangent vector to the curve γ is positive for positive t small enough. It is easy to see that there exists a 1-form ω on RN such that:

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1. The 1-form ω coincides with the 1-form ω on a neighbourhood of the sphere Sε = ∂Bε . has only a finite number of zeros; 2. The vector field X 3. In a neighbourhood of each singular point Q ∈ (V ∩ Bε ) \ {0}, Q ∈ Vi , looks as follows. There exists a (local) analytic dim Vi = k, the 1-form ω diffeomorphism h : (RN , Rk , 0) → (RN , Vi , Q) such that h∗ ω = π1∗ ω 1 + ∗ N 2 , where π1 and π2 are the natural projections π1 : R → Rk and π2 ω 1 is the germ of a 1-form on (Rk , 0) π2 : RN → RN −k respectively, ω with an isolated singular point at the origin, and ω 2 is a radial 1-form on (RN −k , 0). Remark 1. One can demand that the 1-form ω 1 has a non-degenerate singular point (and therefore ind ( ω1 , Rk , 0) = ±1), however, this is not necessary for the definition. Let Sω be the set of singular points of the 1-form ω on V ∩ Bε . For a point k " " ω1 ; R , 0). If k = 0, we set ind(Q) := 1. Q ∈ Sω , let ind(Q) := ind ( Definition 5. The radial index indrad (ω; V, 0) of the 1-form ω on the variety V at the origin is the sum " ind(Q). Q∈Sω

One can show that these notions are well defined (for the radial index of a 1-form see [EG05a]). Just because of the definition, the (radial) index satisfies the law of conservation of number. For a 1-form this means the following: if a 1-form ω with isolated singular points on V is close to the 1-form ω, then indrad (ω ; V, 0) indrad (ω; V, 0) = Q∈Sing ω

where the sum on the right hand side is over all singular points Q of the 1-form ω on V in a neighbourhood of the origin. The radial index generalizes the usual index for vector fields or 1-forms with isolated singularities on a smooth manifold. In particular one has a generalization of the Poincaré-Hopf theorem: Theorem 4 (Poincar´ e-Hopf ). For a compact real analytic variety V and a vector field X or a 1-form ω with isolated singular points on V , one has indrad (X; V, Q) = indrad (ω; V, Q) = χ(V ) Q

Q

where χ(V ) denotes the Euler characteristic of the set (variety) V .


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In the case of radial vector fields, this theorem is due to M.-H. Schwartz [Schw65, Schw86a, Schw86b, Schw91] (see also [BS81, ASV98, KT99]). For a proof of this theorem for the case of 1-forms see [EG05a]. Now let (V, 0) ⊂ (CN , 0) be the germ of a complex analytic variety of pure dimension n. Let ω be a (complex and, generally speaking, continuous) 1-form on a neighbourhood of the origin in CN . In fact there is a one-to-one correspondence between complex 1-forms on a complex manifold M n (say, on CN ) and real 1-forms on it (considered as a real 2n-dimensional manifold). Namely, to a complex 1-form ω one associates the real 1-form η = Re ω; the 1-form ω can be restored from η by the formula ω(v) = η(v) − iη(iv) for v ∈ Tx M n . This means that the index of the real 1-form Re ω is an invariant of the complex 1-form ω itself. However, on a smooth manifold indM n ,x Re ω does not coincide with the usual index of the singular point x of the 1-form ω, but differs n 1from it by the ncoefficient (−1) . (E.g., the index of the (complex analytic) n x dx ((x , . . . , x ) being the coordinates of C ) is equal form ω = j j 1 n j=1 n n to 1, whence the index of the real 1-form Re ω = j=1 uj duj − j=1 vj dvj (xj = uj + ivj ) is equal to (−1)n .) This explains the following definition. Definition 6. The (complex radial) index indC rad (ω; V, 0) of the complex 1form ω on an n-dimensional variety V at the origin is (−1)n times the index of the real 1-form Re ω on V : n indC rad (ω; V, 0) = (−1) indrad (Re ω; V, 0).

3 GSV Index This generalization of the index makes sense for varieties which have isolated complete intersection singularities (in particular, for hypersurfaces with isolated singularities). Let V ⊂ (Cn+k , 0) be an n-dimensional isolated complete intersection singularity (ICIS) defined by equations f1 = . . . = fk = 0 (i.e. V = f −1 (0), where f = (f1 , . . . , fk ) : (Cn+k , 0) → (Ck , 0), dim V = n) and let X = n+k Xi ∂z∂ i be a germ of a (continuous) vector field on (Cn+k , 0) tangent to the i=1

ICIS V . (The latter means that X(z) ∈ Tz V for all z ∈ V \ {0}.) Suppose that the vector field X does not vanish on V in a punctured neighbourhood of the origin. In this situation the following index (called the GSV index after X. G´ omez-Mont, J. Seade, and A. Verjovsky) is defined. Let Bε ⊂ Cn+k be the ball of radius ε centred at the origin with (positive) ε small enough so that all the functions fi (i = 1, . . . , k) and the vector field X are defined in a neighbourhood of Bε , V is transversal to the sphere Sε = ∂Bε , and the vector field X has no zeros on V inside the ball Bε except (possibly) at the origin. Let K = V ∩ Sε be the link of the ICIS (V, 0). The manifold K is (2n − 1)-dimensional and has a natural orientation as the boundary of the complex manifold V ∩ Bε \ {0}.

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Let M(p, q), p ≥ q, be the space of p×q matrices with complex entries and let Dp,q be the subspace of M(p, q) consisting of matrices of rank less than q. The subset Dp,q is an irreducible subvariety of M(p, q) of codimension p − q + 1. The complement Wp,q = M(p, q) \ Dp,q is the Stiefel manifold of q-frames (collections of q linearly independent vectors) in Cp . It is known that the Stiefel manifold Wp,q is 2(p − q)-connected and H2(p−q)+1 (Wp,q ) ∼ = Z (see, e.g., [Hu75]). The latter fact also proves that Dp,q is irreducible. Since Wp,q is the complement of an irreducible complex analytic subvariety of codimension p − q + 1 in M(p, q) ∼ = Cpq , there is a natural choice of a generator of the homology group H2(p−q)+1 (Wp,q ) ∼ = Z. Namely, the (”positive”) generator is the boundary of a small ball in a smooth complex analytic slice transversal to Dp,q at a non-singular point. Therefore a map from an oriented smooth (C ∞ ) closed (2(p − q) + 1)-dimensional manifold to Wp,q has a degree. Define the gradient vector field grad fi of a function germ fi by ∂fi ∂fi grad fi = ,..., ∂z1 ∂zn+k (grad fi depends on the choice of the coordinates z1 , . . . , zn+k ). One has a map (1) Ψ = (X, grad f1 , . . . , grad fk ) : K → Wn+k,k+1 from the link K to the Stiefel manifold Wn+k,k+1 . Definition 7. The GSV index indGSV (X; V, 0) of the vector field X on the ICIS V at the origin is the degree of the map Ψ : K → Wn+k,k+1 . Remark 2. Note that one uses the complex conjugation for this definition and the components of the discussed map are of different tensor nature. Whereas X is a vector field, grad fi is more similar to a covector. This index was first defined in [GSV91] for vector fields on isolated hypersurface singularities. In [SS96] it was generalized to vector fields on ICIS. It is convenient to consider the map Ψ as a map from V to M(n + k, k + 1) defined by the formula (1) (in a neighbourhood of the ball Bε ). It maps the complement of the origin in V to the Stiefel manifold Wn+k,k+1 . This description leads to the following definition of the GSV index. Proposition 2. The GSV index indGSV (X; V, 0) of the vector field X on the ICIS V at the origin is equal to the intersection number (Ψ (V ) ◦ Dn+k,k+1 ) of the image Ψ (V ) of the ICIS V under the map Ψ and the variety Dn+k,k+1 at the origin. Note that, even if the vector field X is holomorphic, Ψ (V ) is not, generally speaking, a complex analytic variety.


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Another definition/description of the GSV index indGSV (X; V, 0) can be given in the following way. Let V = Vt = f −1 (t) ∩ Bε , where t ∈ Ck , 0 < "t" # ε, be the Milnor fibre of the ICIS V , i.e. a smoothing of it. For t small enough in a neighbourhood of the sphere Sε = ∂Bε the manifolds V = V0 and Vt ”almost coincide”. This gives an up to isotopy well defined vector field X on the manifold V in a neighbourhood of its boundary f −1 (t) ∩ Sε (generally speaking, not a complex analytic one). Let us extend this vector field to a on the entire manifold V with only isolated vector field (also denoted by X) zeros. Proposition 3. One has indGSV (X; V, 0) =

indQ X.

Q∈Sing X

This definition can be easily generalized to a germ of a complex analytic variety with an isolated singularity and with a fixed smoothing. For example, J. Seade defined in this way an index for a singular point of a vector field on a complex analytic surface with a normal smoothable Gorenstein singularity [Se87]. For a singularity which is not an ICIS, however, it is possible that such a smoothing does not exist or there may be a number of different smoothings what leads to the situation that the index is not well defined. These difficulties cannot be met for curve singularities. Thus in this situation the corresponding index is well defined (cf. [Gor00]). For a general variety with an isolated singularity, this ambiguity can be avoided for germs of functions (i.e. for their differentials in our terms) by defining the corresponding index as a certain residue, what is done in [IS03]. Now let V be a compact (say, projective) variety all singular points of which are local ICIS and let X be a vector field on V with isolated singular points. One has the following statement. Proposition 4. One has

indGSV (ω; V, Q) = χ(V ),

Q∈Sing ω

where V is a smoothing of the variety (local complete intersection) V . A similar construction can be considered in the real setting. Namely, let V = f −1 (0) ⊂ (Rn+k , 0) be a germ of a real n-dimensional ICIS, f = (f1 , . . . , fk ) : (Rn+k , 0) → (Rk , 0) is a real analytic map, and let n+k Xi dzi be a germ of a (continuous) vector field on (Rn+k , 0) tangent X= i=1

to the variety V outside of the origin. Just as above one defines a map Ψ from R the link K = V ∩ Sεn+k−1 of the ICIS (V, 0) to the Stiefel manifold Wn+k,k+1 n+k R of (k + 1)-frames in R . The Stiefel manifold Wn+k,k+1 is (n − 2)-connected

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R and its first non-trivial homology group Hn−1 (Wn+k,k+1 ; Z) is isomorphic to the group Z of integers for n odd and to the group Z2 of order 2 for n even. Therefore the map Ψ has a degree defined as an integer or as an integer modulo 2 depending on the parity of the dimension n. If the manifold V \ {0} is not connected, the construction can be applied to each connected component of it giving a set of degrees (a multi-degree). This invariant (called real GSV index of the vector field X on the real ICIS V ) was introduced and studied in [ASV98]. In [EG01, EG03a] the notion of the GSV index was adapted to the case of a 1-form. Let ω = Ai dxi (Ai = Ai (x)) be a germ of a continuous 1-form on (Cn+k , 0) which as a 1-form on the ICIS V has (at most) an isolated singular point at the origin (thus it does not vanish on the tangent space TP V to the variety V at all points P from a punctured neighbourhood of the origin in V ). The 1-forms ω, df1 , . . . , dfk are linearly independent for all P ∈ K. Thus one has a map Ψ = (ω, df1 , . . . , dfk ) : K → Wn+k,k+1 .

Here Wn+k,k+1 is the Stiefel manifold of (k + 1)-frames in the space dual to Cn+k . Definition 8. The GSV index indGSV (ω; V, 0) of the 1-form ω on the ICIS V at the origin is the degree of the map Ψ : K → Wn+k,k+1 . Just as above Ψ can be considered as a map from the ICIS V to the space M(n + k, k + 1) of (n + k) × (k + 1) matrices. If the 1-form ω is holomorphic, the map Ψ and the set Ψ (V ) are complex analytic. The obvious analogues of Propositions 2, 3, and 4 hold. There exists an algebraic formula for the index indGSV (ω; V, 0) of a holomorphic 1-form ω on an ICIS V which gives it as the dimension of a certain algebra (see Sect. 6). In [BSS05a], there was defined a generalization of the notion of the GSV index for a 1-form on a complete intersection singularity V = f −1 (0) ⊂ (Cn+k , 0) if the map f = (f1 , . . . , fk ) : (Cn+k , 0) → (Ck , 0), defining the singularity satisfies Thom’s af regularity condition. (This condition implies that the Milnor fibre of the complete intersection singularity V is well-defined.)

4 Homological Index Let (V, 0) ⊂ (CN , 0) be any germ of an analytic variety of pure dimension n with an isolated singular point at the origin. Suppose X is a complex analytic vector field tangent to (V, 0) with an isolated singular point at the origin. k be be the module of germs of differentiable k-forms on (V, 0), Let ΩV,0 i.e. the factor of the module ΩCkN ,0 of k-forms on (CN , 0) by the submodule


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generated by f · ΩCkN ,0 and df ∧ ΩCk−1 N ,0 for all f from the ideal of functions • vanishing on (V, 0). Consider the Koszul complex (ΩV,0 , X): X

X

X

1 n 0 ←− OV,0 ←− ΩV,0 ←− ... ←− ΩV,0 ←− 0 ,

where the arrows are given by contraction with the vector field X. The sheaves i • ΩV,0 are coherent sheaves and the homology groups of the complex (ΩV,0 , X) are concentrated at the origin and therefore are finite dimensional. The following definition is due to X. G´ omez-Mont [Góm98]. Definition 9. The homological index indhom (X; V, 0) = indhom X of the vector field X on (V, 0) is the Euler characteristic of the above complex: indhom (X; V, 0) = indhom X =

n • (−1)j hj (ΩV,0 , X) ,

(2)

j=0 • where hj (ΩV,0 , X) is the dimension of the corresponding homology group as a vector space over C.

The homological index satisfies the law of conservation of number [G´ om98, Theorem 1.2]. In the case when V is a hypersurface, G´ omez-Mont has shown that the homological index is equal to the GSV index. Given a holomorphic 1-form ω on (V, 0) with an isolated singularity, we • , ∧ω): consider the complex (ΩV,0 ∧ω

∧ω

∧ω

1 n −→ ... −→ ΩV,0 −→ 0 , 0 −→ OV,0 −→ ΩV,0

where the arrows are given by the exterior product by the form ω. This complex is the dual of the Koszul complex considered above. It was used by G.M. Greuel in [Gr75] for complete intersections. In [EGS04] the definition of G´ omez-Mont was adapted to this case. Definition 10. The homological index indhom (ω; V, 0) = indhom ω of the 1form ω on (V, 0) is (−1)n times the Euler characteristic of the above complex: indhom (ω; V, 0) = indhom ω =

n • (−1)n−j hj (ΩV,0 , ∧ω) ,

(3)

j=0 • where hj (ΩV,0 , ∧ω) is the dimension of the corresponding homology group as a vector space over C.

In [EGS04], there was proved the following statement. Theorem 5. Let ω be a holomorphic 1-form on V with an isolated singularity at the origin 0.

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(i) If V is smooth, then indhom ω equals the usual index of the holomorphic 1-form ω. (ii) The homological index satisfies the law of conservation of number: if ω is a holomorphic 1-form on V close to ω, then: indhom (ω ; V, x) , indhom (ω; V, 0) = indhom (ω ; V, 0) + where the sum on the right hand side is over all those points x in a small punctured neighbourhood of the origin 0 in V where the form ω vanishes. (iii) If (V, 0) is an isolated complete intersection singularity, then the homological index indhom ω coincides with the GSV index indGSV ω. Statement (ii) follows from [GG02]. Remark 3. We notice that one has an invariant for functions on (V, 0) with an isolated singularity at the origin defined by f → indhom df . By the theorem above, if (V, 0) is an isolated complete intersection singularity, this invariant counts the number of critical points of the function f on a Milnor fibre of the ICIS V . ¯ ¯0) be its normalization. Remark 4. Let (C, 0) be a curve singularity and let (C, 1 ), λ = dim ω /c(Ω Let τ = dim Ker(Ω1C,0 → Ω1C, C,0 ¯ ¯ C,0 ), where ωC,0 is the 0 1 dualizing module of Grothendieck, c : ΩC,0 → ωC,0 is the class map (see [BG80]). In [MvS01] there is considered a Milnor number of a function f on a curve singularity introduced by V.Goryunov [Gor00]. One can see that this Milnor number can be defined for a 1-form ω with an isolated singularity on (C, 0) as well (as dim ωC,0 /ω ∧ OC,0 ) and is equal to indhom ω + λ − τ . The laws of conservation of numbers for the homological and the radial indices of 1-forms together with the fact that these two indices coincide on smooth varieties imply that their difference is a locally constant, and therefore constant, function on the space of 1-forms on V with isolated singularities at the origin. Therefore one has the following statement. Proposition 5. Let (V, 0) be a germ of a complex analytic space of pure dimension n with an isolated singular point at the origin. Then the difference indhom ω − indrad ω between the homological and the radial index does not depend on the 1-form ω. If (V, 0) is an ICIS, this difference is equal to the Milnor number of (V, 0). Together with this fact proposition 5 permits to consider the difference ν(V, 0) := indhom (ω; V, 0) − indrad (ω; V, 0) as a generalized Milnor number of the singularity (V, 0).


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There are other invariants of isolated singularities of complex analytic varieties which coincide with the Milnor number for isolated complete intersection singularities. One of them is (−1)n times the reduced Euler characteristic (i.e., the Euler characteristic minus 1) of the absolute de Rham complex of (V, 0): d

d

d

1 n −→ . . . −→ ΩV,0 −→ 0 : 0 −→ OV,0 −→ ΩV,0

χ(V, ¯ 0) :=

n

• (−1)n−i hi (ΩV,0 , d) − (−1)n .

i=0

In [EGS04] it was shown that: Theorem 6. One has ν(V, 0) = χ(V, ¯ 0) if (i) (V, 0) is a curve singularity, (ii) (V, 0) ⊂ (Cd+1 , 0) is the cone over the rational normal curve in CPd . Statement (i) implies that the invariant ν(V, 0) is different from the Milnor number introduced by R.-O. Buchweitz and G.-M. Greuel [BG80] for curve singularities. Statement (ii) was obtained with the help of H.-Ch. Graf von Bothmer and R.-O. Buchweitz. For d = 4 this is Pinkham’s example [Pi74] of a singularity which has smoothings with different Euler characteristics.

5 Euler Obstruction The idea of the Euler obstruction emerged from [Mac74] where the Euler obstruction of a singular point of a complex analytic variety was defined (the definition was formulated in terms of obstruction theory in [BS81]). The Euler obstruction of an isolated singular point (zero) of a vector field on a (singular) complex analytic variety was essentially defined in [BMPS04] (though formally speaking there it was defined only for holomorphic functions, i.e. for corresponding gradient vector fields). The main result of [BMPS04] (see Theorem 7 below) gives a relation between the Euler obstruction of a function germ on a singular variety V and the Euler obstructions of singular points of V itself. In [BMPS04], there is introduced the notion of the local Euler obstruction of a holomorphic function with an isolated critical point on the germ of a complex analytic variety. It is defined as follows. n-dimensional complex anaLet (V, 0) ⊂ (CN , 0) be the germ of a purely q lytic variety with a Whitney stratification V = i=1 Vi and let f be a holomorphic function defined in a neighbourhood of the origin in CN with an isolated singular point on V at the origin. Let ε > 0 be small enough such that the function f has no singular points on V \ {0} inside the ball Bε . Let

144


grad f be the gradient vector field of f as defined in Sect. 3. Since f has no singular points on V \ {0} inside the ball Bε , the angle of grad f (x) and the tangent space Tx Vi to a point x ∈ Vi \{0} is less than π/2. Denote by ζi (x) = 0 the projection of grad f (x) to the tangent space Tx Vi . Following the construction in [BMPS04], the vector fields ζi can be glued together to obtain a stratified vector field gradV f on V such that gradV f is homotopic to the restriction of grad f to V and satisfies gradV f (x) = 0 unless x = 0. Let ν : V) → V be the Nash transformation of the variety V defined as follows. Let G(n, N ) be the Grassmann manifold of n-dimensional vector subspaces of CN . For a suitable neighbourhood U of the origin in CN , there is a natural map σ : Vreg ∩U → U ×G(n, N ) which sends a point x to (x, Tx Vreg ) (Vreg is the non-singular part of V ). The Nash transform V) is the closure of the image Im σ of the map σ in U × G(n, N ). The Nash bundle T) over V) is a vector bundle of rank n which is the pullback of the tautological bundle on the Grassmann manifold G(n, N ). There is a natural lifting of the Nash transformation to a bundle map from the Nash bundle T) to the restriction of the tangent bundle T CN of CN to V . This is an isomorphism of T) and T Vreg ⊂ T CN over the regular part Vreg of V . The vector field gradV f gives rise to a section ζ) of the Nash bundle T) over the Nash transform V) without zeros outside of the preimage of the origin. Definition 11. [BMPS04] The local Euler obstruction EuV,0 f of the function f on V at the origin is the obstruction to extend the non-zero section ζ) from the preimage of a neighbourhood of the sphere Sε = ∂Bε to the preimage of its interior, more precisely its value (as an element of H 2n (ν −1 (V ∩ Bε ), ν −1 (V ∩ Sε )) ) on the fundamental class of the pair (ν −1 (V ∩ Bε ), ν −1 (V ∩ Sε )). The word local will usually be omitted. Remark 5. The local Euler obstruction can also be defined for the restriction to V of a real analytic function on R2N . In particular, one can take for f the squared distance on V to the origin. In this case, the invariant EuV,0 f is the usual local Euler obstruction EuV (0) of the variety (V, 0) defined in [Mac74, BS81, BLS00]. Denote by Mf = Mf,t0 the Milnor fibre of f , i.e. the intersection V ∩ Bε ∩ f −1 (t0 ) for a regular value t0 of f close to 0. In [BMPS04, Theorem 3.1] the following result is proved. Theorem 7. Let f : (V, 0) → (C, 0) have an isolated singularity at 0 ∈ V . Then * q + EuV (0) = χ(Mf ∩ Vi ) · EuV (Vi ) + EuV,0 f, i=1


145

where EuV (Vi ) is the value of the Euler obstruction of V at any point of Vi , i = 1, . . . , q. One can also define the local Euler obstruction of a stratified vector field on a germ of a complex analytic variety (V, 0) with a Whitney stratification [BSS05b]. Let Vi be a stratum of the Whitney stratification with 0 ∈ Vi and Xi be a vector field on Vi with an isolated singularity at 0. The Proportionality Theorem of [BS81, BSS05b] states that the local Euler obstruction of a radial extension X of Xi at 0 is equal to the local Euler obstruction EuV (0) of (V, 0) times the radial index indrad (X; V, 0) of X. In [EG05a] the definition of the local Euler obstruction of a function was adapted to the case of a 1-form. Let ω be a 1-form on a neighbourhood of the origin in CN with an isolated singular point on V at the origin. Let ε > 0 be small enough such that the 1-form ω has no singular points on V \ {0} inside the ball Bε . The 1-form ω gives rise to a section ω ) of the dual Nash bundle T)∗ over ) the Nash transform V without zeros outside of the preimage of the origin. Definition 12. The local Euler obstruction EuV,0 ω of the 1-form ω on V at the origin is the obstruction to extend the non-zero section ω ) from the preimage of a neighbourhood of the sphere Sε = ∂Bε to the preimage of its interior, more precisely its value (as an element of the cohomology group H 2n (ν −1 (V ∩ Bε ), ν −1 (V ∩Sε )) ) on the fundamental class of the pair (ν −1 (V ∩Bε ), ν −1 (V ∩ Sε )). Remark 6. The local Euler obstruction can also be defined for a real 1-form on the germ of a real analytic variety if the last one is orientable in an appropriate sense. Example 3. Let ω = df for the germ f of a holomorphic function on (CN , 0). Then EuV,0 df differs from the Euler obstruction Euf,V (0) of the function f by the sign (−1)n . The reason is that for the germ of a holomorphic function with an isolated critical point on (Cn , 0) one has Euf,V (0) = (−1)n µf (see [BMPS04, Remark 3.4]), whence EuV,0 df = µf (µf is the Milnor number of xn ) = x21 + . . . + x2n the obstruction Euf,V (0) the germ f ). E.g., for f (x1 , . . . , n n is the index of the vector field i=1 xi ∂/∂xi (which is equal to (−1) n ), but the obstruction EuV,0 df is the index of the (holomorphic) 1-form i=1 xi dxi which is equal to 1. The Euler obstruction of a 1-form satisfies the law of conservation of number (just as the radial index). Moreover, on a smooth variety the Euler obstruction and the radial index coincide. This implies the following statement (cf. Theorem 7). We set χ(Z) := χ(Z) − 1 and call it the reduced (modulo a point) Euler characteristic of the topological space Z (though, strictly speaking, this name is only correct for a non-empty space Z).

146


Proposition 6. Let (V, 0) ⊂ (CN , 0) have an isolated singular point at the origin and let : CN → C be a generic linear function. Then indrad (ω; V, 0) − EuV,0 ω = indrad (d; V, 0) = (−1)n−1 χ(M ), where M is the Milnor fibre of the linear function on V . In particular EuV,0 df = (−1)n (χ(M ) − χ(Mf )). An analogue of the Proportionality Theorem for 1-forms was proved in [BSS05a]. Now let (V, 0) ⊂ (CN , 0) bean arbitrary germ of an analytic variety with q a Whitney stratification V = i=0 Vi , where we suppose that V0 = {0}. For a stratum Vi , i = 0, . . . , q, let Ni be the normal slice in the variety V to the stratum Vi (dim Ni = dim V − dim Vi ) at a point of the stratum Vi and let ni be the radial index of a generic (non-vanishing) 1-form d on Ni : ni = (−1)dim Ni −1 χ(M|Ni ). In particular for an open stratum Vi of V , Ni is a point and ni = 1. The strata Vi of V are partially ordered: Vi ≺ Vj (we shall write i ≺ j) iff Vi ⊂ Vj and Vi = Vj ; i % j iff i ≺ j or i = j. For two strata Vi and Vj with i % j, let Nij be the normal slice of the variety Vj to the stratum Vi at a point of it (dim Nij = dim Vj −dim Vi , Nii is a point) and let nij be the index of a generic 1-form d on Nij : nij = (−1)dim Nij −1 χ(M|Nij ), nii = 1. Let us define the Euler obstruction EuY,0 ω to be equal to 1 for a zero-dimensional variety Y (in particular EuV0 ,0 ω = 1, EuNii ,0 ω = 1). In [EG05a], the following statement was proved. Theorem 8. One has indrad (ω; V, 0) =

q

ni · EuVi ,0 ω.

i=0

To write an ”inverse” of the formula of Theorem 8, suppose that the variety V is irreducible and V = Vq . (Otherwise one can permit V to be reducible, but also permit the open stratum Vq to be not connected and dense; this does obius) inverse of the not change anything in Theorem 8.) Let mij be the (M¨ function nij on the partially ordered set of strata, i.e. nij mjk = δik . ijk

For i ≺ j one has mij =

(−1)r nk0 k1 nk1 k2 . . . nkr−1 kr

i=k0 ≺k1 ≺...≺kr =j

= (−1)dim Vj −dim Vi

i=k0 ≺...≺kr =j

χ(M|Nk

0 k1

) · . . . · χ(M|Nk

r−1 kr

).


147

Corollary 1. One has EuV,0 ω =

q

miq · indrad (ω; Vi , 0).

i=0

In particular EuV,0 df = (−1)dim V −1 × ⎛ q−1 ⎝χ(Mf |V ) + χ(Mf |V ) i

i=0

i=k0 ≺...≺kr =q

⎞ χ(M|Nk

0 k1

) . . . χ(M|Nk

r−1 kr

)⎠ .

In [LT81] the local Euler obstruction of a germ of a variety is related to polar invariants of the germ. A global version of the Euler obstruction of a variety and its connections with global polar invariants are discussed in [STV05a].

6 Algebraic, Analytic, and Topological Formulae for Indices In the introduction, there were mentioned algebraic formulae for the index of an analytic vector field or a 1-form on a smooth manifold: as the dimension of a ring in the complex setting and as the signature of a quadratic form in the real one. It is natural to try to look for analogues of such formulae for vector fields or 1-forms on singular varieties. This appeared to be a rather complicated problem. Other sorts of formulae: analytic (usually as certain residues) or topological ones are of interest as well. A substantial progress in this direction was achieved for the GSV index. Let (V, 0) be the germ of a hypersurface in (Cn+1 , 0) defined by a germ of a holomorphic function f : (Cn+1 , 0) → (C, 0) with an isolated singular point at 0. Let n+1 ∂ Xi X= ∂xi i=1 be a holomorphic vector field on Cn+1 tangent to V with an isolated (in (Cn+1 , 0)) zero at the origin. This implies that Xf = hf for some h ∈ OCn+1 ,0 . Define the following ideals in OCn+1 ,0 : Jf JX J1 J2 J3

:= := := := :=

(f1 , . . . , fn+1 ), (X1 , . . . , Xn+1 ), (h, JX ), (f, Jf ), (f, JX ).

X. G´ omez-Mont [G´ om98] has proved the following formula for the GSV index:

148


Theorem 9. For a vector field X with an isolated zero in the ambient space tangent to the germ of a hypersurface (V, 0) with an isolated singularity at 0, we have: for even n IndGSV (X; V, 0) = dim OCn+1 ,0 /JX + dim OCn+1 ,0 /J1 + dim OCn+1 ,0 /J2 , for n odd IndGSV (X; V, 0) = dim OCn+1 ,0 /J2 + dim OCn+1 ,0 /J3 . In this case the GSV index coincides with the homological index. O. Klehn [Kl03] generalized this formula for the homological index to the case when the vector field has an isolated zero on the hypersurface singularity (V, 0) but not omez-Mont necessarily in (CN , 0). Recently, H.-Ch. Graf von Bothmer, X. G´ and the first author gave formulae to compute the homological index in the case when V is a complete intersection [BEG05]. In [GM97, GM99] algebraic formulae for the index of a real analytic vector field with an algebraically isolated singular point at the origin tangent to a real hypersurface with an algebraically isolated singularity at 0 are derived. The index is expressed as the signature of a certain non-degenerate quadratic form for an even-dimensional hypersurface and as the difference of the signatures of two non-degenerate quadratic forms in the odd-dimensional case. O. Klehn [Kl05] proved that the GSV index coincides with the dimension of a certain explicitly constructed vector space in the case when (V, 0) is an isolated complete intersection singularity of dimension 1 and X is a vector field tangent to V with an isolated zero on V which is deformable in a certain way. He also obtained a signature formula for the real GSV index in the corresponding real analytic case generalizing the Eisenbud-Levine-Khimshiashvili formula. Let (V, 0) ⊂ (Cn+k , 0) be an isolated complete intersection singularity defined by an analytic map f = (f1 , ..., fk ) : (Cn+k , 0) → (Ck , 0) . Let X be a holomorphic vector field on Cn+k defined in a suitable neighbourhood U of the origin, tangent to V , with an isolated singular point at the origin. Let C be the k × k matrix whose entries are holomorphic functions on U such that X ·f = Cf . Assume that (x1 , . . . , xn+k ) is a system of coordinates on U such that when X is written as X=

n+k i=1

Xi

∂ , ∂xi

the sequence (X1 , . . . , Xn , f1 , . . . , fk ) is regular. Let J denote the Jacobian matrix ∂Xi J= . ∂xj


149

We denote by cq the coefficient at tq in the formal power series expansion of √ √ , -−1 −1 −1 J det Ik − t C det In+k − t 2π 2π

in t, where In+k and Ik denote the identity matrices of sizes n + k and k respectively. In [LSS95] the following formula for the GSV index is proved. Theorem 10. Let ε be small enough such that the real hypersurfaces |Xi | = ε are in general position and let Z be the set {f = 0, |Xi | = ε, 1 ≤ i ≤ n}. We assume that Z is oriented so that d(arg X1 ) ∧ d(arg X2 ) ∧ . . . ∧ d(arg Xn ) is positive. Then IndGSV (X; V, 0) = Z

cn dx1 ∧ . . . ∧ dxn

n . i=1 Xi

For generalizations of Theorem 10 and related results see [LS95, Su95, Su98, Su02, IS03]. Now let ω be a holomorphic 1-form on the ICIS (V, 0) = f −1 (0) ⊂ n+k , 0), f = (f1 , . . . , fk ) : (Cn+k , 0) → (Ck , 0), i.e. the restriction to (V, 0) (C of a holomorphic 1-form n+k Ai (x)dxi ω= i=1 n+k

on (C , 0). Assume that ω has an isolated singular point at the origin (on (V, 0)). Let I is the ideal generated by f1 , ..., fk and the (k+1)×(k+1)-minors of the matrix ⎛ ∂f ⎞ ∂f1 1 ∂x1 · · · ∂xn+k ⎜ . .. ⎟ ⎜ . ⎟ . ⎟. ⎜ . ··· ⎜ ∂fk ⎟ k ⎝ ∂x1 · · · ∂x∂fn+k ⎠ A1 · · · An+k Then one has the following formula for the GSV index [EG01, EG03a, EG03b]. Theorem 11. One has indGSV (ω; V, 0) = dim OCn+k ,0 /I. (Note that there is a minor mistake in the proof of this theorem in [EG03a] which is corrected in [EG05b].) This formula was obtained by Lê D.T. and G.-M. Greuel for the case when ω is the differential of a function ([Gr75, Lê74]). T. Gaffney [Ga05] described connections between the GSV index of a holomorphic 1-form on an ICIS and the multiplicity of pairs of certain modules.

150


In [EG06], there was constructed a quadratic form on the algebra A := OCn+k ,0 /I generalizing the Eisenbud-Levine-Khimshiashvili quadratic form defined for a smooth V . This is defined as follows. k M Let F : (Cn+k × CM ε , 0) → (C × Cε , 0) be a deformation of the map n+k k , 0) → (C , 0) (F (x, ε) = (fε (x), ε), f0 = f , fε = (f1ε , . . . , fkε )) f : (C and let ωε be a deformation of the form ω (defined in a neighbourhood of the M −1 origin in Cn+k × CM ε ) such that, for generic ε ∈ (Cε , 0), the preimage fε (0) is smooth and (the restriction of) the form ωε to it has only non degenerate singular points. Let Σ ⊂ (CM ε , 0) be the germ of the set of the values of the parameters ε , 0) such that either the preimage fε−1 (0) is singular or the restriction from (CM ε of the 1-form ωε to it has degenerate singular points. n Let Ai dy i be a 1-form on a smooth complex analytic manifold of dii=1

mension n (in local coordinates y 1 , '. . . , y(n ). At points P where the 1-form ω ∂Ai i j i defines the tensor vanishes, the Jacobian matrix J = ∂A ∂y j ∂y j dy ⊗ dy i,j

of type (0, 2) (i.e. a 'bilinear ( form on the tangent space). The determinant J ∂Ai of the matrix J = ∂yj is not a scalar (it depends on the choice of local coordinates). Under a change of coordinates it is multiplied by the square of the Jacobian of the change of coordinates (since the Jacobian matrix J is transformed to C T J C). Therefore it should be considered as the coefficient in the tensor J(dy 1 ∧ . . . ∧ dy m )⊗2 of type (0, 2m). In this sense the Jacobian of a 1-form is a sort of a ”quadratic ), one can divide this tensor by the tensor differential”. To get a number J(P square of a volume form. Let us fix volume forms on Cn+k and Ck , say, the standard ones σn+k = dx1 ∧ . . . ∧ dxn+k and σk = dz1 ∧ . . . ∧ dzk where x1 , . . . , xn+k and z1 , . . . , zk are Cartesian coordinates in Cn+k and in Ck respectively. There exists (at least locally) an n-form σε on Cn+k such that σn+k = f ∗ σk ∧ σε . Let ε ∈ Σ. The restriction of the form σε to the manifold fε−1 (0) is well defined and is a volume form on fε−1 (0). We also denote it by σε . Let P1 , . . . , Pν be the (nondegenerate) singular points of the 1-form ωε on the n-dimensional manifold / Σ and a germ ϕ ∈ OCn ,0 , let fε−1 (0). For ε ∈ R(ϕ, ε) :=

ν ϕ(Pi ) . Jε (Pi )

(4)

i=1

For a fixed ϕ the function Rϕ (ε) := R(ϕ, ε) is holomorphic in the complement of the bifurcation diagram Σ. In [EG06], it is proved that the function Rϕ (ε) has removable singularities on the bifurcation diagram Σ. This means that Rϕ (ε) can be extended to a holomorphic function on (CM ε , 0) (which we denote by the same symbol). Let


R(ϕ) := Rϕ (0).

151

(5)

This defines a linear function R on OCn ,0 . One can show that this linear function vanishes on the ideal I and hence defines a linear function R on the algebra A. Using this function, we can define a quadratic form Q(ϕ, ψ) = R(ϕ · ψ) on the algebra A generalizing the Eisenbud-Levine-Khimshiashvili quadratic form defined for a smooth V . A similar form can be defined on a module of differential forms associated to the pair (V, ω), see [EG06]. A special case of such a quadratic form (for curve singularities) was considered in [MvS90]. In the smooth case this quadratic form can be identified with the one on the algebra A by an isomorphism of the underlying spaces. This is no longer true for an ICIS. In general, these two quadratic forms have even different ranks. Let g : (Cn+1 , 0) → (C, 0) be a real analytic function (i.e. a function taking real values on Rn+1 ⊂ Cn+1 ) with an isolated singular point at 0. Then there is the following topological formula for the index of the gradient vector field grad g at the origin. Let η be a sufficiently small positive real number. The complex conjugation induces involutions on the Milnor fibres Mg,η and Mg,−η of g. Their actions on the homology groups Hn (Mg,η ) and Hn (Mg,−η ) will be denoted by σ+ and σ− respectively. On these homology groups we have the intersection forms ·, ·. We consider the symmetric bilinear forms Q+ (a, b) := σ+ a, b and Q− (a, b) := σ− a, b on Hn (Mg,η ) and Hn (Mg,−η ) respectively. The following statement was conjectured by Arnold [Ar78] and proved in [Gu84, Va85]. Theorem 12. If n is even then n 1 ind0 grad g = (−1) 2 (sgn Q− − sgnQ+ ) , 2

where grad g is the gradient vector field of the function g on Rn+1 . If Var : Hn (Mg,η , ∂Mg,η ) → Hn (Mg,η ) denotes the variation operator of the singularity of g, then there is another formula for the index in [Gu84]: Theorem 13. One has ind0 grad g = (−1)

n(n−1) 2

sgn Var−1 σ+ .

In [EG99] a generalization of such a formula for the radial index of a gradient vector field on an algebraically isolated real analytic isolated complete intersection singularity in Cn+k was obtained. In [Se95, SS96] formulae are given evaluating the GSV index of a singular point of a vector field on an isolated hypersurface or complete intersection singularity on a resolution of the singularity.

152


A. Esterov has extended the method of A.Khovanski of computation of intersection numbers from complete intersections to determinantal varieties. Using this he gave formulae for the GSV index of a 1-form on an ICIS and for its generalization to collections of 1-forms (see Sect. 8) in terms of Newton diagrams of the equations of the ICIS and of the components of the 1-forms: [?], [Es06] (see also [?] for some preliminary results). In [Kl02] a residue formula for the GSV index of a holomorphic 1-form on an isolated surface singularity in the spirit of Theorem 10 is proved.

7 Indices of Meromorphic 1-Forms Nonzero holomorphic vector fields (as well as holomorphic 1-forms) on a compact complex manifold rarely exist. Therefore it is interesting to consider meromorphic vector fields or 1-forms of which there are a lot (at least for projective manifolds). For a meromorphic vector field defined by a holomorphic section of the vector bundle T M ⊗ L (L is a holomorphic line bundle on M ), the sum of the indices of its zeros is equal to the corresponding characteristic number cn (T M ⊗ L), [M ] of the vector bundle T M ⊗ L (see, e.g., [BB70]). Let α be a meromorphic 1-form on a compact complex manifold M n which means that α is a holomorphic 1-form outside of a positive divisor D and in a neighbourhood of each point of M the form α can be written as α )/F where F = 0 is a local equation of the divisor D and α ) is a holomorphic 1-form. Let L be the line bundle associated to the divisor D, i.e., L has a holomorphic section s with zeros on D. Then ω = sα is a holomorphic section of the vector bundle T ∗ M ⊗L. In some constructions (say, as in [BB70]) one defines a meromorphic 1-form on M simply as a holomorphic section of the tensor product T ∗ M ⊗ L for a holomorphic line bundle L. This definition is somewhat different from the one formulated above. E.g., in this case only the class of the divisor of poles of a meromorphic 1-form is defined, not the divisor itself. Moreover, in this setting, the value of a meromorphic 1-form on a vector field is not a function, but a section of the line bundle L. In the sequel we use the notation ω for a section of the vector bundle T ∗ M ⊗ L, for short calling it a meromorphic 1-form as well. Suppose that the section ω has isolated zeros. Then the sum of their indices is equal to the characteristic number cn (T ∗ M ⊗ L), [M ] and thus depends on L. For a meromorphic 1-form on a smooth compact complex curve M the characteristic number c1 (T ∗ M ), [M ] = −χ(M ) is equal to the number of zeros minus the number of poles counted with multiplicities. Therefore to express the Euler characteristic of a compact manifold in terms of singularities of a meromorphic 1-form one has to take the divisor D of its poles into account as well. If a meromorphic 1-form on a manifold M n is defined simply as a section of T ∗ M ⊗ L, the divisor of poles is not defined and thus one also cannot define singular points of the 1-form on its pole locus.


153

One would like to give a Poincaré-Hopf type formula for meromorphic 1forms, i.e., to express the Euler characteristic of a compact manifold or of a smoothing of a complete intersection in terms of singularities of a meromorphic 1-form. For that it is possible to introduce a suitable notion of an index of a germ of a meromorphic 1-form (with an additional structure) on an ICIS, so that the indices of the singular points sum up to (plus-minus) the Euler characteristic of a smoothing. There are several descriptions of the index of a meromorphic 1-form, one of them being the alternating sum of the dimensions of certain algebras. A Poincaré-Hopf type formula can be considered as one describing a localization of an invariant (say, of the Euler characteristic) of a manifold at singular points of, say, a vector field or a 1-form, i.e., a representation of the invariant as the sum of integer invariants (”indices”) corresponding to singular points. Therefore one first has to define singular points. Let M n be a complex manifold, let L be a line bundle on M , and let ω be a holomorphic section of the bundle T ∗ M ⊗ L. One has zeros of ω on M , but the pole locus of ω is not well defined. In order to discuss singular points of ω on its pole locus (which is necessary as we saw in the example when M was a curve), we have to fix this locus. This means that we have to choose a holomorphic section s = s1 of the line bundle L or to choose its zero divisor D = D1 . One can say that we have to consider α = ω/s, i.e., to proceed with our initial definition of a meromorphic 1-form. For the further setting we suppose that the divisor D of poles of the 1form ω is non-singular (in particular, reduced). This is not very essential for this section, however, this makes the discussion simpler. Since D is a submanifold of M , there is a well defined map T ∗ M |D → T ∗ D and thus (T ∗ M ⊗ L)|D → T ∗ D ⊗ L|D (the restriction of meromorphic 1-forms to D). (Using the 1-form α, one can also define only a section of the vector bundle T ∗ D ⊗ L|D , but not a meromorphic 1-form on D with precisely defined pole locus.) Let ω1 be the restriction of ω to D1 . It is a holomorphic section of the vector bundle T ∗ D ⊗ L|D . Its zeros are well defined and should be considered as singular points of the meromorphic 1-form ω on the pole locus. To discuss its singular points on its pole locus we again have to fix a divisor. Suppose that there exists a (positive) divisor D2 on M which is the zero locus of another section s2 of the same line bundle L and which intersects D1 transversally (in particular, this means that D2 is non-singular at its intersection points with D1 and D1 ∩ D2 is non-singular as well). One has the restriction of ω to D1 ∩ D2 and its zeros there. Going on in this way we arrive at the situation when we have fixed n divisors D1 , . . . , Dn (zeros of sections s1 , . . . , sn of the line bundle L) so that, for each i = 1, . . . n, D1 ∩ . . . ∩ Di is non-singular. The set of singular points of the 1-form ω is the union of the zeros of ω itself and of the restrictions of ω to D1 ∩ . . . ∩ Di for all i = 1, . . . , n. (For i = n the intersection D1 ∩ . . . ∩ Dn is

154


zero-dimensional and all its points should be considered as zeros of the 1-form ω.) One can say that we have to consider a collection of meromorphic 1-forms ω/s1 , . . . , ω/sn on M proportional to each other. There is the following Poincaré-Hopf type formula for meromorphic 1forms. Let M n be a compact complex manifold, let ω be a meromorphic 1-form on M , that is, a holomorphic section of the bundle T ∗ M ⊗ L where L is a holomorphic line bundle with nonzero holomorphic sections. Suppose that D1 = D, D2 , . . . , Dn are zero divisors of holomorphic sections of the line bundle L such that, for each i = 1, . . . , n, D1 ∩. . .∩Di is non-singular. Suppose that the form ω itself and its restrictions to the submanifolds D1 ∩ . . . ∩ Di , i = 1, . . . , n, have only isolated zeros. Let m0 (respectively mi , i = 1, . . . , n) be the number of zeros of the form ω (respectively, of the restriction of ω to the intersection D1 ∩ . . . ∩ Di ) counted with multiplicities. In particular mn is the number of points in D1 ∩ . . . ∩ Dn . Theorem 14. cn (T ∗ M )[M ] = (−1)n χ(M n ) = m0 − m1 + . . . + (−1)n mn . Remark 7. One can show that m0 = (−1)n (χ(M ) − χ(D)) = (−1)n χ(M \ D). Suppose that all zeros of ω are outside of D (one can consider this situation as the generic one). Let s be the holomorphic section of the line bundle L with zeros on D. Then α = ω/s is a holomorphic 1-form on M \ D with simple poles along D. So in this case the number m0 of zeros of the holomorphic 1-form α on M \ D coincides with (−1)n χ(M \ D). This is the relation which holds for holomorphic 1-forms on compact manifolds (M \ D is not compact). Example 4. Let α=

xdy − ydx + dz x2 + 4y 2 + z 2 + 1

be a meromorphic 1-form on the projective space CP3 (x, y, z are affine coordinates). One can see that the zeros of the corresponding ω on CP3 and also the zeros of ω|D , D = D1 = {x2 + 4y 2 + z 2 + 1 = 0}, are isolated and m0 = 0, m1 = 4. To define other singular points one has to choose D2 and D3 (e.g., D2 = {x2 + y 2 + 4z 2 = 0}, D3 = {x2 + y 2 + z 2 = 0}). One has m2 = 8, m3 = 8, 0 − 4 + 8 − 8 = −4 = (−1)3 χ(CP3 ). Note that as a meromorphic 1-form on CP3 with poles on the hypersurface Pd (x, y, z) = 0, deg Pd = d, it is natural to take Ad−2 dx + Bd−2 dy + Cd−2 dz , Pd where Ad−2 , Bd−2 , and Cd−2 are polynomials in x, y, z of degree d − 2 (α is not of this form). However, one can show that such a 1-form has non-isolated zeros on CP3 (at infinity).


155

Now let V n be a compact subvariety of a complex manifold M n+k such that in a neighbourhood of each point V is a complete intersection in M with only isolated singularities (M is not supposed to be compact). Let ω be a meromorphic 1-form on M , i.e., a holomorphic section of the bundle T ∗ M ⊗ L, where L is a line bundle on M with nonzero holomorphic sections. By a smoothing V of V we understand a smooth (C ∞ ) manifold which is obtained from V by smoothing its singular points (which are ICIS) in the usual way (as complex ICIS). (If V is a complete intersection in the projective space CPn+k , a smoothing of V can be obtained as an analytic submanifold of CPn+k as well.) It is possible to formulate a Poincaré-Hopf type formula for the Euler characteristic of V and of its smoothing V in terms of singular points of the 1-form ω on V . As above we suppose that there exist n holomorphic sections s1 , . . . , sn of the line bundle L with zero divisors D1 , . . . , Dn such that, for each i = 1, . . . , n, the intersection V ∩ D1 ∩ . . . ∩ Di is of dimension n − i and has only isolated singularities (and thus is reduced for i < n). We also suppose that the restriction of ω to the non-singular part of V and its restriction to the non-singular part of V ∩D1 ∩. . .∩Di (i = 1, . . . , n) have only isolated zeros. As singular points of the 1-form ω on V we consider all zeros of its restrictions to the non-singular parts of V and of the intersections V ∩ D1 ∩ . . . ∩ Di (i = 1, . . . , n) and all singular points of V and of V ∩ D1 ∩ . . . ∩ Di as well. Let P be a singular point of the 1-form ω on V . Suppose that the divisors D1 , . . . , D (0 ≤ ≤ n) pass through the point P and (if < n) D+1 does not. In a neighbourhood of the point P , in some local coordinates on M centred at the point P , we have the following situation. The variety (ICIS) V (possibly non-singular) is defined by k equations f1 = . . . = fk = 0. Let divisors D1 , . . . , D be defined by equations fk+1 = 0, . . . , fk+ = 0. Let us remind that {f1 = . . . = fk = fk+1 = . . . = fk+i = 0} is an ICIS for each i = 0, 1, . . . , . After choosing a local trivialization of the line bundle L, n+k the 1-form ω can be written as i=1 Ai (x)dxi where Ai (x) are holomorphic. One can say that we consider a collection of meromorphic 1-forms ω/fk+1 , . . . , ω/fk+ on V proportional to each other. Let us denote the set of local data (V, ω, fk+1 , . . . , fk+ ) by Ω. Let ind(0) Ω (respectively ind(i) Ω, i = 1, . . . , ) be the index of the holomorphic 1-form ω on V (respectively on the ICIS V ∩ D1 ∩ . . . ∩ Di ) defined in Sect. 3. Definition 13. The alternating sum indP Ω = ind(0) Ω − ind(1) Ω + . . . + (−1) ind() Ω is called the index of the 1-form ω at the point P with respect to the divisors D1 , . . . , D n . According to the statements from Sect. 3 (see also Sect. 6 for the last point) one has the following three equivalent descriptions of the index indP Ω.

156


1. Let Bδ be the ball of sufficiently small radius δ centred at the origin in Cn+k and let Sδ be its boundary. Let ε = (ε1 , . . . , εk , . . . , εk+ ) ∈ (Ck+ , 0) be small enough and such that ε(k+i) = (ε1 , . . . , εk , . . . , εk+i ) is not a critical value of the map Fk+i = (f1 , . . . , fk , . . . , fk+i ) : (Cn+k , 0) → (Ck+i , 0) for each i = 0, 1, . . . , . The restriction of the 1-form ω to the smooth manifold Vi = −1 Fk+i (ε(k+i) ) has isolated zeros in Bδ . Let mi be the number of them counted with multiplicities. Then ind(i) Ω = mi , indP Ω = m0 − m1 + . . . + (−1) m . 2. Let Ki (i = 0, 1, . . . , ) be the link of the ICIS V ∩ D1 ∩ . . . ∩ Di , i.e., the intersection V ∩ D1 ∩ . . . ∩ Di ∩ Sδ . Let di be the degree of the map (ω, df1 , . . . , dfk , . . . , dfk+i ) : Ki → Wk+i+1 (Cn+k ) (Wk+i+1 (Cn+k ) is the Stiefel manifold of (k + i + 1)–frames in the dual Cn+k ). Then ind(i) Ω = di , indP Ω = d0 − d1 + . . . + (−1) d . 3. Let Ii (i = 0, 1, . . . , ) be the ideal of the ring OCn+k ,0 of germs of holomorphic functions of n + k variables at the origin generated by f1 , . . . , fk , . . . , fk+i and the (k + i + 1) × (k + i + 1)–minors of the matrix ⎞ ⎛ ∂f ∂f1 1 ∂x1 · · · ∂xn+k ⎜ . . .. ⎟ ⎟ ⎜ . .. . ⎟. ⎜ . ⎜ ∂fk+i ∂fk+i ⎟ ⎠ ⎝ ∂x1 · · · ∂x n+k A1 · · · An+k Let νi = dimC OCn+k ,0 /Ii . Then ind(i) Ω = νi , indP ω = ν0 − ν1 + . . . + (−1) ν . One has the following Poincaré–Hopf type formula. Theorem 15.

indP Ω = (−1)n χ(V )

P

where V is a smoothing of the variety V . In order to get the Euler characteristic of the variety V itself one has to correct this formula by taking the Milnor numbers of the singular points of V into account. For P ∈ V , let µP denote the Milnor number of the ICIS V at the point P (see, e.g., [Lo84]). Note that, if V is non-singular at P , then µP = 0. Theorem 16. (−1)n χ(V ) =

(indP Ω − µP ). P

Remark 8. The Milnor number µP can also be written as an alternating sum of indices of holomorphic 1-forms on several ICIS. One can say that the term µP corresponds to the difference between two possible definitions of the index as in [SS98, Proposition 1.4].


157

In some cases it is natural to consider the situation when the pole locus of the meromorphic 1-form ω is a multiple of an irreducible divisor. Examples are: 1) A polynomial 1-form on Cn is a meromorphic 1-form on CPn the pole locus of which is a multiple of the infinite hyperplane CPn−1 ∞ . 2) Let f be a meromorphic function with the pole locus kD where D is an irreducible divisor (i.e., in a neighbourhood of each point f can be written as f/F k where f is holomorphic and F = 0 is a local equation of D). Then its differential df is a meromorphic 1-form with the pole locus (k + 1)D (i.e., in a neighbourhood of each point df can be written as ω /F k+1 where ω is a holomorphic 1-form). A polynomial function of degree k on Cn is a meromorphic function on CPn with the pole locus kCPn−1 ∞ . Let us give a version of Theorem 14 for this case. The fact that the pole divisor of a 1-form is multiple means that the corresponding line bundle is a power of another one. Let ω be a meromorphic 1-form on a compact complex manifold M n , that is, a holomorphic section of the bundle T ∗ M ⊗ L, where L = λk , k > 1. Suppose that D1 = D, D2 , . . . , Dn are zero divisors of holomorphic sections of the line bundle λ such that, for each i = 1, . . . , n, D1 ∩ . . . ∩ Di is non-singular and the form ω itself and its restrictions to D1 ∩ . . . ∩ Di have isolated zeros. Let m0 (respectively mi , i = 1, . . . , n) be the number of zeros of the 1-form ω (respectively of its restriction to D1 ∩. . .∩Di ) counted with multiplicities. Theorem 17. cn (T ∗ M ), [M ] = (−1)n χ(M ) = m0 − km1 + . . . + (−1)n kmn . Remark 9. In [HS98, Theorem 3.1] there is given a formula for the sum of the residues corresponding to the singular points of the foliation on the projective plane CP2 given by df = 0, where f is a polynomial function of degree k on C2 (which defines a meromorphic function on CP2 ). In our terms this is the formula for the number m0 of zeros of the meromorphic 1-form df , the pole locus of which is k + 1 times the infinite line. Thus Theorem 17 can be considered as a generalization of [HS98, Theorem 3.1] to higher dimensions and to meromorphic 1-forms which are not, in general, differentials of functions.

8 Indices of Collections of 1-Forms One can say that all the indices discussed above are connected with the Euler characteristic (e.g. through the Poincaré–Hopf type theorem). If M is a complex analytic manifold of dimension n, then its Euler characteristic χ(M ) is the characteristic number cn (T M ), [M ] = (−1)n cn (T ∗ M ), [M ], where T M is the tangent bundle of the manifold M , T ∗ M is the dual bundle, and cn is the corresponding Chern class. One can try to find generalizations of some notions

158


and statements about indices of vector fields or of 1-forms for other characteristic numbers (different from cn (T M ), [M ] or cn (T ∗ M ), [M ]). This was made in [EG05b] and [EG05c] for the GSV index and for the Euler obstruction. The top Chern class of a vector bundle is the (first) obstruction to existence of a non-vanishing section. Other Chern classes are obstructions to existence of linear independent collections of sections. Therefore instead of 1-forms on a complex variety we consider collections of 1-forms. Let π : E → M be a complex analytic vector bundle of rank m over a complex analytic manifold M of dimension n. It is known that the (2(n − k)-dimensional) cycle Poincaré dual to the characteristic class ck (E) (k = 1, . . . , m) is represented by the set of points of the manifold M where m−k +1 generic sections of the vector bundle E are linearly dependent (cf., e.g., [GH78, p. 413]). Let M(p, q), Dp,q , and Wp,q be the spaces defined sin Sect. 3. Let k = of positive integers with (k1 , . . . , ks ) be a sequence i=1 ki = k. Consider

s = M(m, m − k + 1) and the subvariety Dm,k = the space M m,k i i=1

s D in it. The variety D consists of sets {A m,m−k +1 m,k i } of m×(m−ki +1) i i=1 matrices such that rk Ai < m − ki + 1 for each i = 1, . . . , s. Since Dm,k is irreducible of codimension k, its complement Wm,k = Mm,k \Dm,k is (2k−2)connected, H2k−1 (Wm,k ) ∼ = Z, and there is a natural choice of a generator of the latter group. This choice defines a degree (an integer) of a map from an oriented manifold of dimension 2k − 1 to the manifold Wm,k . Let (V, 0) ⊂ (CN , 0) be an n-dimensional isolated complete intersection singularity (ICIS) defined by equations f1 = . . . = fN −n = 0 (fi ∈ OCN ,0 ). Let f be the analytic map (f1 , . . . , fN −n ) : (CN , 0) → (CN −n , 0) (V = f −1 (0)). (i) Let {Xj } be a collection of vector fields on a neighbourhood of the origin in ki = n) which are tangent to (CN , 0) (i = 1, . . . , s; j = 1, . . . , n − ki + 1; the ICIS (V, 0) = {f1 = · · · = fN −n = 0} ⊂ (CN , 0) at non-singular points of (i) V . We say that a point p ∈ V \ {0} is non-singular for the collection {Xj } (i)

(i)

on V if at least for some i the vectors X1 (p), . . . , Xn−ki +1 (p) are linearly (i)

independent. Suppose that the collection {Xj } has no singular points on V outside of the origin in a neighbourhood of it. Let U be a neighbourhood of the origin in CN where all the functions fr (r = 1, . . . , N − n) and the vector (i) (i) fields Xj are defined and such that the collection {Xj } has no singular points on (V ∩ U ) \ {0}. Let Sδ ⊂ U be a sufficiently small sphere around the origin which intersects V transversally and denote by K = V ∩ Sδ the link of the ICIS (V, 0). The manifold K has a natural orientation as the boundary of a complex analytic manifold. Let ΨV be the mapping from V ∩ U to Mn,k which sends a point x ∈ V ∩ U to the collection of N × (N − ki + 1)-matrices (i)

(i)

{(grad f1 (x), . . . , grad fN −n (x), X1 (x), . . . , Xn−ki +1 (x))},

i = 1, . . . , s.


159

Here grad fr is the gradient vector field of fr defined in Sect. 3. Its restriction ψV to the link K maps K to the subset WN,k . (i)

(i)

Definition 14. The index indV,0 {Xj } of the collection of vector fields {ωj } on the ICIS V is the degree of the mapping ψV : K → WN,k . For s = 1, k1 = n, this index is the GSV index of a vector field on an ICIS (see Sect. 3). Let V ⊂ CPN be an n-dimensional complete intersection with isolated singular points, V = {f1 = . . . = fN −n = 0} where fi are homogeneous (i) functions in (N + 1) variables. Let {Xj } be a collection of continuous vector fields on CPN which are tangent to V . Let V be a smoothing of the complete intersection V , i.e. V is defined by N − n equations f1 = . . . = fN −n = 0 where the homogeneous functions fi are small perturbations of the functions fi and V is smooth. The usual description of Chern classes of a vector bundle as obstructions to existence of several linear independent sections of the bundle implies the following statement. Theorem 18. One has

(i) indp {Xj }

=

s

cki (T V ), [V ],

i=1

p∈V

where V is a smoothing of the complete intersection V . (i)

Now let {ωj } be a collection of (continuous) 1-forms on a neighbourhood ki = n. of the origin in (CN , 0) with i = 1, . . . , s, j = 1, . . . , n − ki + 1, (i) We say that a point p ∈ V \ {0} is non-singular for the collection {ωj } on (i)

V if at least for some i the restrictions of the 1-forms ωj (p) to the tangent (i)

space Tp V are linearly independent. Suppose that the collection {ωj } has no singular points on V outside of the origin in a neighbourhood of it. As above let K = V ∩ Sδ be the link of the ICIS (V, 0) (all the functions fr and the (i) 1-forms ωj are defined in a neighbourhood of the ball Bδ ). Let ΨV be the mapping from V ∩ U to Mn,k which sends a point x ∈ V ∩ U to the collection of N × (N − ki + 1)-matrices (i)

(i)

{(df1 (x), . . . , dfN −n (x), ω1 (x), . . . , ωn−ki +1 (x))},

i = 1, . . . , s.

Its restriction ψV to the link K maps K to the subset WN,k . (i)

(i)

Definition 15. The index indV,0 {ωj } of the collection of 1-forms {ωj } on the ICIS V is the degree of the mapping ψV : K → WN,k .

160

W. Ebeling and S. M. Gusein-Zade (i)

One can easily see that the index indV,0 {ωj } is equal to the intersection number of the germ of the image of the mapping ΨV with the variety DN,k . If (i) all the 1-forms ωj are complex analytic, the mapping ΨV is complex analytic as well. For s = 1, k1 = n, this index is the GSV index of a 1-form (Sect. 3). Let V ⊂ CPN be an n-dimensional complete intersection with isolated singular points, V = {f1 = . . . = fN −n = 0} where fi are homogeneous functions in (N + 1) variables. Let L be a complex line bundle on V and (i) let {ωj } be a collection of continuous 1-forms on V with values in L. Here (i)

this means that the forms ωj are continuous sections of the vector bundle T ∗ V ⊗L outside of the singular points of V . Since, in a neighbourhood of each (i) point p, the vector bundle L is trivial, one can define the index indp {ωj } of (i)

the collection of 1-forms {ωj } at the point p just in the same way as in the local setting above. Let V be a smoothing of the complete intersection V . One can consider L as a line bundle on the smoothing V of the complete intersection V as well (e.g., using the pull back along a projection of V to V ; (i) L is not, in general, complex analytic). The collection {ωj } of 1-forms can also be extended to a neighbourhood of V in such a way that it will define (i) a collection of 1-forms on the smoothing V (also denoted by {ωj }) with (i)

isolated singular points. The sum of the indices of the collection {ωj } on the smoothing V of V in a neighbourhood of the point p is equal to the index (i) indV,p {ωj }. One has the following analogue of Proposition 4 for 1-forms. Theorem 19. One has p∈V

(i)

indp {ωj } =

s

cki (T ∗ V ⊗ L), [V ],

i=1

where V is a smoothing of the complete intersection V . As above, let (V, 0) ⊂ (CN , 0) be the ICIS defined by the equations f1 = (i) · · · = fN −n = 0. Let {ωj } (i = 1, . . . , s; j = 1, . . . , n − ki + 1) be a collection of 1-forms on a neighbourhood of the origin in CN without singular points on (i) V \ {0} in a neighbourhood of the origin. If all the 1-forms ωj are complex (i)

analytic, there exists an algebraic formula for the index indV,0 {ωj } of the (i)

collection {ωj } similar to that from Theorem 11. Let IV,{ω(i) } be the ideal in the ring OCN ,0 generated by the functions j

f1 , . . . , fN −n and by the (N − ki + 1) × (N − ki + 1) minors of all the matrices (i)

(i)

(df1 (x), . . . , dfN −n (x), ω1 (x), . . . , ωn−ki +1 (x)) for all i = 1, . . . , s.


161

Theorem 20. (see [EG05b]) (i)

indV,0 {ωj } = dimC OCN ,0 /IV,{ω(i) } . j

Remark 10. A formula similar to that of Theorem 20 does not exist for collections of vector fields. A reason is that in this case the index is the intersection number with DN,k of the image of the ICIS (V, 0) under a map which is not complex analytic. Moreover, in some cases this index can be negative (see e.g. [GSV91, Proposition 2.2]). Example 5. Let V be the (singular) quadric x21 + x22 + x23 = 0 in the projective space CP3 with the coordinates (x0 : x1 : x2 : x3 ). Let the 1-forms from the (1) (1) (2) (2) collection {{ω1 , ω2 }, {ω1 , ω2 }} be defined in the affine chart C3 = {x0 = (1) (1) (2) 1} of the projective space CP3 as ω1 = dx1 , ω2 = x2 dx3 −x3 dx2 , ω1 = dx2 , (2) ω2 = x1 dx3 − x3 dx1 . They are holomorphic 1-forms on the projective space CP3 with values in the line bundle O(2). It is easy to see that being restricted to the quadric V the indicated collection of 1-forms has no singular points outside of the singular point (1 : 0 : 0 : 0) of V itself. Theorem 19 says that the index of this point is equal to c21 (T ∗ V ⊗ i∗ O(2)), [V ], where V is a non-singular quadric, i∗ O(2) is the restriction of the line bundle O(2) to V . Taking into account that V ∼ = (CP1 )2 , one gets that this characteristic number is equal to 8. Now by Theorem 20 the index of the indicated collection of 1forms on the quadric V at the origin in C3 ⊂ CP3 is equal to dimC OC3 ,0 /x21 + x22 + x23 , x22 + x23 , x21 + x23 which is also equal to 8. There exists a generalization of the notion of the Euler obstruction to collections of 1-forms corresponding to different Chern numbers. Let (V n , 0) ⊂ (CN , 0) be the germ of a purely n-dimensional reduced complex analytic variety at the origin (generally speaking with a non-isolated singularity). Let k = {ki }, i = 1, . . . , s, be a fixed partition of n (i.e., ki are s (i) ki = n). Let {ωj } (i = 1, . . . , s, j = 1, . . . , n − ki + 1) be positive integers, i=1

a collection of germs of 1-forms on (CN , 0) (not necessarily complex analytic; (i) it suffices that the forms ωj are complex linear functions on CN continuously depending on a point of CN ). Let ε > 0 be small enough so that there is a (i) representative V of the germ (V, 0) and representatives ωj of the germs of 1-forms inside the ball Bε (0) ⊂ CN . (i)

Definition 16. A point P ∈ V is called a special point of the collection {ωj } of 1-forms on the variety V if there exists a sequence {Pm } of points from the non-singular part Vreg of the variety V such that the sequence TPm Vreg of the tangent spaces at the points Pm has a limit L (in G(n, N )) and the (i) (i) restrictions of the 1-forms ω1 , . . . , ωn−ki +1 to the subspace L ⊂ TP CN (i)

are linearly dependent for each i = 1, . . . , s. The collection {ωj } of 1-forms

162


has an isolated special point on (V, 0) if it has no special points on V in a punctured neighbourhood of the origin. (i)

Remark 11. If the 1-forms ωj are complex analytic, the property to have an isolated special point is a condition on the classes of these 1-forms in the module 1 = ΩC1 N ,0 /{f · ΩC1 N ,0 + df · OCN ,0 |f ∈ JV } ΩV,0 of germs of 1-forms on the variety V (JV is the ideal of germs of functions vanishing on V ). Remark 12. For the case s = 1 (and therefore k1 = n), i.e. for one 1-form ω, we discussed the notion of a singular point of the 1-form ω on V . One can easily see that a special point of the 1-form ω on V is singular, but not vice versa. (E.g. the origin is a singular point of the 1-form dx on the cone {x2 + y 2 + z 2 = 0}, but not a special one.) On a smooth variety these two notions coincide. (i)

Definition 17. A special (singular) point of a collection {ωj } of germs of 1-forms on a smooth n-dimensional variety V is non-degenerate if the map ΨV : V ∩ U → Mn,k described above is transversal to Dn,k ⊂ Mn,k at a non-singular point of it. Let Lk =

s n−k i +1 i=1

∗ CN ij

j=1

be the space of collections of linear functions on CN (i.e. of 1-forms with constant coefficients). The following statement holds. Proposition 7. There exists an open and dense set U ⊂ Lk such that each (i) collection {j } ∈ U has only isolated special points on V and, moreover, all these points belong to the smooth part Vreg of the variety V and are nondegenerate. (i)

Corollary 2. Let {ωj } be a collection of 1-forms on V with an isolated spe(i)

cial point at the origin. Then there exists a deformation { ωj } of the collection

(i) {ωj }

whose special points lie in Vreg and are non-degenerate. Moreover, (i)

(i)

as such a deformation one can use {ωj + λj } with a generic collection (i)

{j } ∈ Lk . Corollary 3. The set of collections of holomorphic 1-forms with a nonisolated special point at the origin has infinite codimension in the space of all holomorphic collections.


163

(i)

Let {ωj } be a collection of germs of 1-forms on (V, 0) with an isolated special point at the origin. Let ν : V) → V be the Nash transformation of the (i) variety V ⊂ Bε (0) (see Sect. 5). The collection of 1-forms {ωj } gives rise to a section ω ) of the bundle )= T

s n−k i +1 i=1

∗ T)i,j

j=1

∗ where T)i,j are copies of the dual Nash bundle T)∗ over the Nash transform V) )⊂T ) be the set of pairs (x, {α(i) }) where numbered by indices i and j. Let D j

(i) x ∈ V) and the collection {αj } of elements of T)x∗ (i.e. of linear functions on T)x ) (i)

(i)

is such that α1 , . . . , αn−ki +1 are linearly dependent for each i = 1, . . . , s. The ) outside of the preimage ν −1 (0) ⊂ V) image of the section ω ) does not intersect D )\D ) )\D ) → V) is a fibre bundle. The fibre Wx = T of the origin. The map T of it is (2n − 2)-connected, its homology group H2n−1 (Wx ; Z) is isomorphic to Z and has a natural generator (see above). The latter fact implies that the )\D ) → V) is homotopically simple in dimension 2n − 1, i.e. the fibre bundle T fundamental group π1 (V) ) of the base acts trivially on the homotopy group π2n−1 (Wx ) of the fibre, the last one being isomorphic to the homology group H2n−1 (Wx ): see, e.g., [St51]. (i)

Definition 18. The local Chern obstruction ChV,0 {ωj } of the collections of (i)

germs of 1-forms {ωj } on (V, 0) at the origin is the (primary) obstruction )\D ) → V) from the preimage of to extend the section ω ) of the fibre bundle T a neighbourhood of the sphere Sε = ∂Bε to V) , more precisely its value (as an element of the homology group H 2n (ν −1 (V ∩ Bε ), ν −1 (V ∩ Sε ); Z) ) on the fundamental class of the pair (ν −1 (V ∩ Bε ), ν −1 (V ∩ Sε )). (i)

The definition of the local Chern obstruction ChV,0 {ωj } can be reformulated in the following way. Let DVk ⊂ CN ×Lk be the closure of the set of pairs (i) (i) (x, {j }) such that x ∈ Vreg and the restrictions of the linear functions 1 , (i)

. . . , n−ki +1 to Tx Vreg ⊂ CN are linearly dependent for each i = 1, . . . , s. (For s = 1, k = {n}, DVk is the (non-projectivized) conormal space of V [Tei82].) (i) The collection {ωj } of germs of 1-forms on (CN , 0) defines a section ω ˇ of the trivial fibre bundle CN × Lk → CN . Then ChV,0 {ωj } = (ˇ ω (CN ) ◦ DVk )0 (i)

where (· ◦ ·)0 is the intersection number at the origin in CN × Lk . This description can be considered as a generalization of an expression of the local Euler obstruction as a micro-local intersection number defined in [KS90], see also [Sch¨ u03, Sections 5.0.3 and 5.2.1] and [Sch¨ u04].

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Remark 13. On a smooth manifold V the local Chern obstruction ChV,0 {ωj } (i)

(i)

coincides with the index indV,0 {ωj } of the collection {ωj } defined above. Remark 14. The local Euler obstruction is defined for vector fields as well as for 1-forms. One can see that vector fields are not well adapted to a definition of the local Chern obstruction. A more or less direct version of the definition above for vector fields demands to consider vector fields on a singular variety N V ⊂ CN to be sections X = X(x) of T CN |V such that X(x) ∈ Tx V ⊂ Tx C (dim Tx V is not constant). (Traditionally vector fields tangent to smooth strata of the variety V are considered.) There exist only continuous (nontrivial, i.e. with s > 1) collections of such vector fields ”on V ” with isolated special points, but not holomorphic ones. (i)

Remark 15. The definition of the local Chern obstruction ChV,0 {ωj } may also be formulated in terms of a collection {ω (i) } of germs of 1-forms with values in vector spaces Li of dimensions n−ki +1. Therefore (via differentials) it is also defined for a collection {f (i) } of germs of maps f (i) : (CN , 0) → (Cn−ki +1 , 0) (just as the Euler obstruction is defined for a germ of a function). Being a (primary) obstruction, the local Chern obstruction satisfies the law (i) of conservation of number, i.e. if a collection of 1-forms { ωj } is a deformation (i)

of the collection {ωj } and has isolated special points on V , then (i) (i) ChV,0 {ωj } = ChV,Q { ωj } where the sum on the right hand side is over all special points Q of the (i) collection { ωj } on V in a neighbourhood of the origin. With Corollary 2 this implies the following statements. (i)

(i)

Proposition 8. The local Chern obstruction ChV,0 {ωj } of a collection {ωj } of germs of holomorphic 1-forms is equal to the number of special points on V of a generic (holomorphic) deformation of the collection. This statement is an analogue of Proposition 2.3 in [STV05b]. (i)

Proposition 9. If a collection {ωj } of 1-forms on a compact (say, projective) variety V has only isolated special points, then the sum of local Chern (i) obstructions of the collection {ωj } at these points does not depend on the collection and therefore is an invariant of the variety. It is reasonable to consider this sum as ((−1)n times) the corresponding Chern number of the singular variety V . Let (V, 0) be an isolated complete intersection singularity. The fact that (i) both the Chern obstruction and the index of a collection {ωj } of 1-forms satisfy the law of conservation of number and they coincide on a smooth manifold yields the following statement.


165

(i)

Proposition 10. For a collection {ωj } on an isolated complete intersection singularity (V, 0) the difference (i)

(i)

indV,0 {ωj } − ChV,0 {ωj } does not depend on the collection and therefore is an invariant of the germ of the variety. (i)

(i)

Since by Proposition 9 ChV,0 {j } = 0 for a generic collection {j } of linear functions on CN , one has the following statement. Corollary 4. One has (i)

(i)

(i)

ChV,0 {ωj } = indV,0 {ωj } − indV,0 {j } (i)

for a generic collection {j } of linear functions on CN .

Acknowledgements This work was partially supported by the DFG-programme ”Global methods in complex geometry” (Eb 102/4–3), grants RFBR–04–01–00762, NWORFBR 047.011.2004.026.

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Equisingular Families of Projective Curves Gert-Martin Greuel, Christoph Lossen, and Eugenii Shustin 1 2 3

TU Kaiserslautern, Fachbereich Mathematik, Erwin-Schr¨ odinger-Straße, 67663 Kaiserslautern, Germany, [email protected] TU Kaiserslautern, Fachbereich Mathematik, Erwin-Schr¨ odinger-Straße, 67663 Kaiserslautern, Germany, [email protected] Tel Aviv University, School of Mathematical Sciences, Ramat Aviv, Tel Aviv 69978, Israel, [email protected].

Summary. In this survey, we report on progress concerning families of projective curves with fixed number and fixed (topological or analytic) types of singularities. We are, in particular, interested in numerical, universal and asymptotically proper sufficient conditions to guarantee the nonemptyness, T-smoothness and irreducibility of the variety of all projective curves with prescribed singularities in a fixed linear system. We also discuss the analogous problem for hypersurfaces of arbitrary dimension with isolated singularities, and we close with a section on open problems and conjectures.

1 Introduction Let S be a topological or analytic classification of isolated plane curve singularities. Assume that Σ is a non-singular projective algebraic surface over an algebraically closed field of characteristic zero, and D ⊂ Σ is an ample divisor such that dim |D| > 0 and the general member of |D| is irreducible and nonsingular. The part of the discriminant, consisting of irreducible curves on Σ with only isolated singularities, splits into the union of equisingular families irr (S1 , . . . , Sr ), that is, into the varieties of irreducible curves C ∈ |D| (ESF ) V|D| having exactly r isolated singular points of types S1 , . . . , Sr ∈ S, respectively. If Σ = P2 , D = dH, H a hyperplane divisor, we simply write Vdirr (S1 , . . . , Sr ) for the variety of irreducible plane curves of degree d with r singularities of the prescribed types. We focus on the following geometric problems which have been of interest to algebraic geometers since the early 20th century but which are still widely open in general: irr (S1 , . . . , Sr ) non-empty, that is, does there A. Existence Problem: Is V|D| exist a curve F ∈ |D| with the given collection of singularities ? In particular, the question about the minimal degree of a plane curve having a given singularity is of special interest.

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G.-M. Greuel, C. Lossen, and E. Shustin

irr B. T-Smoothness Problem: If V|D| (S1 , . . . , Sr ) is non-empty, is it smooth and of the expected dimension (expressible via local invariants of the singularirr ities) ? More precisely, let C ∈ V|D| (S1 , . . . , Sr ) have singular points z1 , . . . , zr irr of types S1 , . . . , Sr , respectively. We say that the family V|D| (S1 , . . . , Sr ) is T-smooth at C if, for every i = 1, . . . , r, the germ at C of the family of curves C ∈ |D| with a singular point of type Si in a neighbourhood of zi is smooth and has the expected dimension (to be explained later), and, furthermore, all these r germs intersect transversally at C (whence the name T-smooth). irr (S1 , . . . , Sr ) irreducible? C. Irreducibility Problem: Is V|D|

D. Deformation Problem: What are the adjacency relations of ESF in the discriminant? In other words, which simultaneous deformations of the singuirr (S1 , . . . , Sr ) can be realized by a variation of F in |D| ? In larities of F ∈ V|D| fact, this question is closely related to Problem B. For instance, for Σ = P2 , the T-smoothness of Vdirr (S1 , . . . , Sr ) for analytic singularities S1 , . . . , Sr is equivalent to the linear system |dH| inducing a joint versal deformation of all singular points of any member C ∈ Vdirr (S1 , . . . , Sr ). Similarly, the T-smoothness of ESF for semiquasihomogeneous topological singularities S1 , . . . , Sr implies that the independence of simultaneous “lower” (w.r.t. the Newton diagram) deformations of the singularities of C ∈ Vdirr (S1 , . . . , Sr ) (see Sect. 4). Of course, the same questions can be posed for V|D| (S1 , . . . , Sr ), the variety of reduced (but not necessarily irreducible) curves C ∈ |D| with given singularities S1 , . . . , Sr ∈ S. No complete solution to the above problems is known, except for the case of plane nodal curves. However, in this overview article we demonstrate that the approach using deformation theory and cohomology vanishing (developed by the authors of this article over the last ten years) enables us to obtain reasonably proper sufficient conditions for the affirmative answers to the problems stated. More precisely, we intend to give sufficient conditions for a positive answer to the above questions which are • numerical , that is, presented in the form of inequalities relating numerical invariants of the surface, the linear system, and the singularities, • universal , that is, applicable to curves in any ample linear system with any number of arbitrary singularities, and • asymptotically proper (see the definition below). We should like to comment on the latter in more detail. Let D = dD0 with a given divisor D0 ⊂ Σ (e.g., D0 = H if Σ = P2 ). Then the known general irr (S1 , . . . , Sr ) read as restrictions to the singular points of a curve C ∈ V|D| upper bounds to some total singularity invariants by a quadratic function in d. As an example, consider the bound obtained by the genus formula for irreducible curves, r i=1

δ(Si ) ≤

1 2 2 (d D0 + dKΣ D0 ) + 1 , 2

Equisingular Families of Projective Curves

173

where δ(Si ) is the difference between the arithmetic genus and the geometric genus of C imposed by the singularity Si . Sufficient conditions for the “regular” properties of ESF appear in a similar form as upper bounds to total singularity invariants by a linear or quadratic function of the parameter d. In this sense we speak of linear, or quadratic sufficient conditions, the latter revealing the relevant asymptotics. Furthermore, among the quadratic sufficient conditions we emphasize on asymptotically proper ones. We say that the inequality r

a(Si ) ≤ fΣ,D0 (d) ,

i=1

with a(S) a local invariant of singularities S ∈ S, is an asymptotically proper sufficient condition for a regular property (such as non-emptiness, smoothness, irreducibility) of ESF, if (1) for arbitrary r, d ≥ 1 and S1 , . . . , Sr ∈ S it provides the required property irr (S1 , . . . , Sr ), of the ESF V|dD 0| (2) there exists an absolute constant A ≥ 1 such that, for each singularity type S, there is an infinite sequence of growing integers d, r and a (maybe, empty) finite collection of singularities Sd satisfying a(S ) = o(fΣ,D0 (d)) , r · a(S) = A · fΣ,D0 (d) + o(fΣ,D0 (d)), S ∈Sd irr such that the ESF V|dD (r · S, Sd ) does not have that regular property. 0|

If A = 1 in (2), we speak of an asymptotically optimal sufficient condition. In less technical terms, we say that a condition is asymptotically optimal (resp. proper) if the necessary and the sufficient conditions for a regularity property coincide (resp. coincide up to multiplication of the right-hand side with a constant) if r and d go to infinity. Methods and Results: from Severi to Harris T-Smoothness. Already the Italian geometers [Sev68, Seg24, Seg29] noticed that it is possible to express the T-smoothness of the variety of plane curves with fixed number of nodes (A1 -singularities) and cusps (A2 -singularities) infinitesimally. The problem was historically called the “completeness of the characteristic linear series of complete continuous systems” (of plane curves with nodes and cusps). Best known is certainly Severi’s [Sev68] result saying that each non-empty variety of plane nodal curves is T-smooth. But for more complicated singularities (beginning with cusps) there are examples of irreducible curves where the T-smoothness fails (see below). That is, ether the ESF is non-smooth or its dimension exceeds the one expected by subtracting the number of (closed) conditions imposed by the individual

174


singularities from the dimension d(d+3) of the variety of all curves of degree 2 d. In this context, each node imposes exactly one condition. This may be illustrated as follows: a plane curve {f = 0} has a node at the origin iff the 1-jet of f vanishes (= three closed conditions) and the 2-jet is reduced (= one open condition). Allowing the node to move (in C2 ) reduces the number of closed conditions by two. Similarly, it can be seen that each cusp imposes two conditions. Various sufficient conditions for T-smoothness were found. The classical result is that the variety Vdirr (n · A1 , k · A2 ) of irreducible plane curves of degree d with n nodes and k cusps as only singularities is T-smooth, that is, smooth − n − 2k, if of dimension d(d+3) 2 k < 3d .

(1)

For arbitrary singularities, several generalizations and extensions of (1) were found [GK89, GL96, Shu87, Shu91a, Vas90]. All of them are of the form that the sum of certain invariants of the singularities is bounded from above by a linear function in d. On the other hand, the known restrictions for existence and T-smoothness (and the known series of non T-smooth ESF) suggested that an asymptotically proper sufficient condition should be quadratic in d (see below). Restrictions for the Existence. Various restrictions for the existence of plane curves with prescribed singularities S1 , . . . , Sr have been found. First, one should mention the general classical bounds r

δ(Si ) ≤

i=1

(d − 1)(d − 2) , 2

(2)

(for irreducible curves), resulting from the genus formula, respectively r

µ(Si ) ≤ (d − 1)2 ,

i=1

resulting from the intersection of two generic polars and Bézout’s theorem. If C has only nodes and cusps as singularities then the Pl¨ ucker formulas give (among others) the necessary conditions 2 · #(nodes) + 3 · #(cusps) ≤ d2 − d − 2 , 6 · #(nodes) + 8 · #(cusps) ≤ 3d2 − 6d . By applying the log-Miyaoka inequality, F. Sakai sary condition r 2ν · d2 − µ(Si ) < 2ν + 1 i=1

[Sak93] obtained the neces3 d 2

,


175

where ν denotes the maximum of the multiplicities mt Si , i = 1, . . . , r. If S1 , . . . , Sr are ADE-singularities then ⎧ ⎪ ⎪ r ⎨ 3 d2 − 3 d + 2 if d is even , 4 2 µ(Si ) < ⎪ 3 ⎪ 2 i=1 ⎩ d − d + 1 if d is odd , 4 4 is necessary for the existence of a plane curve with r singularities of types S1 , . . . , Sr (cf. [Hir86, Ivi85], resp. [Sak93]). Further necessary conditions can be obtained, for instance, by applying the semicontinuity of the singularity spectrum (see [Var83a]). Methods of Construction. The first method is to construct (somehow) a curve of the given degree which is degenerate with respect to the required curve, and then to deform it in order to obtain the prescribed singularities. For instance, Severi [Sev68] showed that singular points of a nodal curve, irreducible or not, can be smoothed, or preserved, independently. Hence, starting with the union of generic straight lines in the projective plane and smoothing suitable intersection points, one obtains irreducible curves with any prescribed number of nodes, allowed by the genus bound (2), see Fig. 1.

Fig. 1. Constructing irreducible nodal curves.

Attempts to extend this construction to other singularities give curves with a number of singularities bounded from above by a linear function in the degree d (see, for example, [GM88] for curves with nodes, cusps and ordinary triple points), because of the very restrictive requirement of the independence of deformations. The second method consists of a construction especially adapted to the given degree and given collection of singularities. It may be based on a sequence of birational transformations of the plane applied to a more or less simple initial curve in order to obtain the required curve. Or it may consist of an invention of a polynomial defining the required curve. This is illustrated by constructions of singular curves of small degrees as, for instance, in [Wal95], [Wal96], or by Hirano’s [Hir92] construction of cuspidal curves,

176


which led to a series of irreducible cuspidal curves of degree d = 2 · 3k , k ≥ 1 with precisely 9(9k − 1)/8 cusps. Note that in this case the number of conditions imposed by the cusps is d2 /16 + O(d) more than the dimension of the space of curves of degree d. Two main difficulties do not allow to apply this approach to a wide class of degrees and singularities: • for any new degree or singularity one has to invent a new construction, • even if one has constructed a curve with many singularities, it is hard to check that these singular points can be smoothed independently or, at least, that any intermediate number of singularities can be realized (for instance, for Hirano’s examples the latter can hardly be expected). Irreducibility. Severi [Sev68] claimed that all non-empty ESF Vdirr (n · A1 ) of irreducible plane nodal curves are not only T-smooth but also irreducible. However, as was realized later, his proof of the irreducibility was incomplete, and the problem has become known later as “Severi’s conjecture”. For many years, algebraic geometers tried to solve this problem without much progress. A major step forward was made by Fulton [Ful80] and Deligne [Del81], showing that the fundamental group of the complement of each irreducible nodal curve is Abelian (which is a necessary condition for Vdirr (n · A1 ) to be irreducible). Finally, the problem was settled by Harris [Har85a]. He gave a rigorous proof for the irreducibility of the varieties Vdirr (n · A1 ), by inventing a new specialization method and by using the irreducibility of the moduli space of curves of a given genus. Since nodal curves form an open dense subset of each Severi variety [Alb28, Nob84a], that is, of the variety of all irreducible plane curves of a given degree and genus, Harris’ theorem extends to all Severi varieties as well. Later Ran [Ran86] and Treger [Tre88] gave different proofs. Then Ran [Ran89] generalized Harris’ theorem to ESF Vdirr (Om , n · A1 ) with one ordinary singularity Om (of some order m ≥ 1) and any number of nodes n ≤ (d − 1)(d − 2)/2 − δ(Om ). Kang [Kan89, Kan89a] obtained the irreducibility of the families Vdirr (n · A1 , k · A2 ) for k ≤ 3,

or

d2 − 3d + 2 d2 − 4d + 1 ≤n≤ . 2 2

The main idea of the proofs consists of using moduli spaces of curves, special degenerations of nodal curves, or degenerations of rational surfaces. In any case the proofs heavily rely on the independence of simultaneous deformations of nodes for plane curves, which (in general) does not hold for more complicated singularities, even for cusps. Examples of Obstructed and Reducible ESF. Already, Segre [Seg29, Tan84] constructed a series of irreducible plane curves such that the corresponding irr (6m2 · A2 ), m ≥ 3. Similar examples are germs of ESF are non-T-smooth: V6m given in [Shu94]. However, in these examples Vdirr (S1 , . . . , Sr ) is smooth (but of bigger dimension than the expected one). In 1987, Luengo [Lue87] provided


177

the first examples of curves C such that the corresponding ESF is of expected dimension, but non-smooth, e.g. V9irr (A35 ). Concerning reducible ESF, there is mainly one classical example due to Zariski [Zar71]: V6irr (6 · A2 ). More precisely, Zariski shows that there exist exactly two components, both being T-smooth. One component consists of cuspidal sextics C whose singularities lie on a conic. For such curves, Zariski computed the fundamental group π1 (P2 \ C) to be the non-Abelian group Z2 ∗ Z3 . On the other hand, he showed that there exist curves C whose singularities do not lie all on a conic and whose complement in P2 has an Abelian fundamental group. In particular, these curves cannot be obtained by a deformation from curves in the first component. irr (6p2 · A2 ), p ≥ 2, Actually, this example belongs to a series [Shu94]: V6p is reducible. More precisely, for p ≥ 3 there exist components with different dimensions. For p = 2 there exist two different T-smooth components, as in Zariski’s example.

2 Geometry of ESF in Terms of Cohomology In this section, we relate the ESFs V|D| (S1 , . . . , Sr ) to strata of a Hilbert scheme representing a deformation functor. This allows us to deduce geometric properties such as the T-smoothness or the irreducibility from the vanishing of the first cohomology group for the ideal sheaves of appropriately chosen zero-dimensional subschemes of the surface Σ. Let T be a complex space, then by a family of reduced (irreducible) curves on Σ over T we mean a commutative diagram j / Σ×T C = == ww w ϕ {w pr T where ϕ is a proper and flat morphism such that all fibres C t := ϕ−1 (t), t ∈ T , are reduced (irreducible) curves on Σ, j : C → Σ × T is a closed embedding and pr denotes the natural projection. A family with sections is a diagram as above, together with sections σ1 , . . . , σr : T → C of ϕ. To a family of reduced plane curves and a point t0 ∈ T we can associate, in a functorial way, the!deformation (C , z1 ) . . . (C , zr ) → (T, t0 ) of the multigerm (C, Sing C) = i (C, zi ) over the germ (T, t0 ), where C = Ct0 is the fibre over t0 . Having a family with !sections σ1 , . . . , σr , σi (t0 ) = zi , we obtain in the same way a deformation of i (C, zi ) over (T, t0 ) with sections. A family C → Σ × T → T of reduced curves (with sections) is called equianalytic (along the sections) if, for each t ∈ T , the induced deformation of the multigerm (Ct , Sing Ct ) is isomorphic (isomorphic as deformation with section) to the trivial deformation (along the trivial sections). It is called

178


equisingular (along the sections) if, for each t ∈ T , the induced deformation of the multigerm (Ct , Sing Ct ) is isomorphic (isomorphic as deformation with section) to an equisingular deformation along the trivial sections. In other words, a family with sections is equianalytic (resp. equisingular) if the analytic (resp. topological) type of the fibre Ct does not change along the sections (cf. [Wah, GLS06]). Here, the analytic type of a reduced plane curve singularity (C, z) is given by the isomorphism class of its analytic local ring OC,z , while the (embedded) topological type is given by the Puiseux pairs of its branches and their mutual intersection multiplicities or, alternatively, by the system of multiplicity sequences (see [BK86]). The Hilbert functor Hilb Σ on the category of complex spaces defined by Hilb Σ (T ) := { C → Σ × T → T, family of reduced curves over T } is known to be representable by a complex space Hilb Σ (see [Gro61] for algebraic varieties and [Dou] for arbitrary complex spaces). Moreover, the universal family of reduced curves on Σ “breaks!up” into strata with constant Hilbert polynomials, more precisely, Hilb Σ = h∈C[z] Hilb hΣ where the Hilb hΣ are (unions of) connected components of Hilb Σ whose points correspond to curves on Σ with the fixed Hilbert polynomial h. Let Vh (S1 , . . . , Sr ) ⊂ Hilb hΣ denote the locally closed subspace (equisingular stratum) of reduced curves with Hilbert polynomial h having precisely r singularities of types S1 , . . . , Sr ([GL96]). Further, let Vhirr (S1 , . . . , Sr ) ⊂ Vh (S1 , . . . , Sr ) denote the open subspace parametrizing irreducible curves. This notion of (not necessarily reduced) equisingular strata in the Hilbert related to scheme Hilb Σ is closely the ESFs V|D| (S1 , . . . , Sr ) considered be fore: if U ⊂ |C| = P H 0 (OΣ (C)) denotes the open subspace corresponding h to reduced curves, then there exists aunique . morphism U → Hilb Σ which on 0 0 0 the tangent level corresponds to H OΣ (C) H OΣ → H OC (C) . Via this morphism, we may consider U as a locally closed subscheme of Hilb hΣ . In particular, for a regular surface (that is, a surface satisfying H 1 (OΣ ) = 0) the above injection is an isomorphism and U is an open subscheme of Hilb hΣ (see [GL01, GLS06] for details). In the following, we give a geometric interpretation of the zeroth and first cohomology of the ideal sheaves of certain zero-dimensional schemes. We write C ∈ Vh (S1 , . . . , Sr ) to denote either the point in Vh (S1 , . . . , Sr ) or the curve corresponding to the point, that is, the corresponding fibre of the universal family Uh over the Hilbert scheme. Consider the map Φh : Vh (S1 , . . . , Sr ) −→ Symr Σ ,

C −→ (z1 +. . .+zr ) ,

(1)

where Symr Σ is the r-fold symmetric product of Σ and where (z1 +. . .+zr ) is the non-ordered tuple of the singularities of C. Since each equisingular, in particular each equianalytic, deformation of a germ admits a unique singular section (cf. [Tei78]), the universal family over Vh (S1 , . . . , Sr ),


179

Uh (S1 , . . . , Sr ) → Σ × Vh (S1 , . . . , Sr ) → Vh (S1 , . . . , Sr ), admits, locally at C, r singular sections. Composing these sections with the projections to Σ gives a local description of the map Φh and shows in particular that Φh is a well-defined morphism, even if Vh (S1 , . . . , Sr ) is not reduced. We denote by Vh,fix (S1 , . . . , Sr ) the complex space consisting of the disjoint union of the fibres of Φh . Thus, each connected component of Vh,fix (S1 , . . . , Sr ) consists of curves with fixed positions of the singularities in Σ. It follows from the universal property of Vh (S1 , . . . , Sr ) and from the above construction that Vh,fix (S1 , . . . , Sr ), together with the induced universal family on each fibre, represents the functor of equianalytic, resp. equisingular, families of given types S1 , . . . , Sr along trivial sections. Before formulating the main proposition relating the vanishing of cohomology to geometric properties of ESF, we introduce some notation: we write Vh to denote Vh (S1 , . . . , Sr ), resp. Vh,fix (S1 , . . . , Sr ), and V|C| to denote V|C| (S1 , . . . , Sr ), resp. V|C|,fix (S1 , . . . , Sr ). Moreover, we write Z (C) = Z (C, z) z∈Sing(C) ea es to denote one of the 0-dimensional schemes Z ea (C), Zfix (C), Z es (C), Zfix (C), where

• Z ea (C, z) is defined by the Tjurina ideal, that is, the ideal generated by a local equation f ∈ C{u, v} for (C, z) ⊂ (Σ, z) and its partial derivatives; ∂f ea ea • Zfix (C, z) is defined by the ideal Ifix (f ) = f + u, v· ∂f ∂u , ∂v ; es es • Z (C, z) is defined by the equisingularity ideal I (f ) ⊂ C{u, v} as introduced by Wahl [Wah]; es • Zfix (C, z) is defined by the ideal f + εg defines an equisingular deformation es (f ) := g ∈ I es (f ) Ifix of (C, 0) along the trivial section over Tε

(here, Tε denotes the fat point ({0}, C[ε]/ε2 ])). We write JZ (C)/C , resp. JZ (C)/Σ , to denote the ideal sheaf of Z (C) in OC , resp. in OΣ , and J (C) := J ⊗OΣ OΣ (C). Moreover, we write deg Z (C) for the degree of Z (C) as a projective variety, that is, deg Z (C) = dimC OΣ /JZ (C)/Σ . Proposition 1 ([GL01, Prop. 2.6]). Let C ⊂ Σ be a reduced curve with Hilbert polynomial h and precisely r singularities z1 , . . . , zr of analytic or topological types S1 , . . . , Sr . (a) The Zariski tangent space of Vh at C is H 0 JZ (C)/C (C) , while the Za. riski tangent space of V|C| at C is H 0 JZ (C)/Σ (C) H 0 OΣ .

180


(b) h0 JZ (C)/C (C) − h1 JZ (C)/C (C) ≤ dim(Vh , C) ≤ h0 JZ (C)/C (C) . (c1) If H 1 JZ (C)/C (C) = 0 thenVh is T-smooth at C, that is, smooth of the expected dimension h0 OC (C) −deg Z (C) = C 2 +1−pa (C)−deg Z (C) . (c2) If H 1 JZ (C)/Σ (C) = 0 then V|C| is T-smooth at C, that is, smooth of 0 the expected dimension h OΣ (C) − 1 − deg Z (C). (d) If H 1 JZ ea (C)/C (C) = 0 then the natural morphism of germs

r

Hilb hΣ , C −→ Def (C, zi ) i=1

r is smooth of fibre dimension h0 JZ ea (C)/C (C) . Here, i=1 Def (C, zi ) is the Cartesian product of the base spaces of the semiuniversal deformations of the germs (C, zi ). ea es (e) Write Zfix (C) for Zfix (C), respectively Zfix (C). Then the vanishing of 1 H JZfix (C)/C (C) implies that the morphism of germs Φh : Vh (S1 , . . . , Sr ), C → Symr Σ, (z1 +. . .+zr ) is smooth of fibre dimension h0 JZfix (C)/C (C) . We reformulate and strengthen Proposition of 1 in the case plane curves, which is of special interest. Of course, since h1 OP2 = h2 OP2 = 0, there is no difference whether we consider the curves in the linear system |dH|, H the hyperplane divisor, or curves with fixed Hilbert polynomial h(z) = dz − (d2 − 3d)/2. We denote the corresponding varieties by Vd = Vd (S1 , . . . , Sr ), respectively by Vd,fix (S1 , . . . , Sr ). Using the above notation, we obtain: Proposition 2 ([GL01, Prop. 2.8]). Let C ⊂ P2 be a reduced curve of degree d with precisely r singularities z1 , . . . , zr of analytic or topological types S1 , . . . , Sr . . (a) H 0 JZ (C)/P2 (d) H 0 (OP2 ) is isomorphic to the Zariski tangent space of Vd at C. (b) h0 JZ (C)/P2 (d) − h1 JZ (C)/P2 (d) − 1 ≤ dim(V d , C) ≤ h0 JZ (C)/P2 (d) − 1. (c) H 1 JZ (C)/P2 (d) = 0 iff Vd is T-smooth at C, that is, smooth of the expected dimension d(d + 3)/2 − deg Z (C). (d) H 1 JZ ea (C)/P2 (d) = 0 iff the natural morphism of germs 0

r P H OP2 (d) , C → i=1 Def (C, zi ) is smooth (hence surjective) of fibre dimension h0 JZ ea (C)/P2 (d) − 1. ea es (e) Write Zfix (C) for Zfix (C), respectively Zfix (C). Then H 1 JZfix (C)/P2 (d) vanishes iff the morphism of germs


181

Φd : Vd (S1 , . . . , Sr ), C → Symr P2 , (z1 +. . .+zr ) is smooth of fibre dimension h0 JZfix (C)/P2 (d) − 1. In particular, the van ishing of H 1 JZfix (C)/P2 (d) implies that arbitrarily close to C there are curves in Vd (S1 , . . . , Sr ) whose singularities are in general position in P2 .

3 T-Smoothness To show the T-smoothness of V|C| (S1 , . . . ,Sr ) it suffices to show, according to 1 ea Proposition 1 (c2), that H JZ (C)/Σ (C) = 0 (in the case of analytic types), 1 respectively H JZ es (C)/Σ (C) = 0 (in the case of topological types). Note that for Σ = P2 , these conditions are even equivalent to the T-smoothness of Vd (S1 , . . . , Sr ) by Proposition 2 (c). 3.1 ESF of Plane Curves The classical approach to the H 1 -vanishing problem (based on Riemann-Roch and Serre duality) leads to sufficient conditions for the T-smoothness of ESF of plane curves such as the 3d-condition (1) and its extensions mentioned above. In the papers [GLS97, GLS00], we applied two different approaches to the H 1 -vanishing problem, based on the Reider-Bogomolov theory of unstable rank 2 vector bundles (see also [CS97]), respectively on the Castelnuovo function of the ideal sheaf of a zero-dimensional scheme (see also [Dav86, Bar93a]). Both approaches lead to quadratic sufficient conditions for the T-smoothness of ESF of plane curves. Combining both approaches, we obtain: Theorem 1 ([GLS00, GLS01]). Let C ⊂ P2 be an irreducible curve of degree d > 5 having r singularities z1 , . . . , zr of topological (respectively analytic) types S1 , . . . , Sr as its only singularities. Then Vdirr (S1 , . . . , Sr ) is T-smooth at C if r γ (C, zi ) ≤ (d + 3)2 , (2) i=1

γ (C, zi ) = γ es (C, zi ) for Si a topological type, resp. γ (C, zi ) = γ ea (C, zi ) for Si an analytic type. Here, γ es and γ ea are new analytic invariants of singularities which are defined as follows (see [KL05] for a thorough discussion): The γ-invariant. Let f ∈ C{u, v} be a reduced power series, and let I ⊂ ∂f u, v ⊂ C{u, v} be an ideal containing the Tjurina ideal I ea (f ) = f, ∂f ∂u , ∂v . For each g ∈ u, v ⊂ C{x, y}, we introduce ∆(f, g; I) as the minimum among dimC C{u, v}/I, g and i(f, g) − dimC C{u, v}/I, g (where i(f, g) denotes

182


the intersection multiplicity of f and g, i(f, g) = dimC C{u, v}/f, g). By [Shu97, Lemma 4.1], this minimum is at least 1 so that we may define (dimC C{u, v}/I, g + ∆(f, g; I))2 γ f ; I := max , ∆(f, g; I) g∈x,y and, finally,

/ 0 γ ea (f ) := max γ f ; I I ⊃ I ea (f ) a complete intersection ideal , / 0 γ es (f ) := max γ f ; I I ⊃ I es (f ) a complete intersection ideal .

2 Note that γ (f ) ≤ τci (f ) + 1 , where τci (f ) stands for one of / 0 τci (f ) := max dimC C{u, v}/I I ⊃ I ea (f ) a complete intersection ideal ≤ τ (f ) , / 0 es τci (f ) := max dimC C{u, v}/I I ⊃ I es (f ) a complete intersection ideal ≤ τ es (f ) . Here, τ (f ) = dimC C{u, v}/I ea (f ) is the Tjurina number of f and τ es (f ) = dimC C{u, v}/I es (f ) is the codimension of the µ-constant stratum in the semiuniversal deformation of f (see [GLS00, Lemma 4.2]). In general, these bounds for the γ-invariant are far from being sharp. For instance, if f defines an ordinary singularity of order m ≥ 3 (that is, the m-jet of f is a reduced homogeneous polynomial of degree m), then γ es (f ) = 2m2 while (τ es (f ) + 1)2 = 12 m4 + O(m). See [KL05] for the case of semiquasihomogeneous singularities. For ESF of irreducible curves with nodes and cusps, respectively for ESF of irreducible curves with ordinary singularities, we obtain: Corollary 1. Vd (n · A1 , k · A2 ) is T-smooth or empty if 4n + 9k ≤ (d + 3)2 .

(3)

Corollary 2. If S1 , . . . , Sr are ordinary singularities of order m1 , . . . , mr , then Vd (S1 , . . . , Sr ) is T-smooth or empty if 4 · #(nodes) + 2 · m2i ≤ (d + 3)2 . (4) mi ≥3

In particular, it follows that Theorem 1 is asymptotically proper for ordinary singularities, since the inequality r

mi (mi − 1) ≤ (d − 1)(d − 2)

i=1

is necessary for the existence of an irreducible curve with ordinary singularities of multiplicities m1 , . . . , mr . More generally, by constructing series of ESF where the T-smoothness fails (see [Shu97, GLS97, Los02]), we proved the asymptotic properness of condition (2) in Theorem 1 for the case of semiquasihomogeneous singularities.


183

3.2 ESF of Curves on Smooth Algebraic Surfaces Let Σ be a smooth projective surface and D0 an effective divisor on Σ. In this situation, for lack of a generalization of the Castelnuovo function approach, so far only the Reider-Bogomolov approach leads to a quadratic sufficient condition for the T-smoothness of (non-empty) ESF V|dD0 | (S1 , . . . , Sr ). Set A(Σ, D0 ) :=

2 (D0 .KΣ )2 − D02 KΣ , 4

where KΣ is the canonical divisor on Σ. By the Hodge index theorem, this is a non-negative number if D0 or KΣ is ample. Theorem 2 ([GLS97, GLS06a]). Let C ⊂ Σ be an irreducible curve with precisely r singular points z1 , . . . , zr of topological or analytic types S1 , . . . , Sr , (C, z1 ) ≥ . . . ≥ τci (C, zr ). Assume that C and C − KΣ are ordered such that τci 2 2 ample and that C ≥ max {KΣ , A(Σ, C)}. If r

τci (C, zi )
2. Then, for any n > 2, there exists a hypersurface W ⊂ Pn of degree d having only isolated singular points, such that τ (W ) = τ (C) and h1 JZ ea (C)/P2 (d) = h1 JZ ea (W )/Pn (d) . Notice that, in view of the obstructed families given in [DPW00], this also yields that the inequality (15) cannot be improved by adding a constant.

4 Independence of Simultaneous Deformations The T-smoothness problem for equisingular families is closely related to the independence of simultaneous deformations of isolated singular points of a curve on a surface, or of a hypersurface in a smooth projective algebraic variety, and we present here sufficient conditions for the independence of simultaneous deformations which are analogous to the T-smoothness criteria of Sect. 3. 4.1 Joint Versal Deformations Let W be a hypersurface with r isolated singularities of analytic types S1 , . . . , Sr , lying in a smooth projective algebraic variety X. The obstructions to the versality of the joint deformation of the singularities of W induced by the linear system |W | lie in the group H 1 (JZ ea (W )/X (W )). In turn, the latter group is the obstruction to the T-smoothness of the germ at W of the ESF of all hypersurfaces W ∈ |W | having precisely r singularities of analytic types S1 , . . . , Sr (see Proposition 1 for the case that X is a surface). Thus, the aforementioned sufficient conditions for T-smoothness, formulated in Theorems 1, 2, 3, 4, 5 and in Corollaries 1, 3, 4, 6, are also sufficient conditions for the versality of the joint deformations of the singular points: Theorem 7. Let W be a hypersurface with only isolated singular points z1 , . . . , zr in a smooth projective algebraic variety X of dimension n ≥ 2. Then the germ at W of the linear system |W | induces a joint versal deformation of all the singularities of W , if one of the following conditions holds:

188


(i) X = P2 , and W = C is an irreducible curve of degree d such that (2) is satisfied with γ = γ ea ; if C has n nodes and k cusps as only singularities, condition (2) can be replaced by (3); (ii) X = Σ is a surface with Neron-Severi group L · Z, L being ample, and with canonical divisor KΣ = kΣ · L; W = C ∼ d · L is an irreducible curve, satisfying d ≥ max{kΣ + 1, −kΣ } and condition (14) with γ = γ ea ; (iii) X = Σ is a surface, W = C is an irreducible curve such that C, C − KΣ 2 are ample and C 2 ≥ max {KΣ , A(Σ, C)}, and Σ, C satisfy conditions (5) and (6) with τci = τci ; (iv) X = Σ is a surface with −KΣ nef, W = C is an irreducible curve such that C ∼ dD0 , D0 an ample divisor on Σ such that D02 ≥ A(Σ, D0 ), d > 0 2 /D02 and condition (7) with τci = τci (the latter condisatisfying d2 ≥ KΣ tion reduces to (8) if C has n nodes and k cusps as only singularities); (v) X = Σ ⊂ P3 is a hypersurface of degree e ≥ 5, W = C is an irreducible curve such that C ∼ d(e − 4)H, where H is a hyperplane section, d ∈ Q, d > 1, such that maxi τci (C, zi ) ≤ d − 2, and such that the conditions (10), = τci (the latter two conditions turn into (13), (11) are satisfied with τci if C has n nodes and k cusps as only singularities); (vi) X = Pn , n ≥ 2, W is a reduced hypersurface of degree d, and condition (15) is fulfilled. 4.2 Independence of Lower Deformations Also the T-smoothness of a topological equisingular family has a deformation theoretic counterpart: the independence of lower deformations of isolated singularities. Let W be a hypersurface with only isolated singular points z1 , . . . , zr in a smooth projective algebraic variety X of dimension n ≥ 2. For sake of simplicity, we assume that the singular points of W are all semiquasihomogeneous (SQH). That is, for each singular point zi , there are local analytic coordinates x = (x1 , . . . , xn ) on X such that the germ (W, zi ) is given by α Fi = A(i) (18) α x ∈ C{x} = C{x1 , . . . , xn } , i (α)≥ai

n

(i)

(i)

(i)

wj αj for some w(i) = (w1 , . . . , wn ) ∈ (Z>0 )n , the New (i) ton polygon of Fi is convenient and Fi,0 = i (α)=ai Aα xα defines an isolated singularity at the origin. We call Fi a SQH representative of (W, zi ) with principal part Fi,0 . Note that the class of SQH singularities includes all simple singularities. We fix SQH representatives F1 , . . . , Fr for (W, z1 ), . . . , (W, zr ). Then by a deformation pattern for (W, zi ) (respectively for Fi ), we denote any affine hypersurface of Cn given by a polynomial α α Gi = A(i) A(i) α x = Fi,0 + α x where i (α) =

j=1

0≤i (α)≤ai

0≤i (α) 1, such that maxi τci es = τci ; (11) are satisfied with τci n (vi) X = P , n ≥ 2, W is a reduced hypersurface of degree d, and condition (17) is fulfilled with q = 0.

5 Existence In the following, we describe two methods which lead to general numerical sufficient conditions for the existence of projective hypersurfaces with prescribed singularities. Both approaches are based on the reduction of the existence problem to an H 1 -vanishing problem for the ideal sheaves of certain zerodimensional schemes associated with topological, respectively analytic, types


191

of singularities. One way is to associate directly a zero-dimensional scheme corresponding to the r prescribed singularity types S1 , . . . , Sr (fixing the position of the singular points) and to produce a sufficient condition by using an appropriate H 1 -vanishing criterion. Another way is to construct, first, a projective hypersurface with ordinary singularities (in general position) and then to deform it into a hypersurface with the prescribed singularities using the patchworking construction ([Shu98, Shu05]). The sufficient conditions obtained by each of the two approaches do not cover the conditions obtained by the other approach in general. Hence, both methods are needed. 5.1 ESF of Plane Curves With a reduced plane curve germ (C, z) ⊂ (P2, z) we associate the following zero-dimensional schemes of P2 (with support {z}): • Z s (C, z), the singularity scheme, defined by the ideal / 0 )(q) for each q ∈ T ∗ (C, z) , I s (C, z) := g ∈ OP2 ,z mt g)(q) ≥ mt C where T ∗ (C, z) denotes the tree of essential infinitely near points, and g)(q) )(q) ) is the total transform of g (resp. of (C, z)) at q (see [GLS98a] (resp. C or [GLS06] for details); s (C, z) := Z s (CL, z), where L is a curve which is smooth and transversal • Zst to (the tangent cone of) C at z; • Z a (C, z), the scheme defined by the ideal I a (C, z) ⊂ OP2 ,z encoding the analytic type (see [GLS00] for a definition); a • Zst (C, z), the scheme defined by the ideal mz I a (C, z), where mz denotes the maximal ideal of OP2 ,z .

H 1 -Vanishing Approach. The following proposition allows us to deduce the existence of plane curves with prescribed singularities from an H 1 -vanishing statement: Proposition 3 ([Shu04]). (1) Given a zero-dimensional scheme Z ⊂ P2 , a point z ∈ P2 outside the support of Z and a reduced curve germ (C, z) ⊂ (P2, z) satisfying (20) H 1 JZ∪Zsts (C,z)/P2 (d) = 0 . 0 Then there exists a curve D ∈ H JZ∪Z s (C,z)/P2 (d) such that the germ of D at z is topologically equivalent to (C, z).Moreover, these curves D form a dense open subset in H 0 JZ∪Z s (C,z)/P2 (d) . (2) In the previous notation, let H 1 JZ∪Zsta (C,z)/P2 (d) = 0 .

(21)

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Then there exists a curve D ∈ H 0 JZ∪Z a (C,z)/P2 (d) such that the germ of D at z is analytically equivalentto (C, z). These curves D form a dense open subset in H 0 JZ∪Z a (C,z)/P2 (d) . Together with the H 1 -vanishing theorem for generic zero-dimensional schemes given in [Shu04] (using a Castelnuovo function approach), Proposition 3 yields: Theorem 10 ([Shu04]). Let (C1 , z1 ), . . . , (Cr , zr ) be reduced plane curve germs, let n be the number of nodes, k the number of cusps and t the number of A2m singularities, m ≥ 2, among the singularities (Ci , zi ), i = 1, . . . , r. (1) If 6n + 10k +

49 625 t+ 6 48

δ(Ci , zi ) ≤ d2 − 2d + 3 ,

(22)

(Ci ,zi )=A1 ,A2

then there exists a reduced, irreducible plane curve of degree d having r singular points topologically equivalent to (C1 , z1 ), . . . , (Cr , zr ), respectively, as its only singularities. (2) If 6n + 10k +

(Ci ,zi )=A1 ,A2

2 7µ(Ci , zi ) + 2δ(Ci , zi ) ≤ d2 − 2d + 3 , 6µ(Ci , zi ) + 3δ(Ci , zi )

(23)

then there exists a reduced, irreducible plane curve of degree d having r singular points analytically equivalent to (C1 , z1 ), . . . , (Cr , zr ), respectively, as its only singularities. See [Shu04] for a slightly stronger result. Note that condition (23) can be weakened to the following simple form (in view of δ ≤ 3µ/4 for reduced plane curve singularities different from nodes): r i=1

µ(Ci , zi ) ≤

1 2 (d − 2d + 3) . 9

(24)

Next, we pay special attention to the case of curves with exactly one singular point, because such curves are an essential ingredient for the patchworking approach to the existence problem. Curves with one Singular Point and Order of T-existence. Let (C, z) be a reduced plane curve singularity. Denote by es (C, z), resp. ea (C, z), the minimal degree d of a plane curve D ⊂ P2 whose singular locus consists of a unique point w such that (D, w) is topologically (resp. analytically) equivalent to (C, z) and which satisfies the condition H 1 JZ es (D)/P2 (d − 1) = 0 , resp. H 1 JZ ea (D)/P2 (d − 1) = 0 . (25) We call es (C, z) (resp. ea (C, z)) the order of T-existence for the topological (resp. analytic) singularity type represented by (C, z).


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Lemma 2. (1) Let D be a plane curve as in the definition of the order of Texistence, and let L be a straight line which does not pass through the singular point w of D. Then the germ at D of the family of curves of degree d having in a neighbourhood of w a singular point which is topologically (respectively analytically) equivalent to (C, z) is smooth of the expected dimension, and it intersects transversally the linear system 4 3 G ∈ H 0 OP2 (d) G ∩ L = D ∩ L . (2) Let L ⊂ P2 be a straight line. Then the set of d-tuples (z1 , . . . , zd ) of distinct points on L for which there is a curve D of degree d as in the definition of the order of T-existence satisfying D ∩ L = {z1 , . . . , zd } is Zariski open in Symd (L). Combining Theorem 10 and the existence result for plane curves with simple singularities in [Los99], we get the following estimates for es and ea : Theorem 11. If (C, z) is a simple plane curve singularity then ⎧ √ ⎪ if (C, z) of type Aµ , µ ≥ 1 , ⎨ ≤ 2(√µ + 5) es (C, z) = ea (C, z) ≤ 2( µ + 7) + 1 if (C, z) of type Dµ , µ ≥ 4 , ⎪ ⎩ = (µ/2) + 1 if (C, z) of type Eµ , µ = 6, 7, 8. If (C, z) is not simple, then 25 5 es (C, z) ≤ √ δ(C, z) − 1 , 4 3 5 7µ(C, z) + 2δ(C, z) − 1 ≤ 3 µ(C, z) − 1 . ea (C, z) ≤ 5 6µ(C, z) + 3δ(C, z) For simple singularities with small Milnor number, these estimates are far from being sharp. For instance, it is well-known that ⎧ ⎧ 3 if µ = 4 , ⎪ ⎪ 4 if 3 ≤ µ ≤ 7 , ⎨ ⎨ 4 if µ = 5 , s s e (Aµ ) = 5 if 8 ≤ µ ≤ 13 , e (Dµ ) = 5 if 6 ≤ µ ≤ 10 , ⎩ ⎪ ⎪ 6 if 14 ≤ µ ≤ 19 , ⎩ 6 if 11 ≤ µ ≤ 13 . Moreover, es (E6 ) = es (E7 ) = 4, es (E8 ) = 5. Curves with many Singular Points (Patchworking Approach). The following proposition is a special case of Proposition 5 below, which is proved by a reasoning based on patchworking:

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Proposition 4. Let (C1 , z1 ), . . . , (Cr , zr ) be reduced plane curve singularities. Let mi := es (Ci , zi ) (resp. mi := ea (Ci , zi )), and assume that H 1 JZ(m)/P2 (d − 1) = 0 , (26) where Z(m) is the fat point scheme supported at the (generic) points p1 , . . . , pr i of P2 and defined by the ideals mm pi . Moreover, let d > max es (Ci , zi ) , resp. d > max ea (Ci , zi ) . 1≤i≤r

1≤i≤r

Then there exists a reduced, irreducible plane curve D ⊂ P2 of degree d with Sing(D) = {p1 , . . . , pr } such that each germ (D, pi ) is topologically (resp. analytically) equivalent to (Ci , zi ), i = 1, . . . , r. Applying the H 1 -vanishing criterion of [Xu95, Thm. 3], we immediately derive Corollary 7. Let (C1 , z1 ), . . . , (Cr , zr ) be reduced plane curve singularities such that es (C1 , z1 ) ≥ . . . ≥ es (Cr , zr ). If es (C1 , z1 ) + es (C2 , z2 ) ≤ d − 1 , as r ≥ 2 , e (C1 , z1 ) + . . . + es (C5 , z5 ) ≤ 2d − 2 , as r ≥ 5 , r 9 (d + 2)2 , (es (Ci , zi ) + 1)2 < 10 i=1 s

then there exists a reduced, irreducible plane curve C of degree d with exactly r singular points p1 , . . . , pr , such that each germ (C, pi ) is topologically equivalent to (Ci , zi ), i = 1, . . . , r. The same statement holds true if we replace es by ea and the topological equivalence relation by the analytic one. Comparing the sufficient conditions of Theorem 10 and of Corollary 7, we see that the existence criterion of Theorem 10 is better for non-simple singularities, whereas the criterion obtained from Corollary 7 and the estimates in Theorem 11 is better for simple singularities. 5.2 ESF of Curves on Smooth Projective Surfaces In [KT02], the following sufficient criterion for the existence of curves with prescribed singularities is proved: Proposition 5 ([KT02]). Let Σ be a smooth projective algebraic surface, D a divisor on Σ, and L ⊂ Σ a very ample divisor. Let (C1 , z1 ), . . . , (Cr , zr ) be reduced plane curve singularities. Let H 1 JZ(m)/Σ (D − L) = 0 , (27) max mi < L(D − L − KΣ ) − 1 , (28) 1≤i≤r


195

where Z(m) ⊂ Σ is the fat point scheme supported at some (generic) points s i p1 , . . . , pr ∈ Σ and defined by the ideals mm pi , with mi = e (Ci , zi ), respectively a mi = e (Ci , zi ), i = 1, . . . , r. Then there exists an irreducible curve C ∈ |D| such that Sing(C) = {p1 , . . . , pr } and each germ (C, pi ) is topologically, resp. analytically, equivalent to (Ci , zi ), i = 1, . . . , r. Combining this with the H 1 -vanishing criterion of [KT02, Cor. 4.2] and with the estimates for es , ea in Theorem 11, we get the following explicit numerical existence criterion: Theorem 12 ([GLS06a]). Let Σ be a smooth projective algebraic surface, D a divisor on Σ with D − KΣ nef, and L ⊂ Σ a very ample divisor. Let (C1 , z1 ), . . . , (Cr , zr ) be reduced plane curve singularities, among them n nodes and k cusps. (1) If 18n + 32k +

625 24

δ(Ci , zi ) ≤ (D − KΣ − L)2 ,

(29)

δ(Ci ,zi )>1

5 25 √ max δ(Ci , zi ) + 1 < (D − L − KΣ ).L , 4 3 1≤i≤r

(30)

and, for each irreducible curve B with B 2 = 0 and dim |B|a > 0, 5 25 √ max δ(Ci , zi ) < (D − KΣ − L).B + 1 , 1≤i≤r 4 3

(31)

then there exists a reduced, irreducible curve C ∈ |D| with r singular points topologically equivalent to (C1 , z1 ), . . . , (Cr , zr ), respectively, as its only singularities. (2) If

18n + 32k + 18

µ(Ci , zi ) ≤ (D − KΣ − L)2 ,

(32)

µ(Ci ,zi )>2

3 max

1≤i≤r

5 µ(Ci , zi ) + 1 < (D − L − KΣ ).L ,

and, for each irreducible curve B with B 2 = 0 and dim |B|a > 0, 5 3 max µ(Ci , zi ) < (D − KΣ − L).B + 1 , 1≤i≤r

(33)

(34)

then there exists a reduced, irreducible curve C ∈ |D| with r singular points analytically equivalent to (C1 , z1 ), . . . , (Cr , zr ), respectively, as its only singularities. Here |B|a means the family of curves algebraically equivalent to B. A discussion of the hypotheses of Theorem 12 for specific classes of surfaces as well as concrete examples can be found in [KT02].

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5.3 ESF of Hypersurfaces in Pn For singular hypersurfaces in Pn , n ≥ 3, no general asymptotically proper sufficient condition for the existence of hypersurfaces with prescribed singularities (such as (24) in the case of plane curves) is known. But, restricting ourselves to the case of only simple singularities, in [SW04] even an asymptotically optimal condition is given. To formulate this result, we need some notation: let S be a finite set of analytic types of isolated hypersurface singularities in Pn . Define αn (S) = lim sup d→∞ Wd

τ (Wd ) , dn

where Wd runs over the set of all hypersurfaces Wd ⊂ Pn of degree d whose singularities are of types S ∈ S and which belong to the T-smooth component of the corresponding ESF. Here, τ (Wd ) stands for the sum of the Tjurina numbers τ (Wd , z) over all points z ∈ Sing(Wd ). By αnR (S) we denote the respective limit taken over hypersurfaces having only real singular points of real singularity types S ∈ S. Clearly, αnR (S) ≤ αn (S) ≤ 1/n!. Theorem 13 ([SW04, Wes03, Wes04]). Let n ≥ 2. (1) For each finite set S of simple hypersurface singularities in Pn , we have αnR (S) = αn (S) =

1 . n!

(2) For each finite set S of analytic types of isolated hypersurface singularities of corank 2 in Pn , we have αn (S) ≥ αnR (S) ≥

1 . 9n!

The proof exploits again the patchworking construction. It is based on the following fact: for each simple singularity type S and each n ≥ 2, there exist an n-dimensional convex lattice polytope ∆n (S) of volume µ(S)/n! and a polynomial F ∈ C[x1 , . . . , xn ] (resp. F ∈ R[x1 , . . . , xn ]) with Newton polytope ∆n (S) which defines a hypersurface in the toric variety Tor(∆n (S)) (associated to ∆n (S)) having precisely one singular point of type S in the torus (C∗ )n (resp. in the torus (R∗ )n ) and being non-singular and transverse along the toric divisors in Tor(∆n (S)).

6 Irreducibility The question about the irreducibility of ESF V|C| (S1 , . . . , Sr ) is more delicate than the existence and smoothness problem, in particular, if one tries to find sufficient conditions for the irreducibility. The results are by far not that


197

complete as for the other two problems. The irreducibility problem is of special topological interest, since it is connected with the problem of having within the same ESF different fundamental groups of the complement of a plane algebraic curve. As pointed out in the introduction, even the case of plane nodal curves (Severi’s conjecture) appeared to be very hard. The examples of reducible ESF listed below indicate that for more complicated singularities, beginning with cusps, possible numerical sufficient conditions for the irreducibility should be rather different with respect to their asymptotics to the necessary existence conditions (as discussed in Sect. 5). Approaches to the Irreducibility Problem. (1) One possible approach (for ESF of plane curves) consists of building for any two curves in the ESF a connecting path, using explicit equations of the curves, respectively of projective transformations. This method works for small degrees only. Besides the classical case of conics and cubics, this method has been used to prove that all ESF of quartic and quintic curves are irreducible (cf. [BG81, Wal96]). But for degrees d > 5, this is no longer true and the method is no more efficient (except for some very special cases). (2) Arbarello and Cornalba [AC83] suggested another approach. It consists of relating the ESF to the moduli space of plane curves of given genus, which is known to be irreducible (cf. [DM69]). This gave some particular results on families of plane nodal curves and plane curves with nodes and cusps. Namely, Kang [Kan89] proved that the variety Vdirr (n · A1 , k · A2 ) is irreducible whenever d2 − 4d + 1 (d − 1)(d − 2) ≤ n ≤ , 2 2

k ≤

d+1 . 2

(35)

(3) Harris introduced a new idea to the irreducibility problem, which completed the case of plane nodal curves ([Har85a]). This new idea was to proceed inductively from rational plane nodal curves (whose family is classically known to be irreducible) to any family of plane nodal curves of a given genus. Further development of this idea lead to new results by Ran [Ran89] and by Kang [Kan89a]: if Om denotes an ordinary singularity of order m ≥ 2, then Ran showed that, for each n ≥ 0, the variety Vdirr (n · A1 , 1 · Om ) is irreducible (or empty). Kang’s result says that, for each n ≥ 0, k ≤ 3, the variety Vdirr (n · A1 , k · A2 ) is irreducible (if non-empty). However, the requirement to study all possible deformations of the considered curves does not allow to extend such an approach to more complicated singularities, or to a large number of singularities different from nodes. (4) Up to now, there is mainly one approach which is applicable to equisingular families of curves of any degree with any quantity of arbitrary singularities (and even to projective hypersurfaces of any dimension). The basic idea is to find an irreducible analytic space M(S1 , . . . , Sr ) and a dominant morphism

198


V|D| (S1 , . . . , Sr ) −→ M(S1 , . . . , Sr ) with equidimensional and irreducible fibres. It turns out, that in such a way proving the irreducibility of V|D| (S1 , . . . , Sr ) can be reduced to an H 1 vanishing problem: let Z(C) = Z s (C) (resp. Z(C) = Z a (C)) be the zerodimensional schemes encoding the topological (resp. analytic) type of the singularities (see Sect. 5). Then the variety V|D| (S1 , . . . , Sr ) is irreducible if 1 H JZ(C)/Σ (D) = 0 for each C ∈ V|D| (S1 , . . . , Sr ). For a detailed discussion of the latter approach, we refer to [GLS00, Kei03]. Combining this approach with [Xu95, Thm. 3] and another H 1 -vanishing theorem based on the Castelnuovo function approach, we obtain: Theorem 14 ([GLS00]). Let S1 , . . . , Sr be topological or analytic types of plane curve singularities, and d an integer. If max τ (Si ) ≤ (2/5)d − 1 and i=1..r

2 25 10 · #(nodes) + 18 · #(cusps) + · τ (Si ) + 2 < d2 , 2 9 τ (Si )≥3

then Vdirr (S1 , . . . , Sr ) is non-empty and irreducible. Here, τ (Si ) stands for τ es (Si ) if Si is a topological type and for τ (Si ) if Si is an analytic type. In particular, Corollary 8. Let d ≥ 8. Then Vdirr (n · A1 , k · A2 ) is irreducible if 25 n + 18k < d2 . 2

(36)

Corollary 9. Let S1 , . . . , Sr be ordinary singularities of order m1 , . . . , mr , and assume that max mi ≤ (2/5) d. Then Vdirr (S1 , . . . , Sr ) is non-empty and irreducible if m2 (mi + 1)2 25 i · #(nodes) + < d2 . (37) 2 4 mi ≥3

Reducible Equisingular Families. In [GLS00], we apparently gave the first series of reducible ESF of plane cuspidal curves, where the different components cannot be distinguished by the fundamental group of the complement of the corresponding curves. If this happens, we say that the ESF has components which are anti-Zariski pairs. The following proposition gives infinitely many ESFs with anti-Zariski pairs: Proposition 6 ([GLS00]). Let p, d be integers satisfying 6 p ≥ 15, 6p < d ≤ 12p − 32 − 35p2 − 15p + 14 .

(38)


199

Then the variety Vdirr (6p2 · A2 ) of irreducible plane curves of degree d with 6p2 cusps has components of different dimensions. Moreover, for all curves C ∈ Vdirr (6p2 · A2 ) the fundamental group of the complement is π1 (P2 \ C) = Z/dZ. irr (1350 · A2 ) is reducible, and it has components For instance, the variety V91 which are anti-Zariski pairs. Further, in [GLS01], we gave a series of reducible ESF of plane curves with only ordinary singularities having components which are anti-Zariski pairs:

Proposition 7 ([GLS01]). Let m ≥ 9. Then there is an integer 0 = 0 (m) such that for each ≥ max{0 , m} and for each s satisfying * 7 + 2 3 −1 ≤ s ≤ 1− − 2 m 2 irr (2 · Om ) of plane irreducible curves of degree m + s having the variety Vm+s 2 ordinary singularities of order m as only singularities is reducible. irr (2 · Om ) has at least two components, one regular More precisely, Vm+s component (of the expected dimension) and one component of higher dimension. And, for each curve C belonging to any of the components, we have π1 (P2 \ C) = Z/(m + s)Z.

7 Open Problems and Conjectures Though some results discussed above are sharp, others seem to be far from a final form, and here we start with a discussion and conjectures about the expected progress in the geometry of families of singular curves. Further discussion concerns possible generalizations of the methods and open questions. 7.1 ESF of Curves Existence of Curves with Prescribed Singularities. A natural question about the existence results for algebraic curves given in Sect. 3 concerns a possible improvement of the asymptotically proper conditions to asymptotically optimal ones: How to improve the constant coefficients in the general sufficient conditions for the existence ? Concerning our method based on H 1 -vanishing for the ideal sheaves, a desired improvement would come from finding better H 1 -vanishing conditions for generic zero-dimensional schemes. For instance, from proving the HarbourneHirschowitz conjecture, which gives (if true) the best possible H 1 -vanishing criterion for ideal sheaves of generic fat schemes Z(m).

200


Another type of questions concerning curves with specific singularities is the following: the known sufficient and necessary conditions for the existence of singular plane curves are formulated as bounds to sums of singularity invariants. But it seems that there cannot be a general condition of this type which is sufficient and necessary at the same time. The simplest question of such kind is: Are there k and d such that a curve of degree d with k cusps does exist, but with k < k cusps does not ? A candidate could be Hirano’s series of cuspidal curves mentioned in the introduction. T-Smoothness and Versality of Deformations. The following conjecture about the asymptotic properness of the sufficient conditions for the Tsmoothness of topological ESF of plane curves given in Sect. 3 seems to be quite realistic (and holds for semiquasihomogeneous singularities): Conjecture 1. There exists an absolute constant A > 0 such that for each topological singularity type S there are infinitely many pairs (r, d) ∈ N 2 such that Vdirr (r · S) is empty or non-smooth or has dimension greater than the expected one and r · γ(S) ≤ A · d2 . We propose a similar conjecture for analytic ESF of plane curves, though it is confirmed only for simple singularities (in which case it coincides with the conjecture for topological ESF). A closely related question, belonging to local singularity theory, concerns the γ-invariant: Find an explicit formula, or an algorithm to compute γ es (f ), γ ea (f ). Find (asymptotically) close lower and upper bounds for these invariants. Is γ es a topological invariant ? Irreducibility Problem. Our sufficient irreducibility conditions seem to be far from optimal ones. We state the problem: Find asymptotically proper sufficient conditions for the irreducibility of ESF of plane curves (or show that the conditions in Sect. 6 are asymptotically proper). We also rise the following important question: Does there exist a pair of plane irreducible algebraic curves of the same degree with the same collection of singularities, which belong to different components of an ESF but are topologically isotopic in P2 (anti-Zariski pair) ? The examples in Sect. 6 provide candidates for this – reducible ESF, whose members have the same (Abelian) fundamental group of the complement.


201

7.2 Hypersurfaces in Higher-Dimensional Varieties One can formulate the existence, T-smoothness, and irreducibility problems for families of hypersurfaces with isolated singularities, belonging to (very) ample linear systems on projective algebraic varieties. To find a relevant approach to these problems is the most important question. For hypersurfaces of dimension > 1 there exists no infinitesimal deformation theory for topological types. So, we restrict ourselves to analytic types here. Constructions of curves with prescribed singularities as presented in Sect. 5 can, in principle, be generalized to higher dimensions. An expected analogue of the results for curves could be Conjecture 2. Given a very ample linear system |W | on a projective algebraic variety X of dimension n, there exists a constant A = A(X, W ) > 0 such that, for each collection S1 , . . . , Sr of singularity types and for each positive integer d r satisfying i=1 µ(Si ) < Adn , there is a hypersurface Wd ∈ |dW | with exactly r isolated singularities of types S1 , . . . , Sr , respectively. In view of the patchworking approach, to prove the conjecture, it is actually enough to consider the case of ordinary singularities and to answer the following analogue of one of the above questions affirmatively: Does there exist some number A(n) > 0 such that, for each analytic type S of isolated hypersurface singularities in Pn , there exists a hypersurface of Pn of degree d ≤ A(n) · µ(S)1/n which has a singularity of type S and no nonisolated singularities ? This is known only for simple singularities (see [Wes03, Shu04]). Hypersurfaces with specific singularities (such as nodes) attract the attention of many researches, mainly looking for the maximal possible number of singularities (see, for instance, [Chm92]). We would like to raise the question about an analogue of the Chiantini-Ciliberto theorem for nodal curves on surfaces (see [CC99]) as a natural counterpart, concerning the domain with regular behaviour of ESF: Given a projective algebraic variety X and a very ample linear system |W | on it with a non-singular generic member. Prove that, for any r ≤ dim |W | there exists a hypersurface Wr ∈ |W | with r nodes as its only singularities such that the germ at W of the corresponding ESF is T-smooth. 7.3 Related Problems Enumerative Problems. Recently, the newly founded theories of moduli spaces of stable curves and maps, Gromov-Witten invariants, quantum cohomology, as well as deeply developed methods of classical algebraic geometry and algebraic topology have led to a remarkable progress in enumerative geometry, notably for the enumeration of singular algebraic curves (see, for

202


example, [KM94, CH98, CH98a, GP98] for the enumeration of rational nodal curves on rational surfaces, see [Ran89a, CH98b] for the enumeration of plane nodal curves of any genus, see [Kaz03, Liu00] for counting curves with arbitrary singularities). We point out that the questions to which this survey has been devoted, such as on the existence of certain singular algebraic curves, on the expected dimension and on the transversality of the intersection of ESF, are unavoidable in all of the above approaches to enumerative geometry. The affirmative answers to such questions are necessary for attributing an enumerative meaning to the computations in the aforementioned works. We pose the problem to find links between the methods discussed above and the methods of enumerative geometry, and we expect that this would lead to a solution for new enumerative problems and to a better understanding of known results. As an example, we mention the tropical enumerative geometry [Mik03, Mik05, Shu05], in which the patchworking construction and, more generally, the deformation theory play an important role. Non-Isolated Singularities. None of the problems discussed above is even well-stated for non-reduced curves, or hypersurfaces with non-isolated singularities. We simply mention this as a direction for further study.

Acknowledgments Work on the results presented in this paper has been supported by the DFG Schwerpunkt “Globale Methoden in der komplexen Geometrie” and by the Hermann Minkowski – Minerva Center for Geometry at Tel Aviv University.

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Critical Points of the Square of the Momentum Map Peter Heinzner and Henrik St¨ otzel Fakult¨ at f¨ ur Mathematik Ruhr Universit¨ at Bochum Universit¨ atsstrasse 150 D-44780 Bochum [email protected], [email protected]

1 Introduction Let Z be a Kählerian manifold endowed with a holomorphic action of a complex linear reductive group. As a convenient definition in our context we call a complex Lie group reductive if it is the universal complexification of a compact Lie group U and denote it by U C . We also assume that the action of U on Z is Hamiltonian. By this we mean that the K¨ ahlerian form ω on Z is U -invariant and that there is a momentum map µ : Z → u∗ . Here u denotes the Lie algebra of U and u∗ its dual. Besides of U -equivariance the defining condition for µ is given by dµξ = ıξZ ω. Here ıξZ ω denotes the one form on Z which is obtained by contracting ω with the vector field ξZ on Z whose one-parameter group of transformations is given by (t, z) → exp tξ · z and µξ (z) := µ(z), ξ := µ(z)(ξ). Our principal case of interest here concerns the geometry of the action of a real form G of U C on Z. By definition this means that G is a subgroup of U C which is a union of connected components of the set (U C )σ˜ := {g ∈ U C ; σ ˜ (g) = ˜ is an antiholomorphic involution. A priori g} of σ ˜ fixed points in U C , where σ G and U are not related in an obvious way. But using basic results in Lie theory it turns out that there is an anti-holomorphic Lie group involution θ : U C → U C such that U = (U C )Θ = {g ∈ U C ; θ(g) = g} and an antiholomorphic involution σ : U C → U C which commutes with θ such that G is conjugate to an open subgroup of (U C )σ . For simplicity we will always assume here that σ ˜ = σ. This in particular implies that G is compatible with the Cartan decomposition of U C . By definition an arbitrary Lie subgroup G

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of U C is said to be compatible with the Cartan decomposition of U C if the map K × p → G, (k, ξ) → k exp ξ is a diffeomorphism. Here K := G ∩ U and p = g ∩ iu where g denotes the Lie algebra of G. For a compatible Lie subgroup G we have the inclusion ip ⊂ u and by restriction we obtain a K-equivariant map µip : Z → (ip)∗ . As we will see µip contains a lot of information about the G-action on Z. For example, it gives the notion of semistable points with respect to G and as we will explain here they play a similar role as the set of semistable points in the sense of Mumfords Geometric Invariant Theory. The paper does not contain complete proofs. If not stated otherwise they are given in [HeSch05] or [HeSt05].

2 Elementary Properties of Hamiltonian Actions and Examples Let Z be a Kählerian manifold endowed with a holomorphic action of a complex reductive group U C and let µ : Z → u∗ be a momentum map. Before we state some elementary properties of µ let us consider some examples. Example 1. Consider the case where the Kählerian form is given by a global U invariant potential, i.e. ω = −ddc ρ for a strictly plurisubharmonic U -invariant function ρ on Z where dc ρ := dρ ◦ J and J denotes the complex structure on Z. Then the U -action is Hamiltonian and a momentum map is given by ∂ µξ (z) = dc ρ(ξZ )(z) = ρ(exp(itξ) · z). ∂t 0 Example 2. Let R : U C → GL(V ) be a holomorphic representation on a Hermitian vector space (V, , ). By integrating over U we may assume that the representation of U is unitary. Then the squared norm function ρ(v) := 12 v, v is U -invariant and strictly plurisubharmonic and therefore induces a K¨ ahlerian form on V . Using Example 1 we obtain the momentum map µξ (v) = i · ξV (v), v. Example 3. Consider the projective space P(V ) with a U C -action which is induced by a representation on V . As in Example 2 we assume that V is equipped with a U -invariant Hermitian product , . It induces the FubiniStudy metric on P(V ) which defines an up to multiplication with a positive constant unique K¨ ahlerian form invariant under the action of the unitary (v),v group of V . In this setting a momentum map is given by µξ (π(v)) = i· ξVv,v where π : V → P(V ) denotes the canonical projection.

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Example 4. Let Z = U C and assume that U is realized as a subgroup of the unitary group U (n). The standard Hermitian product on the space of complex n×n matrices is given by z, w = tr(zwt ). The function ρ(k exp(ξ)) := 12 ξ, ξ where k ∈ U and ξ ∈ iu is strictly plurisubharmonic on Z and consequently defines a K¨ ahlerian form. The actions (u, z) → u · z and (u, z) → z · u−1 of U on Z given by left multiplication and by right multiplication are Hamiltonian since ρ is invariant with respect to this U × U -action. Computing the momentum map by using the formula in Example 1, we get µ(k exp(ξ)) = −i Ad(k)(ξ) for the left multiplication and µ(k exp(ξ)) = iξ for the right multiplication. Here we have identified u with its dual by , . Example 5. The groups SLn (R), SU(n) and SU(p, q) where p + q = n are real forms of SLn (C). Other compatible subgroups of SLn (C) are SOn (R), SOn (C), Spn (R) and Spn (C). All these groups are compatible subgroups of GLn (C) = U (n) exp(iu(n)). Remark 1. Compatible subgroups of U C are not necessarily closed. For example, a dense one-parameter subgroup of the torus S 1 × S 1 is a compatible subgroup of C∗ × C∗ . Remark 2. Any connected compatible subgroup G of a complex reductive group U C is a product G = G0 · G1 · G2 of a group G0 which is a real form of a semisimple complex reductive subgroup (U0 )C of U C , a semisimple complex reductive subgroup G1 of U C and a subgroup G2 which lies in the center of G. The groups G0 , (U0 )C , G1 and G2 are compatible subgroups of U C and the pairwise intersections of G0 , G1 and G2 are finite subgroups of the center of G. Since we are interested in the relation between the momentum map and the action of G rather than the action of U C , we consider for a subspace m of u which usually will be ip the map µm : Z → m∗ which is given by composing µ with the adjoint of the inclusion map m → u. We call it the restriction of µ to m. We fix a U -invariant inner product , on u which is used to identify u equivariantly with its dual u∗ . The restriction of µ to m is then obtained by the orthogonal projection of u onto m. If G is a real form of U C which is compatible with the Cartan decomposition , then the direct sum decomposition µ = µk + µip is called the Cartan decomposition of the momentum map µ. Note that µk is a momentum map with respect to the action of K on Z. We will deal with the K-equivariant map µip in the first place. For a subspace n of uC let n · z := {ξZ (z); ξ ∈ n} ⊂ Tz (Z) and nz := {ξ ∈ n; ξZ (z) = 0}. For a subset V of Tz (Z) let V ⊥ be the perpendicular of V with respect to the Riemannian metric given by the K¨ ahlerian form ω of Z. We collect a few elementary properties of the restricted momentum map. Lemma 1. Let m ⊂ u be a subspace. Then ker dµm (x) = (im · x)⊥ .

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Lemma 2. The function γ(t) := µξ (exp(itξ) · z) is strictly monotone increasing for all z ∈ Z and all ξ ∈ u, ξ ∈ / uz . For β ∈ ip∗ let Mip (β) := µ−1 ip (β) and since β = 0 plays a prominent role we set Mip := Mip (0). Using Lemma 2 and the K-equivariance of µip , one can proof the following Corollary 1. Let z ∈ Mip . Then 1. G · z ∩ Mip = K · z. 2. The G-isotropy group Gz of z is compatible with the Cartan decomposition of U C . The decomposition is given by Gz = Kz exp(pz ). Example 6. Let Z be the projective space Pn (C) equipped with the natural action of U C = SLn+1 (C) and the momentum map of Example 3. Then G := SU(1, n) is a compatible real form of UC with G = K exp(p) where K = t S(U (1) × U (n)) ∼ = Cn ∪ = U (n) and p = { z0 z0 ; z ∈ Cn }. If we write Z ∼ n Pn−1 (C) where the inclusion ι : C → Pn (C) is given by ι(z) = [1, z] then a computation shows Mip = {0} ∪ Pn−1 (C). Notice that the K-orbits in Z are Pn−1 (C) and the spheres in Cn . The isotropy of G at 0 is given by K and isomorphic ( to U (1, n − 1). Concretely, the isotropy at points of Pn−1 (C) is ' 0 −1 ; A ∈ U (1, n − 1)}. This for z0 := [0, . . . , 0, 1] we get Gz0 = { A0 det(A) example will be continued in Sects. 5 and 8.

3 Linearization Let Z be a Kählerian manifold with a holomorphic U C -action and U -equivariant momentum map µ : Z → u∗ . A very important property of the U C -action is that it is linearizable at fixed points. More precisely, let x ∈ Z and let L := Ux denote the isotropy group of U at x. Let LC = L exp il denote the complexification of L. Note that LC ⊂ (U C )x is a complex Lie subgroup of the isotropy group of U C at x. The group LC acts on the tangent space Tx (Z) by the isotropy representation, (g, v) → g∗ (x) · v where g∗ (x) : Tx (Z) → Tx (Z) denotes the derivative of z → g · z for g ∈ LC . In fact the LC -action on Tx (Z) is a local model for the LC -action on Z in an open LC -stable neighborhood of x. This is the content of the following Theorem 1 (Linearization Theorem). There exists an LC -stable open neighborhood S of 0 ∈ Tx (Z), an LC -stable open neighborhood Ω of x in Z and an LC -equivariant biholomorphic map Ψ : S → Ω such that Ψ (0) = x. Corollary 2. If x ∈ Mip then S and Ω are Gx -invariant and Ψ is Gx equivariant. Proof. The isotropy Gx is compatible with the Cartan decomposition of U C (Corollary 1). Since the action of U C is holomorphic, Gx is a subgroup of LC .


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Remark 3. In general (Ux )C is a proper subgroup of (U C )x and there is no (U C )x -equivariant linearization map Ψ . Example 7. Let U := SU2 (C). Then U C = SL2 (C) / acts on the 0projective line 1 ] we get (U )C = ( t 0 ) ; t2 = 1 and (U C ) = by (g, [z]) → [gz]. For x = [ P x x 0 t /1 t u 0 0 ( 0 t ) ; t2 = 1 and u ∈ C . The tangent space Tx (P1 ) is one dimensional and the isotropy representation of (U C )x is given by ( 0t ut ) → t12 idTx (P1 ) .

4 The Momentum Norm Function Recall that for a subspace m of u we have the restriction µm : Z → m∗ of µ to m which is induced by the orthogonal projection of u onto m with respect to a U -invariant inner product , on u. The associated norm square function will be denoted by ηm (z) := 12 "µm "2 . Lemma 3. Let x ∈ Z and set βm := µm (x). The following are equivalent 1. (βm )Z (x) = 0 2. dµβmm (x) = 0 and 3. dηm (x) = 0. Example 8. Let G = K exp(p) be compatible with the Cartan decomposition of U C and let βip be a value of µip which is critical for ηip . Then the lemma above applies. In fact this is our main example. For a fixed x ∈ Z we consider now Tx (Z) as the isotropy representation space of (U C )x . Let ξ ∈ iux and fix an open (Ux )C -stable neighborhood S of 0 ∈ Tx Z and an open (Ux )C -equivariant embedding Ψ : S → Z such that Ψ (0) = x holds (Linearization Theorem). Note that ξ acts as a self-adjoint operator on Tx (Z), i.e., the action of ξ gives a direct sum decomposition of Tx (Z) into one-dimensional eigenspaces with real eigenvalues. Let Tx (Z) = V− ⊕ V0 ⊕ V+ be the decomposition of Tx (Z) into the sum of negative, zero and positive eigenspaces of the action of ξ. It should also be noted that V− ∪ V+ ⊂ S holds automatically. Proposition 1. Let v ∈ Tx (Z) be an eigenvector of ξ = iβ ∈ iux with eigenvalue λ(ξ), set γv (t) := tv and let γ = Ψ ◦ γv . Then 2

d β 1. dt ◦ γ)(0) = λ(ξ)"v"2 2 (µ 2. If ξ = iµm (x) where x is a critical point of ηm , then

d2 (ηm ◦ γ)(0) = λ(ξ)"v"2 + "dµm (x)(v)"2 dt2

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The proposition is seen as follows. The vector field ξTx (Z) given by ξ on Tx (Z) is linear and therefore ξTx (Z) (tv) = tλ(ξ)γ˙ v (t) holds. Here γ˙ v ded notes the tangent curve of γv and similarly below. This gives dt (µβ ◦ γ)(t) = 2 . The first statement tω(−iλ(ξ)γ(t), ˙ γ(t)) ˙ = tλ(ξ)ω(γ(t), ˙ iγ(t)) ˙ = tλ(ξ)"γ(t)" ˙ is obtained by differentiating this equality. d2 d2 βm ◦γ)(0)+dµ(x0 )(v), dµ(x0 )(v) For ξ = iβm we get dt 2 (ηm ◦γ)(0) = dt2 (µ and therefore the second statement. In general, if ξ = iβ acts on a finite dimensional vector space W such that all eigenvalues of ξ are real and the representation is completely reducible, i.e., ξ acts on W as a self-adjoint operator with respect to some inner product, then there is a ξ-stable decomposition W = W− ⊕ W0 ⊕ W+ where W− denotes the sum of eigenspaces of ξ which are strictly negative, W0 is the zero eigenspace and W+ denotes the sum of eigenspaces of ξ which are strictly positive. In the next example we apply this to the adjoint action of U C on its Lie algebra uC and give directly the interpretation on the group level. Example 9. Note that for any ξ = iβ ∈ p we have β ∈ uβ where U C acts on uC by the adjoint representation. Let (U C )− = {g ∈ U C ; limt→+∞ exp(tξ)g exp(−tξ) = e}, (QC )− := {g ∈ U C ; limt→+∞ exp(tξ)g exp(−tξ) exists}, (U C )+ := Θ((U C )− ), (QC )+ := Θ((QC )− ) and (U C )0 = (QC )− ∩ (QC )+ . We also set G− = (U C )− ∩ G, Q− := (QC )− ∩ G, G+ = Θ(G− ), Q+ = Θ(Q− ), G0 = Q− ∩ Q+ . From the definition we obtain G0 = ZG (ξ) = {g ∈ G; Ad(g)(ξ) = ξ} = {g ∈ G; ξG (g) = 0}. Moreover G0 is compatible with the Cartan decomposition of U C . More precisely we have K0 = ZK (ξ) = {g ∈ K; Ad(g)(ξ) = ξ} = {g ∈ K; ξG (g) = 0} , p0 = p ∩ gξ = {η ∈ p; [ξ, η] = 0} and G0 = K0 exp p0 . The map p+ : Q+ → G0 , g → limt→−∞ exp(tξ)g exp(−tξ), is a group homomorphism with kernel G+ .

5 Retraction and Semistability Let G = K exp p be a Lie subgroup of U C which is compatible with the Cartan decomposition of U C . For a subset Q of Z we set SG (Q) := {z ∈ Z; G · z∩Q = ∅}. Here G · z denotes the closure of G · z in Z. Recall that Mip is the zero fiber of µip by definition. We call SG (Mip ) the set of semistable points of Z with respect to µip and the G-action. The following basic result has been shown in [HeSt05]: Theorem 2. The set SG (Mip ) is open in Z. Example 10. Assume that the momentum map is given by a strictly plurisubharmonic function ρ as explained in Example 1. If ρ is in addition an exhaustion then it has a minimum on the closure of each G-orbit and this minimum is contained in Mip . Consequently SG (Mip ) = Z.


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Example 6. We continue Example 6 of Sect. 2, i.e. Z = Pn (C) and G = SU(1, n). Let π : Cn+1 \ {0} → Pn (C) denote the canonical projection. Then two points π(x) and π(y) are contained in the same G-orbit if and only if the restrictions of the product < v, w >:= −v0 w0 + v1 w1 + . . . + vn wn to the lines C · x and C · y have the same signature. Thus, Z is the union of three G-orbits, namely the open ball with radius 1 in Cn , its boundary and the complement of these two. In particular, SG (Mip ) is the union of the two open orbits, i.e. equals Z without the unit sphere in Cn . Remark 4. If G is commutative, the openness of SG (Mip ) is proved in [HeSch05] by using an existence theorem of strictly plurisubharmonic exhaustions and by shifting of the momentum map. The idea of the proof of Theorem 2 is to retract a neighborhood of the zero fiber Mip along the G-orbits onto Mip . This is sufficient since SG (Mip ) is G-stable. Consider the gradient flow of the negative of the squared norm ηip := 12 "µip "2 of the momentum map µip , i.e the R-action whose orbits are the integral curves of the gradient vector field grad(−ηip ). We have dηip (z) · v = µip (z), dµip (z) · v for all v ∈ Tz (Z) and therefore grad(ηip )(z) = (iµip (z))Z (z) for all z ∈ Z. Since iµip (z) ∈ p the field grad(ηip ) is tangent to the G-orbits and if Mip is not empty then it coincides with the set of global minima of ηip . Hence the gradient flow of −ηip is a good candidate for the retraction. However it is a priori neither clear that the integral curves γz (t) and their limit points exist for t → ∞ nor that they are global minima of ηip . The first problem can be solved by a reduction to the commutative case and by using the geometry of the momentum map there. For this let a ⊂ p be a maximal commutative subspace and let A := exp(a) be the corresponding Lie subgroup of G. The required properties of the momentum map µia are summarized in the following Proposition. Proposition 2. The image µia (A · z) is an open convex subset of an affine subspace of ia∗ and the momentum map µia : A · z → ia∗ is a diffeomorphism onto its image. The proof uses existence of Kählerian potentials in the commutative case and is an adaption of the proof of the convexity theorem in [HeHu96] (see [HeSt05] for the details). Denote the integral curves of grad(−ηia ) where ηia = 12 "µia "2 by γ˜z (t). In order to prove the existence of γ˜z (t) for t ≥ 0 and of the limit points z∞ := limt→∞ γ˜z (t) one constructs relatively compact neighborhoods Up of points p ∈ Mia such that γ˜z (t) ∈ Up for z ∈ Up and t ≥ 0. For this take any relatively compact neighborhood Vp of p in Mia and intersect SA (Vp ) with the −1 −1 ([0, )). Then Proposition 2 guarantees that Up := SA (Vp ) ∩ ηia ([0, )) set ηia is relatively compact for sufficiently small . More precisely is any number

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smaller than the minimal value of ηia on a certain part of the boundary of an arbitrary relatively compact neighborhood of the closure of Vp . Due to Proposition 2 the set Up is contained in this relatively compact neighborhood. In particular, can be chosen such that Up is arbitrary small. The set Up is open since SA (Mia ) is open. Since this procedure applies for any compact subset of Mia and not only for a point, this result can be generalized to the non-commutative case using the decomposition G = KAK. The precise statement in [HeSt05] is Proposition 3. Let C be a compact K-stable subset of Mip and let W be an open neighborhood of C in Z. Then there exist a G-stable open set Ω in Z −1 ([0, )) ⊂ W holds. and an > 0 such that C ⊂ Ω ∩ ηip This proves that for z in a neighborhood of Mip the integral curve γz (t) exists for t ≥ 0 and that there exits a sequence tn ≥ 0 with limn→∞ tn = ∞ such that the limit z∞ := limn→∞ γz (tn ) exists. For the proof of Theorem 2 it remains to show that z∞ ∈ Mip . It is easy to see that z∞ is a critical point of ηip . Using again the geometry of the momentum map in the commutative case, one can show that the set of critical points of ηia in SA (Mia ) coincides with the zero fiber Mia of µia . The idea then is to generalize this to the non-commutative case by using again the decomposition G = KAK. This turns out 8 to be possible only in a neighborhood of Mip , namely inside the open set k∈K k · SA (Mia ). But we may 8 assume that the closure of the open set W in Proposition 3 is contained in k∈K k · SA (Mia ). Then z∞ is contained therein as well for points z in the constructed neighborhood of Mip . This concludes the proof of Theorem 2. 8 Remark 5. If Z is compact one can show that SG (Mip ) equals k∈K k · SA (Mia ). So in this case the critical set of ηip in SG (Mip ) coincides with the zero fiber Mip . To summarize the argument, we have imposed two conditions on a neighborhood V of Mip to obtain a set theoretical map Ψ : [0, ∞] × V → V with ψ(∞, V ) = Mip and Ψ (t, z) ∈ G · z. However we have not shown that Mip is a deformation retract of V , since we have no statement concerning continuity of the map Ψ . Even worse, we have not proven that the limit z∞ for z ∈ V is unique. If G is complex reductive and its action on Z is a representation on a vector space it is proved in [Ne85] and [Sch89] that the zero fiber of the momentum map is a strong deformation retract of Z. The method used there leads to another proof of the property z∞ ∈ Mip but only under the additional assumption that ηip is real analytic. Of course the existence of a neighborhood 0 is still required, but there is of Mip where the gradient flow exists for t ≥ 8 no need to restrict to the possibly smaller set k∈K k · SA (Mia ) as above. Now we sketch this method, following the argumentation in [Ne85] (see [Sch89] or [HeHu94]). The main ingredient is L ojasiewiczs inequality:


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Proposition 4 (Lojasiewicz). Let Z be a real analytic manifold with C ∞ metric g. For z0 ∈ Z and a real analytic function f defined in a neighborhood of z0 with f (z0 ) = 0, there exists a neighborhood Uz0 of z0 , a constant c > 0 and Θ with 0 < Θ < 1 so that " grad(f )(z)" ≥ c|f (z)|Θ for all z ∈ Uz0 . Let V be a neighborhood of Mip such that γz (t) exists for all z ∈ V and t > 0. Applying L ojasiewiczs inequality to the function ηip , one can show that ηip (γz (t)) ≤ c · t− 1− 1

for all z ∈ V , t > 0 and 0 < ≤ 12 . This immediately implies z∞ ∈ Mip . Furthermore this inequality can be used to show that Ψ : [0, ∞) × V → V , Ψ (t, z) = γz (t) is uniformly continuous and thus extends to a continuous map Ψ : [0, ∞] × V → V . As a consequence we obtain the following Theorem 3 (Retraction Theorem). Assume that µip is real analytic. Then Ψ : [0, ∞]×V → V is a K-equivariant strong deformation retraction of V onto Mip = Ψ (∞, V ) which stabilizes closures of G-orbits, i.e. Ψ (t, z) ∈ G · z for all t ∈ [0, ∞). We do not know if Theorem 3 also holds without the analyticity assumption for a general compatible group G. But if G is complex reductive, e.g., G = U C , then one may use the existence of local normal forms for ω and µ in order to deduce that for every x ∈ Mip there is a U -stable open neighborhood Ω such that the restriction µip |Ω is real analytic up to a diffeomorphism (see the discussion in [Sj98]). This is sufficient to show that for a complex reductive group G no analyticity assumption is necessary in the statement of the retraction theorem. Question 1. Let G be a compatible Lie subgroup of U C and x ∈ Mip . Does there exist a diffeomorphism Φ defined in a K-stable open neighborhood of K · x such that Φ∗ µip = µip ◦ Φ is real analytic? Following [HeSt05], it is possible to generalize Theorem 2 to arbitrary fibers Mip (β) = µ−1 ip (β) of µip . Theorem 4. The set SG (Mip (β)) is open in Z. The proof requires some facts about co-adjoint orbits. It will therefore be postponed to the end of the next section.

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6 Co-adjoint Orbits In this section we consider actions of real forms of U C on U -homogeneous K¨ ahlerian manifolds more closely. By Z we will denote a U -homogeneous K¨ ahlerian manifold with a fixed U -invariant K¨ ahlerian form ω and a fixed U equivariant momentum map µ : Z → u∗ . Since Z is U -homogeneous the image µ(Z) is a co-adjoint orbit of U and since Z is a compact complex manifold the complexification U C of U acts by holomorphic transformations on Z. In order to avoid technical complications we will also assume that U is a connected semisimple Lie group. Co-adjoint orbits are symplectic manifolds. By the natural symplectic form on a co-adjoint U -orbit O we mean here the symplectic form ωO on O which is defined by ωO (β)(ξO (β), ηO (β)) = β([ξ, η]) for ξ, η ∈ u. The corresponding momentum map on O is given by the inclusion map O → u∗ . Since Z is U -homogeneous and for every z ∈ Z ω(ξZ (z), ηZ (z)) = ωO (ξO (µ(z)), ηO (µ(z))) holds, µ is a symplectic map. Any symplectic map is an immersion and since µ is U -equivariant it is a covering map. As is well known co-adjoint orbits are simply connected. This shows Lemma 4. For a U -homogeneous K¨ ahlerian manifold Z the momentum map µ : Z → u∗ is a symplectic diffeomorphism onto its image. Corollary 3. For every x ∈ Z we have Ux = Uβ = ZU (β) where β = µ(x). We will now compute the isotropy group of the complexified group U C at x ∈ Z = U · x. Proposition 5. For x ∈ Z let ξ := iβ := iµ(x). Then (U C )x = (QC )+ = {g ∈ U C ;

lim exp(tξ)g exp(−tξ) exists}.

t→−∞

The proof of Proposition 5 requires some preparation. First we recall the notion of Example 9. Let (QC )+ be defined as above and let (qC )+ denote its Lie algebra. We have the direct sum decomposition uC = (uC )− ⊕(uC )0 ⊕(uC )+ where (uC )0 = (uβ )C = (ux )C . We also have (qC )− = (uC )− ⊕ (ux )C where (qC )− is the Lie algebra of (QC )− etc. Now let v ∈ Tx (Z) be an eigenvector with eigenvalue λ(ξ) where ξ acts on Tx (Z) by the isotropy representation (see Sect. 3). Since η is constant on Z we obtain from Proposition 1 0 = λ(ξ)"v"2 + "dµ(x) · v"2 . Since dµ(x) is an isomorphism this implies that λ(ξ) < 0. From this we obtain Tx (Z) = (uC )− · x and (uC )0 · x ⊕ (uC )+ · x ⊂ (uC )x . On the infinitesimal


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level it remains to show that the linear map (uC )− → Tx (Z), η → ηZ (x) is injective. Let η ∈ (uC )− and assume that ηZ (x) = 0. Since θ(η) ∈ (uC )+ we also have (θ(η))Z (x) = 0 and consequently ηZ (x) + (θ(η))Z (x) = 0. Therefore η + θ(η) ∈ ux ∩ ((uC )− ⊕ (uC )+ ) = 0. This implies η = 0. As a consequence we obtain at the group level that (QC )+ is an open subgroup of (U C )x . Since U C is connected and and Z is simply connected we get (QC )+ = (U C )x . Remark 6. It should be underlined that for any β ∈ u∗ the coadjoint orbit is endowed with the symplectic form ωO defined above. It follows that the inclusion O → u∗ is a momentum map for the U -action on O. Moreover, if we define (U C )+ as above with respect to the isotropy representation of iβ on Tβ (U C · β) ⊂ (uC )∗ and (uC )+ denotes its Lie algebra then the map r+ : (uC )+ → Tx (O), η → (η + θ(η))O (x) is a linear isomorphism. If we endow O with the U -invariant complex structure given by multiplication with i on the complex vector space (uC )+ then O becomes a Kählerian manifold. Moreover if we replace (uC )+ by (uC )− and r+ by r− : (uC )− → Tx (O), η → (η +θ(η))O (x) ahlerian and the momentum map is in the above construction, then −ωO is K¨ given by α → −α. We now fix a real form G of U C which is given by an anti-holomorphic involution σ of U C which commutes with the Cartan involution θ of U C . Although the main object to be considered here is the set of semistable points of a U -homogeneous space Z with respect to µip it should be noted that every x ∈ Mip is for trivial reasons a minimum of the square of the momentum norm function. We therefore investigate the critical points of the norm square of the momentum map more closely. For a linear subspace m of u let Cm := {x ∈ Z; x is a critical point of ηm }. In the U -homogeneous case the main property of Cip is summarized in the following Proposition 6. We have Ck = Cip = {x ∈ Z; [µk (x), µip (x)] = 0}. Proof. A point x is critical for ηip , if and only if it is critical for ηk since η = ηk + ηip is constant. Using Lemma 3 we get µk (x), µip (x) ∈ ux = uµ(x) = zu (µ(x)) which implies Cip ⊂ {x ∈ Z; [µk (x), µip (x)] = 0} since µ(x) = µk (x) + µip (x). For the reverse inclusion note that [µk (x), µip (x)] = 0 implies that [µip (x), µ(x)] = 0 and consequently µip (x) ∈ uµ(x) = ux . Now Lemma 3 implies Cip = {x ∈ Z; [µk (x), µip (x)] = 0}. We will now consider some special cases. Assume that x = β ∈ ip∗ and let Z = O be the co-adjoint U -orbit through β endowed with the complex structure such that −ω is K¨ ahlerian. In this case (QC )+ is the isotropy group C of U at β and since "β" ≥ ηip (z) for all z ∈ Z we also have that ηip (β) is a maximal value of ηip . Let σ : uC → uC be the anti-holomorphic involution of Lie algebras defined by σ|g = idg and σ|ig = − idig . Since β ∈ ip∗ , the

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Lie algebra qC of QC is stable with respect to σ. Therefore σ induces an antiholomorphic involution on the tangent space Tβ (U · β) ∼ = uC /qC which we C also denote by σ. We have Tβ (U · β) = u · β = k · β + ip · β = ik · β + p · β. Since σ|(g · β) = idg·β and σ|(ig · β) = − idig·β , it follows that k · β = p · β. From this we obtain Tβ (G · β) = Tβ (K · β) and also the the following Lemma 5. If β ∈ ip∗ , then G · β = K · β. As another extreme case assume that Z is U -homogeneous and that z ∈ Mip . From k ⊂ ker dµip (z) = (p · z)⊥ we obtain g · z = k · z ⊕ p · z and u = k ⊕ ip. Then dimR Z = dimR u · z = dim k · z + dim(ip) · z ≤ dimR g · z. This shows Lemma 6. If z ∈ Mip , then G · z is open in Z. Now we come to the proof of Theorem 4, i.e we show that the set SG (Mip (β)) is open in Z for arbitrary β ∈ ip∗ . Proof. Applying the decomposition of G in Remark 2, we may assume that G is a real form of U C . Let O := U · β be the co-adjoint orbit of β. The product Z × O is a K¨ ahlerian manifold such that the map µ ˆ(z, α) = µ(z) − α is a momentum map with respect to the diagonal action of U . By Theorem 2 µ−1 îp is open. the set SG (ˆ ip (0)) of semistable points of Z × O with respect to µ Then its intersection S with Z × G · β is open in Z × G · β. Using the fact that the G-orbit G · β is a K-orbit (Lemma 5), one can show that the image of S under the canonical projection Z × G · β → Z equals SG (Mip (β)). Since the projection is an open map this proves the theorem. Remark 7. If Z is a smooth connected projective algebraic variety and ω is an integral K¨ ahlerian form, then it is shown in [Sch06] that µip (Z) ∩ ia∗+ is a convex polytope. Here a∗+ ∼ = a+ denotes a positive Weyl chamber in a ⊂ p which can be considered as a fundamental domain for the K-action on p. As a consequence the set of x ∈ Z such that ηip attains its minimal value at x is mapped by µip onto a single K-orbit K · β. But this shows that K · Mip (β) coincides with the set of minimal values of ηip . In the case where Z = U · α is a co-adjoint U -orbit in u∗ this can be used to show that 1. G · x is open for all x ∈ Mip (β) and 2. every open G-orbit intersects Mip (β) in a single K-orbit. In particular SG (Mip ) coincides with the union of all open G-orbits. It should also be noted that in the co-adjoint case the set K · Mip (β) is exactly the union of K-orbits which are complex submanifolds of Z = U · α.

7 The Slice Theorem Let H be a closed subgroup of the Lie group G and S an H-space. The twisted product G ×H S is the quotient (G × S)/H with respect to the H-action


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(h, (g, x)) → (gh−1 , h · x). Since the G-action on G × S given by multiplication from the left on the first factor commutes with the H-action there is an induced G-action on G×H S. We use the notation [g, x] for H ·(g, x) ∈ G×H S. Note that G ×H S is a G-fiber bundle over G/H with fiber S associated to the H-principal bundle G → G/H. Now let X be a G-space. An H-stable subspace S of X is said to be a global H-slice if the natural map G×H S → X, [g, z] → g · z is an isomorphism. If x ∈ S, S is Gx -stable, G · S is open in X and G ×Gx S → G · S is an isomorphism, then S is called a geometric G-slice at x. Theorem 5 (Slice Theorem). Let Z be as above and x ∈ Mip . Then there is a geometric G-slice at x, i.e., there is a Gx -invariant locally closed real analytic submanifold S of Z, x ∈ S, such that G · S is open in Z and such that the natural map G ×Gx S → G · S, [g, s] → g · s, is a real analytic isomorphism. Remark 8. It is important to note that S can be chosen to be equivariantly isomorphic to an open Gx -stable neighborhood of zero in a Gx -representation space. The first step of the proof of the Slice Theorem is to construct a candidate for S. The starting point for this is the Linearization Theorem (Theorem 1). It says that an open Gx -stable neighborhood of x ∈ Z can be identified with a Gx -stable open neighborhood of zero in the Gx -representation space Tx (Z). Now one has to show that the Gx -representation Tx (Z) is completely reducible. This in particular implies that we have a splitting Tx (Z) = g · x ⊕ Nx . Consequently there exists an open Gx -stable neighborhood of zero in Nx which is identified with a Gx -stable locally closed submanifold S˜ of Z such that 0 ∈ Nx corresponds to x. In order to prove the Slice Theorem we show that after possibly shrinking S˜ the natural map Φ : G ×Gx S˜ → Z, [g, s] → g · x is an open embedding. It is an elementary observation that in general the set of points where a Gequivariant map is a local diffeomorphism is an open G-stable set. Moreover an open G-stable subset of G ×Gx S˜ → Z is of the form G ×Gx S → Z where ˜ Consequently we may assume that Φ is a local diffeomorphism S is open in S. at every point. In particular its image G · S˜ is open in Z. The difficult part in the proof of the Slice Theorem is to show that after replacing S˜ by a sufficiently small open Gx -stable neighborhood S of x in S˜ the restricted map Φ : G ×Gx S → Z is injective. This can be reformulated as an extension problem. Let ψ denote the inverse of Φ which is defined in some open K-stable neighborhood V of x and maps x to [e, x]. We have to show that ψ extends to a G-equivariant map Ψ defined on G · V , at least after possibly shrinking V . One important property of ψ is local equivariance, i.e., for every y ∈ V there is an open neighborhood W ⊂ G × V of [e, x] such that ψ(g · z) = g · ψ(z) holds for all (g, z) ∈ W . In addition, the map ψ is K-equivariant. If G is closed

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in U C which is assumed here, then there is a commutative subspace a of p such that G = K · A · K where A := exp a. Now the extension property of ψ follows from the following basic Proposition 7. Each compact subset C in µ−1 ia (0) has arbitrary small open neighborhoods W such that for z ∈ W and η ∈ a the set {t ∈ R; exp tη·z ∈ W } is connected. Let us first see how the Proposition gives the desired extension of ψ. Since −1 µ−1 ip (0) ⊂ µia (0) we can apply the proposition to the compact set K · x ⊂ µ−1 ia (0). Let W ⊂ V be as in the proposition. Note that ψ satisfies ψ(a · z) = a · ψ(z) for all a ∈ A and z ∈ V such that a · z ∈ W . Since W contains K · x we have an open K-stable neighborhood V1 of K · x such that V1 ⊂ W ⊂ V holds. From this we obtain that Ψ : G · V1 → G ×Gx S, Ψ (g · z) = g · ψ(z) where z ∈ V1 is a well defined map. We will now comment on the proof of Proposition 7. The statement concerns only actions of compatible subgroups of commutative complex reductive groups, since we always may replace U C with the complex algebraic Zariski closure of A in U C . For simplicity we consider here only the case where (U C ) = exp(ia) exp a, i.e. the case where K = U = exp(ia). In this case µ = µia holds. Moreover, at every y ∈ M := µ−1 (0) we have a candidate for a slice S(y) at y and a U C -equivariant locally biholomorphic map C Φ : U C ×(Uy ) S(y) → Z. But, compared to the case of a general compatible group G, the situation is now much more simple. The main new ingredient is that the orbit U · y is isotropic, i.e. ω(ξZ (y), ηZ (y)) = 0 holds for ξ, η ∈ u. This follows directly from dµξ (ηZ (y)) = ω(ξZ (y), ηZ (y)) and the invariance of µ : Z → u∗ . Remark 9. Also in the non-commutative case the U -orbits through points y ∈ µ−1 (0) are isotropic submanifolds of Z. The analogous statement for a general compatible subgroup G of U C would be that (ip) · y is an isotropic linear subspace of Ty (Z) for y ∈ Mip . But this is far from being the case in general. For general G we only have that k·y and (ip)·y are perpendicular with respect to the symplectic form ω(y) : Ty (Z) × Ty (Z) → R. The proof of Proposition 7 is now carried out using K¨ ahlerian potentials as follows. Let ωΦ = Φ∗ ω and µΦ = µ ◦ Φ denote the pull back of ω and C Φ to ZΦ = U C ×(Uy ) S(y). After an appropriate shrinking of S(y) one sees that ωΦ = −ddc ρΦ for some invariant smooth function ρΦ . Here one has to use the fact that U · x is isotropic, has to choose S(y) such hat U · [e, y] is a deformation retract of ZΦ and is a Stein manifold. The retraction property guaranties that ωΦ is exact and the Stein property implies then the existence of ρΦ . The next step is to show that the sets ∆Φ (ε, Ω) := {z ∈ ZΦ ; ρΦ (z) < ρΦ (y) + ε} ∩ Ω where ε > 0 and Ω is an U C -stable open neighborhood of [e, y] in ZΦ give a basis of neighborhoods of U · y ∼ = U · [e, y] in ZΦ . In particular if these are sufficiently small Φ maps each such set biholomorphically onto


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an open neighborhood ∆(ε, Ω, y) of y in Z. One then has to show that the open neighborhood W of C = K · x in Proposition 7 is given by a union of sufficiently small chosen sets ∆(ε, Ω, y), y ∈ C.

8 Quotients We define a relation ∼ on a G-invariant subset X ⊂ Z by setting x ∼ y if and only if G · x ∩ G · y = ∅ where the closure is taken in X. If this defines an equivalence relation on X then we call the quotient X/ ∼ the topological Hilbert quotient of X by G and denote it by X//G. In the case where G is complex reductive, i.e. G = U C , the following theorem is proved in [HeLo94]. Theorem 6. Let M := Miu = µ−1 (0). 1. The topological Hilbert quotient π : SU C (M) → SU C (M)//U C exists. 2. Each fiber of π contains a unique closed U C -orbit which is the unique orbit of minimal dimension in that fiber. 3. For every z ∈ M, the orbit U C · z is closed in SU C (M). 4. The inclusion M → SU C (M) induces a homeomorphism M/U ∼ = SU C (M)//U C . Example 11. Let U C := C∗ act on Z = C2 by (z, (w1 , w2 )) → (zw1 , z −1 w2 ) and let µ be the momentum map introduced in Example 2. Then M = {(w1 , w2 ) ∈ C2 ; |w1 | = |w2 |} and SU C (M) = C2 . The fiber π −1 (π(0)) of π : SU C (M) → SU C (M)//G is the union of the orbits (0, 0), U C · (1, 0) and U C · (0, 1) and on the complement C2 \ π −1 (π(0)) the categorical quotient coincides with the geometric quotient. The quotient M/U ∼ = SU C (M)//U C is isomorphic to C. In [HeSch05] Theorem 6 is generalized to arbitrary closed subgroups G of U C which are compatible with the Cartan decomposition. The crucial property used for this is the content of the following Lemma. Lemma 7. Let C1 , C2 ⊂ Mip be closed, disjoint and K-stable. Then there exist disjoint G-invariant open neighborhoods Ω1 of C1 and Ω2 of C2 . With Lemma 7 one can prove Proposition 8. Let z ∈ Z. Then G · z ∩ Mip = ∅ if and only if G · z ∩ Mip = K · z0 for some z0 ∈ Mip . From Proposition 8 it follows that the topological Hilbert quotient SG (Mip )//G of SG (Mip ) by G exists and that the inclusion of Mip into Z induces a bijection of Mip /K with SG (Mip )//G. Using Lemma 7 again it is possible to show that the quotient map SG (Mip ) → SG (Mip )//G maps closed G-invariant sets onto closed sets. Thus, we have:

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Theorem 7. The bijection Mip /K → SG (Mip )//G is a homeomorphism. In particular, SG (Mip )//G is metrizable and locally compact. Using the Slice Theorem (Theorem 5), one can prove Lemma 8. 1. Let z ∈ SG (Mip ). Then G · z is closed in SG (Mip ) if and only if it intersects Mip . 2. Let z ∈ SG (Mip ), y ∈ G · z ∩ Mip and assume G · z = G · y. Then the orbit G · z has strictly larger dimension than G · y. So each fiber of the quotient map SG (Mip ) → SG (Mip )//G contains exactly one closed orbit, namely the orbit intersecting Mip . Every other orbit in the fiber has strictly larger dimension and closes up in this orbit. Example 6. We continue Example 6 of Sects. 2 and 5, i.e. Z := Pn (C) and G = SU(1, n). Here the orbits intersecting Mip are the open orbits and the quotient SG (Mip )//G coincides with the geometrical quotient SG (Mip )/G and consists of two points.

References [HeHu94] P. Heinzner, A. Huckleberry Invariant plurisubharmonic exhaustions and retractions, manuscripta math. 83 (1994) [HeHu96] P. Heinzner, A. Huckleberry K¨ ahlerian potentials and convexity properties of the moment map, Invent. math. 126 (1996), 65–84. [HeLo94] P. Heinzner, F. Loose Reduction of complex Hamiltonian G-spaces, Geom. Funct. Anal. 4 (1994), no. 3, 288–297 [HeSch05] P. Heinzner, G. Schwarz Cartan decomposition of the moment map [HeSt05] P. Heinzner, H. St¨ otzel Semistable points with respect to real forms [Ne85] A. Neeman The topology of quotient varieties, Ann. of Math. 103 (1985), 419–459 [Sch06] P. Sch¨ utzdeller P. Sch¨ utzdeller Convexity properties of the moment map for real forms acting on K¨ ahlerian manifolds, Dissertation RuhrUniversit¨ at Bochum, in preparation [Sch89] G. Schwarz The topology of algebraic quotients, Topological methods in algebraic transformation groups (New Brunswick, NJ, 1988), 135–151, Progr. Math., 80, Birkh¨ auser Boston, 1989 [Sj98] R. Sjamaar Convexity properties of the moment mapping re-examined, Adv. math. 138 (1998), no. 1, 46–91

Actions on Flag Manifolds: Related Cycle Spaces Alan Huckleberry Ruhr Universit¨ at Bochum Institut f¨ ur Mathematik D-44780 Bochum, Germany [email protected]

1 Introduction Starting with classical analysis on function spaces on the unit disk, bounded Hermitian symmetric spaces D and their quotients by discrete groups have played a major role in the representation theory of real semisimple Lie groups G0 of Hermitian type. From the point of view of the present paper such a domain D is just one example of a G0 –orbit in a flag manifold Z which is a projective algebraic homogeneous manifold of its complexification G. In this bounded domain case, Z is just the compact dual of D. In the mid 1960s, e.g., with the foundational work ([W1]) and the ensuing results of Schmid (see e.g. [S1, S2, S3]), it became clear that open orbits D of arbitrary real semisimple groups G0 in arbitrary flag manifolds Z = G/Q are interesting from a complex geometric viewpoint and that their complex analytic properties play an important role in the G0 –representation theory. In hindsight one sees that the full action of G0 on Z is of great importance. Due to the fact that a large class of these orbits D are “measurable” and therefore possess natural maximally invariant exhaustions which are q-convex in the sense of Andreotti and Grauert, the first results in the representation theory had to do with the representations on cohomology spaces H q (D, E) of homogeneous vector bundles E with appropriate curvature. It soon became clear, however, that the role of cohomology spaces in this dimension can be explained by the fact that every maximal compact subgroup K0 of G0 possesses a unique complex orbit C0 in D. Thus one is led to focus on the space of compact complex cycles Cq (D). As will be explained in the sequel, we are now beginning to understand certain properties of this “full cycle” space. The cycle space that was initially

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seen as being important for representation theory and which was defined in ([WW]) is in fact quite often smaller (see §8). This is the group–theoretically defined space MD which is the connected component of the intersection of the orbit G.C0 of the base cycle with the full cycle space Cq (D). It is a closed complex submanifold of Cq (D). In fact, associated to an arbitrary G0 –orbit γ in an arbitrary flag manifold Z = G/Q, there is a natural group–theoretically defined cycle space C(γ). Our work in this area has been devoted to applying methods of complex geometry together with certain combinatorial methods of Lie theory to give a complete description of these cycle spaces. The relevant original works are ([AkG,F,FH,GM,HoH,H,HN,HSB,HW2,HW3,HW4,W2,WZ]) and a systematic presentation has recently appeared in book form ([FHW]). This research also deals with the transform of the cohomology spaces to interesting spaces of sections of holomorphic vector bundles on the Stein cycle spaces which arise. Here we primarily restrict our attention to the description of the group– theoretically defined cycle spaces C(γ). Our goal is to give a detailed Leitfaden for the proof of the fact that, except for orbits of special Hermitian type where C(γ) agrees with the bounded symmetric space associated to G0 , for every G0 –orbit γ in an arbitrary G–flag manifold the cycle space C(γ) is naturally biholomorphically equivalent to a certain remarkable universal domain U. Often we refer the reader to the technical details in ([FHW]). However, proofs which lead to a conceptual understanding of relevant methods are given. We also use this opportunity to correct a certain point in [FHW] (see the remark after Proposition 13). Roughly speaking, the first half of this paper is devoted to a nontechnical presentation of all necessary background for this subject. We hope that this will be useful to complex geometers who might like to enter this area to work on the wealth of open questions where complex analytic and/or algebraic geometric phenomena are clearly playing an important role.

2 Basic Notions In our considerations of cycle spaces all groups and group actions which arise are algebraic. In this section we collect the basic background information which will be used in the sequel. 2.1 Generalities on Actions of Algebraic Groups If V is a complex vector space, then GL(V ) denotes the group of complex linear isomorphisms of V . It carries an affine structure which is compatible with the group operations by the closed embedding in V ⊕ C, defined by

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T → (T, det(T )−1 ). A complex linear algebraic group, or briefly an algebraic group, is a subgroup of some GL(V ) which at the same time is a closed subvariety. If X is a complex algebraic variety and G is an algebraic group, then an algebraic G–action on X is a regular morphism G × X → X, (g, x) → g(x), so that the identity Id ∈ G fixes all points of X, i.e., Id(x) = x for all x ∈ X, and g(h(x)) = (gh)(x) for all g, h ∈ G and all x ∈ X. We denote the G–orbit of a point x ∈ X by G.x and let Gx := {g ∈ G; g(x) = x} denote the isotropy group at x. Assuming that the action is algebraic, Gx is an algebraic subgroup of G. Chevalley’s theorem, which states that the image under a regular morphism of a constructible set is constructible, implies that if G × X → X is an algebraic action and x ∈ X, then G.x is Zariski open it its closure cl(G.x) = G.x ∪˙ Y . In particular, Y is a union of G–orbits which are of smaller dimension that that G.x, and it follows that there is a closed G–orbit in the closure of every G–orbit. A G–orbit G.x is naturally identified with the coset space G/Gx by the map G → G.x, g → g(x), which naturally factors through the map G → G/Gx to the quotient space of G by the action of Gx on G which is defined by right– multiplication. The quotient space G/Gx carries a canonical structure of an algebraic variety so that G acts algebraically on it by left–multiplication, and the quotient map G → G/Gx is an algebraic Gx –principal bundle. The identification of G/Gx with the orbit G.x is algebraic. 2.2 Complex Lie Algebras and Lie Groups The radical R of a real Lie group G is by definition the maximal, connected, normal Lie subgroup of G. It is automatically closed and the radical of the quotient G/R is trivial. Assuming G is connected, there is a closed subgroup S of G which has trivial radical with G = SR, and where S ∩ R is discrete. If G is simply–connected, then it is just a semidirect product G = R S, i.e., S ∩ R consists of just the identity. A connected Lie group S with trivial radical is called semisimple. The semisimple subgroups which occur in the above decompositions are called Levi-Malcev factors. Their existence and uniqueness up to conjugation is guaranteed by the Levi-Malcev theorem. If G is a complex algebraic group, then both R and S are complex algebraic. In fact, a complex semisimple Lie group S carries an intrinsic algebraic group structure which is compatible with its structure as a complex Lie group.

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˜ is the universal cover of a Lie group G, then G = G/Γ ˜ , Note that if G where Γ is a discrete central subgroup of G. Since the center of a complex semisimple group G is finite (It is a discrete algebraic subgroup!), the univer˜ → G is a finite morphism. Therefore in most contexts we may lift sal cover G ˜ and assume that G is simply–connected. our considerations G Simply–connected Lie groups are completely determined by their Lie algebras. We say that a complex Lie algebra g is semisimple if the associated simply–connected group G is semisimple. To understand the structure of g means to understand its adjoint representation. Detailed information on the structure of semisimple Lie algebras plays an important role in our considerations. We sketch the basics of this below, but before doing so, let us review the notions of (finite–dimensional) Lie algebra and Lie group representations. If G is a Lie group, then a finite–dimensional (complex) representation of G is a Lie group morphism ρ : G → GL(V ) with values in the general linear group of a complex vector space V . If G is complex, then, unless otherwise stated, we assume that ρ is holomorphic. If G is a complex semisimple group equipped with its canonical affine algebraic structure, then holomorphic morphisms of G are automatically algebraic. Thus, a representation ρ : G → GL(V ) is an algebraic morphism, and consequently the resulting G–action on V is also algebraic. The Lie algebra g = Lie(G) of a (real) Lie group G is by definition the tangent space TId G at the identity. A tangent vector ξ ∈ TId G can be uniquely extended to a left– or right–invariant vector field ξˆ on the manifold G which is globally integrated by a 1–parameter subgroup t → exp ξt of G. For example, the left–invariant field is given by ˆ )(g) = d f (g exp ξt) . ξ(f dt t=0 ˆ ηˆ] = [ξ, The Lie algebra structure of g is well–defined by [ξ, η]. If (g, [ , ]) is any real Lie algebra and x ∈ g, then the operator ad(x) on g is defined by ad(x) := [x, ·]. This defines a linear map ad : g → End(g) which is a morphism of Lie algebras. (Here End(g) is regarded as the Lie algebra of GL(g).) It is a basic example of a representation of g which is a Lie algebra morphism to any End(V ). This adjoint representation has the additional property, which is equivalent to the Jacobi identity, that ad(x) is a derivation, ad(x)([y, z]) = [ad(x)(y), z] + [y, ad(x)(z)] . To every real Lie algebra g there is a uniquely associated Lie group G with Lie(G) = g. Note that the derivative ϕ∗ : g → h of a Lie group morphism


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ϕ : G → H is a morphism of Lie algebras. Conversely, if ψ : g → h is any Lie algebra morphism, then there is a uniquely associated morphism ϕ : G → H of the simply–connected Lie groups so that ϕ∗ = ψ. The adjoint representation of g is the derivative of the adjoint representation Ad : G → GLR (g) of any Lie group with Lie(G) = g. This representation of the group is defined by observing that for every g ∈ G the inner automorphism int(g) : G → G, x → gxg −1 , fixes the identity and therefore its derivative Ad(g) = int(g)∗ is a linear map Ad(g) : TId G → TId G which defines the representation Ad : G → GLR (g). One checks that Ad∗ = ad. Let us now return to the semisimple case. A fundamental property of both semisimple Lie algebras and Lie groups is that their representations are completely reducible. For example, if ρ : G → GL(V ) is a representation of semisimple group, then every G–invariant subspace W of V has a G–invariant ˜ so that V = W ⊕ W ˜ is a G–decomposition. Notice that at complement W the Lie algebra level an ideal s ⊂ g is by definitiion a g–invariant subspace in the adjoint representation. Thus complete reducibility implies that a semisimple Lie algebra is a direct sum of simple ideals. At the Lie group level this means that G is almost a direct project of simple Lie groups, i.e., Lie groups with no positive–dimensional normal subgroups. Here the almost means that G = G1 · . . . · Gm , where the Gi are simple normal subgroups and where Gi ∩ Gj is at most a finite (central) subgroup. This decomposition, which up to order is unique, allows us to reduce most of our considerations to the case of simple Lie groups.

2.3 Structure of Complex Semisimple Lie Algebras A transformation T ∈ End(V ) is diagonalizable if V can be decomposed into a direct sum V = V1 ⊕ . . . ⊕ Vm of its eigenspaces. A set t of such operators is said to be simultaneously diagonalizable if there is such a decomposition which is an eigenspace decomposition for every T ∈ t. If t is simultaneously diagonalizable, then the operators in t commute with each other. A maximal torus t in a complex semisimple Lie algebra g is a maximal Abelian subalgebra consisting of elements x ∈ g such that ad(x) is diagonalizable. The reason for this notation is that the associated subgroup T in any associated semisimple Lie group G is an algebraic subgroup which is isomorphic to (C∗ )r which is the complexification of a compact r–dimensional torus. Maximal toral algebras are unique up to the action of Ad(G) and maximal toral subgroups are unique up to G–conjugation. One can show that the 0–eigenspace of t, i.e., the centralizer of t in g is just t itself. The remaining t–eigenspaces are called root spaces and are denoted

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by gα , where α ∈ t∗ is the linear function such that ad(ξ)(x) = α(ξ) · x for all x ∈ gα . Here are the first basic properties of root systems: 1. 2. 3. 4.

The root spaces gα are 1–dimensional. If α is a root, then so is −α. If β = −α, then either [gα , gβ ] = gα+β or gα and gβ commute. The commutator tα := [gα , g−α ] is a maximal toral algebra in the Lie subalgebra sl2 (α) = g−α ⊕ tα ⊕ gα which is isomorphic to the Lie algebra of SL2 (C).

As a result of these properties one can divide the set Φ of roots (in a number of ways) into sets of positive and negative roots, i.e., Φ = Φ+ ∪˙ Φ− where −Φ+ = Φ− . We define the nilpotent Lie algebra u+ = ⊕ gα α∈Φ+

and note that the associated subgroup U + is an algebraic unipotent group which is isomorphic as a complex variety to some affine space Cm . The analogous definitions and remarks hold for u− and U − . By definition t acts on u+ and we let b := t u+ and refer to it as the associated Borel subalgebra. It is a maximal solvable subalgebra of g. Any two maximal solvable subalgebras are conjugate by an element of Ad(G). Analogously, a Borel subgroup of G is by definition a maximal connected solvable subgroup of G. Borel subgroups are automatically complex algebraic subgroups of G and any two are conjugate by an element of G. The unipotent subgroups U + and U − are maximal unipotent subgroups and, just as in the case of Borel subgroups, any two such are conjugate. Thus a choice of a maximal torus and then a choice of the notion of positivity for the root system leads to a decomposition g = u− ⊕ t ⊕ u+ of the semisimple Lie algebra g. Given these choices, we let b+ := t u+ and b− := t u− . At the group level we use the analogous notation, i.e., B + and B − . Much more (very precise) information is known about such root systems so that it is fair to say that a complete combinatorial picture of complex semisimple Lie algebras is available.

2.4 Real Forms of Complex Semisimple Lie Groups In this work we are mainly concerned with actions of real forms of complex semisimple Lie groups where the combinatorial picture at the Lie algebra level is more complicated than that for complex semisimple Lie algebras.


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A real structure on a complex Lie algebra g is C–antilinear Lie algebra morphism τ : g → g which satisfies τ 2 = Id. Its fixed point set Fix(τ ) is the real Lie algebra under consideration. The anti–holomorphic involution of associated simply–connected complex Lie group G is also denoted by τ . One should keep in mind that, as exemplified by GLn (R), its fixed point set may not be connected. So when we refer to a real form of a Lie group we mean the connected component containing the identity. Real forms of complex semisimple Lie algebras and Lie groups are classified (see e.g. [Hel]). As we noted above, it is only necessary to classify the simple real forms, but it is important to note that simple complex Lie groups G occur as real forms of products G × G. For this observe that if σ : G → G is an antiholomorphic involution of G, then τ : G × G → G × G, (x, y) → (σ(y), σ(x)), is a real structure on G × G with Fix(τ ) being the copy of G which is antiholomorphically embedded in the product by the map x → (x, σ(x)). A semisimple complex Lie group G possesses a unique (up to equivalence by G–conjugacy) compact real form which we denote by Gu . For example, if G = SL2 (C), then one normally chooses Gu = SU2 . In the case of SL2 (C) the only other real form is G0 = SU (1, 1). At the Lie algebra level the defining −1 , involutions are σ : g → g, A → −A¯t , for gu and τ : g → g, A → E1,1 σ(A)E1.1 for g0 . Here one regards 1 0 E1.1 = 0 −1 as being the matrix which defines the standard Hermitian form of signature (1,1) on C2 . Note that we have been able to choose σ and τ so that they commute. Thus σ stabilizes the fixed point set g0 of τ and analogously τ stabilizes gu . Since we are primarily interested in the the noncompact real form, we focus on the restriction θ of σ to g0 . Now θ2 = Idg0 . Therefore, as is true for any involution, θ is diagonalizable and we decompose g0 into its ±1–eigenspaces. Let g0 = k0 ⊕ p0 denote this decomposition, where the +1–eigenspace k0 is the Lie algebra g0 ∩ gu . We see directly that the corresponding subgroup K0 is the compact group of diagonal matrices Diag(eiϕ , e−iϕ ). Recall that every connected Lie group L contains maximal compact subgroups M and that any two such subgroups are conjugate by an element of L. Furthermore, the quotient L/M is diffeomorphic to a cell so that all of the topology of L comes from M ; in particular, M is also connected. In the case of the above example, Gu = SU2 is a maximal compact subgroup of G = SL2 (C) and K0 is a maximal compact subgroup of G0 = SU (1, 1).

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The above example reflects the following general setup. Let g be a complex semisimple Lie algebra and τ : g → g define a noncompact real form g0 . In order to avoid notational complications we assume that g0 is simple. It can be shown that there is a compact real form gu defined by an involution σ which commutes with τ so that the fixed point subalgebra k0 of the restriction θ of σ to g0 is the Lie algebra of a maximal compact subgroup K0 of G0 . Here we have chosen G0 to be a subgroup of the linear algebraic group G and therefore K0 is indeed compact. If we had regarded G0 as being an abstract Lie group with Lie algebra g0 , then this might not have been the case. For example, the universal cover of SU(1, 1), which is contractible, is also a Lie group associated to the Lie algebra g0 of our basic example.

2.5 Symmetric Spaces If σ, τ and θ are as above, then θ : g0 → g0 is referred to as a Cartan involution of g0 , and the resulting eigenspace decomposition g0 = k0 ⊕ p0 is called the associated Cartan decomposition. We now regard θ as a Lie group involution and consider the homogeneous manifold M0 = G0 /K0 . Since θ fixes K0 , it defines a diffeomorphism θ : M0 → M0 whose differential at the neutral point x0 is −Id. Since K0 is compact, we may choose a positive–definite inner product on Tx0 M0 which is invariant with respect to the K0 –representation on Tx0 M0 . Using the G0 –action on M0 we transfer this to all tangent spaces of M0 to obtain a Riemannian metric which is G0 – and θ–invariant. By conjugating θ by an appropriate element of G0 , given any point x ∈ M0 we obtain an isometry of M0 which fixes x and whose differential there is −Id. Thus M0 is a Riemannian symmetric space. A general connected, simply–connected Riemannian symmetric space M is uniquely a product M = Mpos × Mflat × Mneg , where Mpos is of positive curvature, Mflat is isometrically equivalent to an Rn equipped with the standard Euclidean metric, and Mneg is of negative curvature. The positively curved manifolds Mpos are homogeneous spaces of compact semisimple groups, and the negatively curved Mneg are homogeneous manifolds of noncompact semisimple groups. Corresponding to the decompositions of the semisimple groups into simple factors, these manifolds split isometrically. Thus one may reduce to the case of simple Lie groups, where the manifolds Mneg are exactly the symmetric spaces of the form M0 above. As might be expected, the simple factors of compact type are constructed analogously. Namely, instead of restricting σ to g0 we restrict τ to gu to obtain the compact symmetric space Mu = Gu /K0 which we regard as being dual to M0 .


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Now all of this transpires inside of the complex semisimple Lie group G, and it is therefore appropriate to (uniquely) extend θ from g0 to a C–linear involution θ : g → g. Its fixed point set k defines the connected algebraic subgroup K of G which is the complexification of K0 . Lie groups which are complexifications of their maximal compact subgroups are said to be reductive. As we have seen above, complex semisimple groups G are complexifications of their maximal compact real forms Gu . Reductive groups are slightly more general than semisimple groups. For example, GLn (C) is the complexification of Un . However, the center of GLn (C), i.e., the group of diagonal matrices, is positive–dimensional. The decomposition GLn (C) = C∗ SLn (C) reflects the general situation where a reductive group is uniquely the product (C∗ )n G (with possible finite intersection) of a central subgroup group and a complex semisimple group G. A reductive complex Lie group also has unique compatible linear algebraic group structure and holomorphic morphisms, e.g., finite–dimensional representations, are algebraic. If G is reductive and H is a closed complex subgroup, then the complex homogeneous manifold X = G/H is Stein if and only if H is reductive. In this case X has a unique structure of an affine manifold so that the G–action on X is algebraic. Applying this to our situation where X = G/K and K is the complexification of K0 , we see that G/K is an affine G–manifold which is symmetric in the sense that the holomorphic involution θ : G → G along with the transitivity of the G–action defines holomorphic symmetries at each of its points. We refer to X = G/K as an affine symmetric space. Recall that Gu and G0 are simple Lie groups, but that G might be a product of a simple complex Lie group with itself such that G0 is an antiholomorphic diagonal. In this case, G = K × K and X = G/K is just the complex Lie group K equipped with the group G = K × K which acts by left– and right–multiplication. In summary, we have the the following beautiful situation which was at least implicitly recognized by E. Cartan. The complex semisimple Lie group G has simple real forms Gu and G0 which are defined by σ and τ respectively. These involutions commute and define Cartan decompositions g0 = k0 ⊕ p0 and gu = k0 ⊕ ip0 which in turn define the indecomposible symmetric space M0 = G0 /K0 of noncompact type and its dual Mu = Gu /K0 of compact type. Defining θ to be the restriction of σ to g0 and extending it to g as a holomorphic involution by the same name, it defines the affine symmetric space X = G/K. If x0 ∈ X is the neutral point, then the orbit G0 .x0 = G0 /K0 is the Riemannian symmetric space M0 embedded as a closed half–dimensional totally real submanifold of X. In that sense X is the complexfication of M0 . The same is true for the

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compact dual Mu = Gu .x0 = Gu /K0 . Thus, in one picture we have the affine symmetric space X as the complexification of M0 and Mu . Except for flat symmetric spaces, up to finite coverings or quotients of Mu every Riemannian symmetric space occurs in this way. 2.6 Iwasawa Decomposition Let G0 be a simple real Lie group of noncompact type and as above let K0 be a maximal compact subgroup defined by a Cartan involution θ : g0 → g0 . We consider the symmetric space M0 = G0 /K0 and, using the Cartan decomposition g0 = k0 ⊕ p0 , we identify p0 with the tangent space Tx0 M0 at its neutral point. Now the fact that θ is a Lie algebra morphism implies that [k0 , p0 ] ⊂ p0 . This corresponds to the fact that p0 is stabilized by the restriction AdK0 of the adjoint representation of G0 to K0 . This in turn is just the tangent representation of K0 on Tx0 M0 ∼ = p0 . It should be mentioned that since M0 has been assumed to be indecomposible, i.e., G0 is simple, it follows that this representation of K0 is irreducible. Representations of K0 which occur in this way have been completely classified. Whenever one has an action of a compact group K0 on a manifold M , he has at least a chance of determining a slice Σ which is a closed submanifold such that K0 · Σ = M so that generically for x ∈ Σ the orbit K0 .x is transversal to Σ. In the case at hand there is a perfect slice for the K0 –action on M0 . For this we first note that the restriction of the exponential map of g0 to p0 is a diffeomorphism of p0 onto a cell P0 which is a closed submanifold of G0 . In fact the map K0 × P0 → G0 , (k0 , exp(ξ)) → k0 exp(ξ), defines a diffeomorphism of K0 × P0 and G0 which allows us to identify the symmetric space M0 = G0 /K0 with P0 or, perhaps more conveniently, with the tangent space p0 . This identification is K0 –equivariant, and therefore we look for a slice Σ for the adjoint representation of K0 on p0 . At this point it is important to note that p0 is contained in the igu part of the complex Lie algebra g = gu ⊕ igu . Now we can always embed G in some GL(V ) where the complex vector space V is equipped with a unitary structure which is compatible with the involution σ. In other words, by choosing a unitary basis, σ is given at the Lie algebra level by A → −A¯t . Thus the elements of igu are Hermitian operators and consequently the symmetric space M0 can also be regarded as a manifold of Hermitian operators. One key point is therefore that if x ∈ p0 , then ad(x) is the restriction of a Hermitian operator to g0 and is therefore diagonalizable over the reals! Hence, if we consider a maximal commutative subspace a0 of p0 , which is of course automatically a Lie subalgebra of g0 , the restriction of the adjoint representation of g0 to a0 can be simultaneously diagonalized.


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One can show that any two such maximal commutative subspaces are conjugate by a transformation in AdK0 , and furthermore that A0 := exp(a0 ) is a slice for the K0 –action on P0 , i.e., on the the symmetric space M0 , or equivalently for the K0 –representation on p0 . It is therefore important to analyze the role that a0 (resp. A0 ) plays in determining the structure of g0 (resp. G0 ). Since the elements of a0 are simultaneously diagonalizable, it lies in at least one maximal torus h of g. To find such a torus one first considers the centralizer z = zg0 (a0 ) of a0 in g0 . From the maximality of a0 and the structure of the Cartan decomposition of g0 it follows that this splits into a direct sum z = m0 ⊕ a0 , where m0 is the centralizer of a0 in k0 . Now M0 is a (compact) subgroup of K0 (not related to the symmetric space by the same name) and it can be shown that if t0 is an arbitrary maximal (compact) torus in m0 , then h0 := t0 ⊕ a0 is a real form of a maximal complex torus h in g. In fact, corresponding to the direct sum h0 = t0 ⊕ a0 , the group H0 splits into a direct product H0 = (S 1 )s × (R>0 )r of real algebraic subgroups. The integer r := dim a0 is called the rank of the real form g0 or of the symmetric space G0 /K0 . As sketched above, we have the root theory of the complex Lie algebra g defined by the maximal torus h. Since the elements of a0 are simultaneously diagonalizable over R, we may restrict this to a0 to obtain a decomposition of g0 . Following the above notation, the 0-eigenspace is m0 ⊕ a0 , and we denote the remaining eigenspaces by gλ0 . The roots λ ∈ a∗0 in this case are the real linear functions which arise as restrictions of the roots in h∗ for the eigenspace decomposition of g with respect to h. It is important to keep in mind that two different roots on h may restrict to the same root on a0 and that the root spaces gλ0 are quite often more than 1-dimensional. Nevertheless we have the analogous decomposition + g0 = n− 0 ⊕ m0 ⊕ a0 ⊕ n0

with respect to the restricted roots. Just as in the complex case, the nilpotent Lie algebras n± 0 are defined by a notion of positivity which is available due to the fact that if α is a restricted root, then so is −α. One checks that the − Cartan involution interchanges them, θ(n+ 0 ) = n0 . Restricted root systems are completely classified using, e.g., Satake diagrams. Without using this rather complicated combinatorial picture, one can show that k0 lies diagonally with respect to the direct sum of the nilpotent Lie algebras and that in fact g0 = k0 ⊕ a0 ⊕ n0 ,

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where we choose n0 to be n+ 0 . This Iwasawa decomposition can be exponentiated to a decomposition G0 = K0 A0 N0 at the group level, where the components intersect only at the identity element of G0 , i.e., this is really a product. Thus the Iwasawa decomposition of G0 shows that the symmetric space G0 /K0 can be identified with the real solvable group A0 N0 . 2.7 Flag Manifolds and their Schubert Varieties The main purpose of this article is to present a survey of recent results on certain aspects of the complex geometry related to the G0 –actions on the flag manifolds Z = G/Q of its complexification G. In the present section we begin by introducing the notation and sketching some background for these manifolds. Let us define a flag manifold Z of a semisimple complex Lie group G to be a (compact) projective algebraic variety Z on which G is acting transitively. One can prove that the G–action on Z is automatically algebraic and thus Z = G/Q, where Q is an algebraic subgroup of G. In general the automorphism group of a projective manifold Z acts in a nontrivial fashion on Pic(Z). However, in the case of a complex semisimple group, if we assume that G is simply–connected, which is essentially no assumption at all, then given a line bundle L → Z on a compact complex G–manifold there is a unique lifting of the G–action Z to a group of holomorphic bundle transformations of L. Briefly stated, every line bundle is a G–bundle. Applying this in our particular case where Z = G/Q and L → Z is some very ample bundle, we have the associated G–representation on Γ (Z, L) and an equivariant embedding ϕL : Z → P(V ) in the projective space of the dual representation V := Γ (Z, L)∗ . In this way, a G–flag manifold is simply a closed G–orbit in the projective space P(V ) of some representation. Let B be a Borel subgroup of the semisimple subgroup G and consider its action on a flag manifold Z = G/Q. Choose a very ample G–bundle L → Z, as above, which realizes Z as an orbit in P(V ). In this setting we may apply the following Fixed Point Theorem of A. Borel. If S is a connected, solvable Lie group acting linearly on a closed subvariety X of a projective space P(V ), then FixX (S) = ∅. Proof. We proceed by induction on dim P(V ) where we may assume that X is not contained in a proper linear projective subspace. The beginning step where dim P(V ) = 0 requires no proof. If P(V ) is higher–dimensional, then we apply Lie’s flag theorem to obtain a hyperplane H ⊂ P(V ) which is S–invariant.


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Now X is compact and positive–dimensional and therefore X1 := X ∩ H is nonempty. Since S is connected, it stabilizes the connected components of X1 and the desired result follows by induction. Corollary 1. Every Borel subgroup B of G has a fixed point in every G–flag manifold Z = G/Q. Given B and a flag manifold Z = G/Q we may therefore assume that B ⊂ Q. We choose B as above so that we have the decomposition b = t u, where u = u+ is the direct sum of the positive root spaces in some root system defined by t. Since q contains b, it follows that it is obtained by simply adding certain negative root spaces to b. One refers to the subgroups Q (resp. subalgebras q) which contain a Borel subgroup B (resp. subalgebra b) as being parabolic. Here is a summary of basic properties which we formulate at the group level: 1. A subgroup Q of G is parabolic if and only if Z = G/Q is a flag manifold. In particular, parabolic subgroups are algebraic. 2. The only fixed point of a parabolic subgroup Q in Z = G/Q is that which corresponds to the neutral point. In other words, the normalizer NG (Q) is Q itself. 3. Parabolic subgroups Q are connected and flag manifolds Z = G/Q are simply connected. The key input in the proofs of these properties is the following basic fact. Proposition 1. Let Z = G/Q be a flag manifold where Q ⊃ B and b = tu+ as above. Let u− be the direct sum of the negative root spaces and define U := U − to be the associated Lie subgroup of G. Then U is an algebraic subgroup of G and its orbit U.z0 of the neutral point in Z = G/Q is a Zariski open subset of Z which is algebraically equivalent to affine space. In particular, Z is a projective rational manifold. Since any two Borel subgroups are conjugate, and the maximal torus T of B − : T U − is contained in the isotropy group at z0 , we can summarize this as follows. Corollary 2. If Z = G/Q is an n–dimensional G–flag manifold, then every Borel subgroup B has an open orbit B.z1 = O ∼ = Cn in Z. One can therefore view flag manifolds in the following more general context. Definition 1. If G is a reductive algebraic group, then a G–variety X is said to be spherical if a Borel subgroup of G has an open orbit in X. The following is a useful characterization of spherical varieties. Theorem 1. A G–variety X is spherical if and only if every Borel subgroup of G has only finitely many orbits in every G–equivariant compactification of X.

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In the case of flag manifolds Z = G/Q the (finitely many) orbits of a Borel subgroup B of G are of a very special nature. Theorem 2. If B = T U is a Borel subgroup of G and O = B.z is an arbitrary B–orbit in a G–flag manifold Z = G/Q, then T has a unique fixed point in O = B.z = U.z ∼ = Cm(O) . The isomorphism O ∼ = Cm(O) is algebraic, and we denote the Zariski closure cl(O) by S and refer to it as a B–Schubert variety in Z. The orbit O itself is called a Schubert cell. Schubert varieties S can be quite singular along the boundary Y of the U –orbit O. Of course Y is the union of finitely many lower–dimensional U –orbits and it is an algebraic hypersurface in S, but it may not be Cartier. For a fixed Borel subgroup B of G, we let S = S(B, Z) be the set of B– Schubert varieties in the flag manifold Z = G/Q. Since Schubert varieties are closures of cells and there are no cells in odd dimensions, the above theorem has the following consequence. Corollary 3. The set S of Schubert varieties in a flag manifold Z freely generates the homology H∗ (Z, Z).

3 Orbit Duality and Related Cycles Here we begin our discussion of the main point of this work, namely the complex geometric aspects of actions of noncompact real forms on flag manifolds Z = G/Q. As above, we may assume that the complex group G is either simple or is the product of a complex simple group with itself. The given noncompact real form G0 may be assumed to be simple. Since the Z is simply–connected, the compact real form Gu acts transitively on it and ahler metrics therefore Z comes equipped with (numerous) Gu –invariant K¨ h = g + iω. The symplectic form ω is by definition Gu –invariant and Gu is semisimple. Thus we have a uniquely associated moment map µ : Z → g∗u . This is a Gu –equivariant diffeomorphism of Z onto an orbit Gu .α in the coadjoint representation. Here α can be chosen to be in the dual space t∗u of a maximal compact torus of Gu which is embedded in g∗u by the duality defined by the invariant Killing form. It is not necessarily a root function, but the case where it is a root function important. The above shows that from the point of view of symplectic geometry we may regard Z as a coadjoint orbit with ω being its canonically defined symplectic form. In our case Z comes equipped with an ω–compatible Gu –invariant complex structure J which can be described in terms of the root picture. In


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this section we discuss a basic duality theorem in this sympletic context. This is a special case of the much more general setting considered in the work of P. Heinzner and H. St¨ otzel which appears in this volume. 3.1 Critical Points of Energy Functions and Orbit Duality The starting point for considerations of the complex geometry of G0 –orbits in a flag manifold Z = G/Q is that there are only finitely many such orbits and therefore that there are open orbits. Thus, without going to the seemingly more delicate level of CR-geometry, we can analyze these with methods of classical complex analysis. Inspection of examples shows that such an open orbit D normally possesses only the constant holomorphic functions, a phenomenon which can usually be explained by the fact that D is usually covered by compact subvarieties. In this context the following observation of J. A. Wolf ([W1]) is of particular interest. Theorem 3. In every open G0 –orbit D in a flag manifold Z = G/Q the maximal compact subgroup K0 of G0 has a unique orbit which is a complex manifold. Here we shall indicate how this reflects a duality between the set OrbZ (K) of K–orbits and the set OrbZ (G0 ) of G0 –orbits in Z. The main result can be described as follows. ahler form on Z which is defined by regarding Let ω be the Gu –invariant K¨ Z as a coadjoint orbit equipped with its canonical symplectic structure and a Gu –invariant complex structure as discussed above. In this way the Gu – moment map is just the identity, but by restriction we have the K0 –moment map µK0 : Z → k∗0 which is of great interest. Define E := "µK0 "2 to be its energy function computed with respect to a K0 –invariant Killing norm, and let ∇E be its gradient field computed with respect to the associated K¨ ahler metric. One proves that the critical set C = {z ∈ Z | ∇E = 0} is a finite union of K0 –orbits κ0 . Let CritZ (K0 ) be the set of all such K0 –orbits κ0 . If z ∈ κ0 , define κ(κ0 ) := K(z) and γ(κ0 ) := G0 (z). The basic duality theorem states that these maps are bijective correspondences: OrbZ (K) ∼ = CritZ (K0 ) ∼ = OrbZ (G0 ). In fact, κ = κ(κ0 ) and γ = γ(κ0 ) are dual to each other if and only if κ∩γ = κ0 . Furthermore, the (K0 –equivariant) gradient flow of ∇E is tangent to all K– and G0 –orbits. It is hyperbolic in the sense that if (κ, γ) is a dual pair with κ ∩ γ = κ0 , then the flow of ∇E realizes κ0 as a strong deformation retract of

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γ and the flow of −∇E realizes κ0 as a strong deformation retract of κ. Here is a precise result that sets up the Bott-Morse theory. Theorem 4. Let κ ∈ OrbZ (K) and γ ∈ OrbZ (G0 ), and suppose that z ∈ κ∩γ is a critical point of the energy function E. Then the degeneracy of the Hessian of the energy function E at z is the tangent space Tz κ0 at z of κ0 = K0 (z). The induced forms on the quotients Tz κ/Tz κ0 and Tz γ/Tz κ0 are, respectively, positive and negative definite. Using Bott-Morse theory one shows that the duality indicated above holds. If κ ∈ OrbZ (K) and γ ∈ OrbZ (G0 ) correspond to each other in this duality, then we say that (κ, γ) is a dual pair. Theorem 5. The following are equivalent: 1. 2. 3. 4.

κ ∩ γ consists of exactly one K0 –orbit κ0 . κ ∩ γ contains an isolated K0 –orbit κ0 . (κ, γ) is a dual pair. κ ∩ γ is compact.

Note that as a consequence of the duality and of the fact that the critical set of the energy function consists of only finitely many K0 –orbits, it follows that OrbZ (G0 ) and OrbZ (K) are finite. Furthermore, since open orbits of complex Lie groups are Zariski open in their closures, there is a unique open orbit κop ∈ OrbZ (K). In this case duality means that the dual G0 –orbit γcl is a compact G0 –orbit which is contained in κop . Since the G0 –action is algebraic and K0 is a maximal compact subgroup of G0 , it acts transitively on every closed G0 -orbit γ. Thus if γ is closed, then it is dual to κop , and it follows that therefore γcl is the unique closed orbit in OrbZ (G0 ). This duality, which is often called Matsuki duality, was origanally proved by combinatorial methods (see the bibliography for the relevant works of T. Matsuki). The approach reflected in the above presentation was introduced in ([MUV]). The duality theorem was proved there in the case of G = G/B. The general case was handled in ([BL], see §8 in [FHW]). The following technical refinement of Theorem 5 is also quite useful. Here t → e(t) is the 1-parameter group which integrates ∇E. Corollary 4. Suppose that (κ, γ) is not a dual pair and that κ ∩ γ = ∅. Then for every w ∈ κ ∩ γ the gradient ∇E is nowhere tangent to K0 .w, and for ε sufficiently small, the action map (−ε, ε) × K0 .w → κ ∩ γ, by (t, z) → e(t)(z) is a diffeomorphism onto its image M , which is a locally closed submanifold of κ ∩ γ.


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One of the basic properties of orbit duality is that it reverses the partial ordering on orbits which is defined by orbit closure. For a precise statement of this fact, if γ1 , γ2 ∈ OrbZ (G0 ), we write γ1 < γ2 whenever γ2 ⊂ c(γ1 ) \ γ1 . Similarly, κ1 < κ2 means that κ2 ⊂ c(κ1 ) \ κ1 for κ1 , κ2 ∈ OrbZ (K). Theorem 6. If (γ1 , κ1 ) and (γ2 , κ2 ) are dual pairs, then γ1 < γ2 if and only if κ2 < κ1 . 3.2 Cycles Associated to Dual Pairs Given the real form G0 and having fixed the maximal compact subgroup K0 , we see that to every orbit γ ∈ OrbZ (G0 ) we have a canonically associated complex analytic object, the dual K–orbit κ. If γ is an open orbit (we always denote such by D), then the associated dual K–orbit κ is a compact complex submanifold in D. If κ is q–dimensional, then we either regard it as a point C0 in the cycle space Cq (D) or in Cq (Z). It is of interest to understand how C0 moves inside of D, either by transformations in G or otherwise. In other words, it is relevant to study the complex geometry of Cq (D) with respect to the transformation groups at hand. In the case where γ is not open, κ is not contained in γ and in a certain sense the base cycle is the CR–manifold κ0 = κ ∩ γ which, as in the case of open orbits, is the distinguished K0 –orbit in γ. It would be of interest to understand this situation from the point of view of CR–geometry. However, for our initial considerations we proceed in a simpler way and define the associated cycle to be the closure cl(κ) which is also a point in the cycle space Cq (Z). Here we are also interested in moving this cycle, and in this case the condition that should be preserved is the compactness of κ ∩ γ. Thus in this case the appropriate object is really the pair consisting of cycle cl(κ) together with bd(κ). Below we briefly recall the relevant background on cycle spaces, introduce the spaces of cycles that are related to the G0 –orbits in Z, and explain certain techniques which will be of use in giving explicit descriptions of these spaces. We also explain various methods of transforming complex analytic data from the G0 –orbits to the cycle spaces.

4 Cycle Spaces and Transforms 4.1 Background on Cycle Spaces An n–dimensional cycle in a complex space X is a formal sum C = n1 C1 + . . . + nm Cm , where the coefficients are positive integers, and the Cj are irreducible n–dimensional compact analytic subsets of X. Its support |C| is

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defined to be the union of the varieties Cj . The space of all n–dimensional cycles in X is denoted by Cn (X). The space Cn (X) has a canonical complex structure. Near a cycle C it is defined by setting up local charts that are called scales. An essential step for this is to set up a local analytic set as an analytic cover of, for example, a polydisk. For this, let A be an analytic subset of pure dimension n in an open subset W of Cn+p . For simplicity it may be assumed that the origin 0 is the base point in A. Then there are linear subspaces Cn and Cp and polydisks U ⊂ Cn and B ⊂ Cp such that c(U ) × c(B) ⊂ W and A ∩ (c(U ) × bd(B)) = ∅. Replacing A by A ∩ (U × B), we consider the projection π : A → U onto the first factor. This is an analytic cover representation of A near the base point. It is a finite–fibered, proper, surjective holomorphic map. Outside a nowhere dense analytic subset E ⊂ U it is a k–sheeted unramified covering map. That defines a holomorphic map U \ E → Symk (B) into the k-fold symmetric product of B which extends holomorphically, by Riemann’s Theorem on removable singularities, to a holomorphic map ψ : U → Symk (B). In a na¨ıve way we may think of ψ as a local coordinate representation of a piece A of a cycle, and moving ψ in Hol(U, Symk (B)) gives us candidates for local charts in nearby cycles. Definition 2. A scale in X is a triple E = (U, B, f ) such that U ⊂ Cn and B ⊂ Cp are relatively compact polydisks, and f : XE → Cn+p is an embedding of an open subset XE of X into an open neighborhood of c(U ) × c(B) in Cn+p . If C ∈ Cn (X) is a cycle and the analytic cover condition f (|C| ∩ XE ) ∩ (c(U ) × bd(B)) = ∅ is fulfilled, then one says that E is adapted to C. If E is a scale which is adapted to an n–cycle C, then C induces a ramified cover of degree degE C = kE onto U and its associated map U → SymkE (B). The following definition gives the correct condition for gluing local ramified covering representations in Hol(U, Symk (B)). Definition 3. Let S be a complex space and let {Cs }s∈S be a family of n– cycles in X. 1. The family {Cs } is called analytic if, for every s0 ∈ S and every scale E = (U, B, f ) adapted to Cs0 , there exists an open neighborhood SE of s0 in S such that a) E is adapted to Cs for all s ∈ SE , b) degE Cs = degE Cs0 for all s ∈ SE , and c) the map gE : SE × U → SymkE (B), defined by the condition that gE (s, ·) is the holomorphic map induced by Cs , is holomorphic.


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2. The family is called an analytic family of cycles if for every s0 ∈ S and every open neighborhood W of |Cs0 | there exists an open neighborhood S of s0 such that |Cs | ⊂ W for all s ∈ S . The graph XS of a family {Cs } of n–cycles is defined by the natural incidence relation XS := {(s, x) ∈ S × X : x ∈ |Cs |}. It is an analytic subset of S × X. The maps p : XS → S and π : XS → Cn (X) are defined by the respective projections. The topological condition for {Cs } to be an analytic family of cycles is that π be proper. The following basic existence theorem is due to D. Barlet [B1]. Theorem 7. The set Cn (X) carries a complex structure such that the following conditions are fulfilled. 1. The family {Cs }s∈Cn (X) is an analytic family of compact n–cycles. 2. The maps p and π are holomorphic and π is proper. 3. If {Cs }s∈S is an analytic family of n–cycles, the map S → Cn (X), s → Cs , is holomorphic. The complex space Cq (X) is locally finite–dimensional and, if X is a compact K¨ ahlerian space, its irreducible components are compact and Cq (X) is itself K¨ ahlerian. If X is projective algebraic, then Cq (X) is the underlying complex space of the Chow scheme and in particular its components are projective algebraic. 4.2 Incidence Varieties and Trace Transforms Here we introduce the trace transform in its simplest version and discuss those of its properties that are needed for our work. For our applications it is enough to consider a complex manifold Z, a closed analytic set S in Z of pure codimension q and a closed analytic subset Y of S of pure codimension 1. Let Γ (S, O(∗Y )) denote the space of meromorphic functions on S with poles contained in Y . The incidence variety IY is defined by IY := {C ∈ Cq (Z) : C ∩ Y = ∅}, and the trace transform is a canonically defined linear map Tr : Γ (S, O(∗Y )) → Γ (Cq (Z), O(∗IY )). For its definition we consider the set Ξ := {C ∈ Cq (Z) | C ∩ Y = ∅ and C ∩ T is finite} which can be shown to be Zariski open in Cq (Z). For f ∈ Γ (S, O(∗Y )) define the function Tr(f ) : Ξ → C by Tr(f )(C) = f (zj ) zj ∈C∩T

(counting multiplicities). The following is the main result required for our applications (see the Appendix in [HSB]).

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Proposition 2. Let X be a closed irreducible subspace of Cq (Z) such that X ∩ Ξ = ∅ so that in particular X ∩ Ξ is a dense Zariski open set in X. Then Tr(f ) is holomorphic on X ∩ Ξ and extends meromorphically to X. Using the above result it is shown in [BK] that Tr(f ) is holomorphic on X \IY . In our applications this is clear because, if S ∩ Y is positive dimensional, then C ∩ Y = ∅. In any case, as stated above, if we replace IY by X ∩ IY , then Tr : Γ (S, O(∗Y )) → Γ (X, O(∗IY )). Before going further we would like to compare the trace–transform with an integral transform considered by Andreotti and Norguet ([AN]). For this let Ω q be the sheaf of holomorphic q–forms and H q (Z \ Y ; Ω q ) the associated Dolbeault cohomology space. A class ξ = [α] in H q (Z \ Y ; Ω q ) defines a holomorphic function AN (ξ) on Cq (Z \ Y ) by AN (ξ)(C) = C ω. Now a function f ∈ Γ (S, O(∗Y )) defines a Dolbeault class ξ in a very natural way. For this, let cS be the fundamental class of S (see [B1]). This is the class of the integration current [S], and for f as above we may regard f cS as an element of H q (Z \ Y ; Ω q ). It then follows that Tr(f ) = AN (f cS ) (see the Appendix in [HSB]). The above discussion would be of no interest if Γ (S, O(∗Y )) contained only the constant functions. Hence, we now restrict to a situation where we are guaranteed many nonconstant meromorphic functions. Let us say that Y has an ample Cartier structure in S if S is compact and there is a holomorphic embedding of S in a projective space P(V ) so that Y is the intersection of S with a projective hyperplane H. Theorem 8. Suppose that Y has an ample Cartier structure, and let X be a closed irreducible subset of the cycle space Cq (Z). Let {Cn } be a sequence of cycles in X ∩ Ξ such that there exist points pn ∈ Cn ∩ S with the property that pn → p ∈ Y . Then, after going to a subsequence there exists f ∈ Γ (S, O(∗Y )) such that lim |T r(f )(Cn )| = ∞. In particular, IY = {C ∈ X | C ∩ Y = ∅} is an analytic hypersurface. Proof. Embed S in a projective space Pm so that Y is the intersection with a hyperplane H, and regard the complement as being Cm with coordinates (z1 , . . . , zm ). Since pn → p ∈ Y , by going to a subsequence we may assume that z1 (pn ) → ∞. Furthermore, it may be assumed that the number of points in Cn ∩ S is constant, say k. We write this intersection as {p1n , . . . , pkn }, where pn = p1n . Thus for each n we have the k–tuple vn = (z1 (p1n ), . . . , z1 (pkn )) ∈ Ck . Now let sn be the point in Symk (C) associated to vn . Since vn is a divergent sequence in Ck , it follows that sn diverges in Symk (C), and consequently there is a regular function f on Symk (C) with lim sup |f (sn )| = ∞. We regard f as a regular function on Ck that is invariant under the permutation group Sk . The ring of Sk –invariant functions on Ck is the polynomial


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algebra generated by the Newton polynomials N0 , . . . , Nk . Thus, there is a Newton polynomial N such that lim sup |N (vn )| = ∞, i.e., lim sup z1 (pjn ) = ∞. j

Hence we let f (z1 , . . . , zk ) = z1 , and it follows that lim sup | Tr(f )(Cn )| = ∞. 4.3 Cycle Spaces Associated to G0 –Orbits Here we return to our basic setup where G0 is a simple noncompact real form of G and Z = G/Q is a flag manifold. Associated to every orbit γ ∈ OrbZ (G0 ) a cycle space C(γ) is defined which is closely related to the group actions at hand. For this it is useful to keep the following fact in mind. Proposition 3. The G–action on Cq (Z) is algebraic. In particular, every orbit G.C in Cq (Z) is Zariski open in its closure. Given γ ∈ OrbZ (G0 ), we consider the dual K–orbit κ ∈ OrbZ (K) which is Zariski open in its closure cl(κ) = C0 in Z and define MZ (γ) to be the G– orbit G.C0 . It is Zariski open in its closure in Cq (Z). If the context is clear, we drop the notational dependence on γ and simply write MZ . If H is the G–isotropy group at C0 in MZ , then we may regard MZ as the G–homogeneous space MZ = G/H. Since C0 is the closure of a K–orbit, it is immediate that H ⊃ K. Thus all possibilities for H are known. In particular, ˜ or G0 is of HermiH is either a finite extension of K, which we denote by K, tian type and H is either K or one of the parabolic groups P± which contains K. In the latter case X± = G/P± are the compact symmetric spaces dual to the bounded Hermitian symmetric domains B± defined by the G0 –invariant complex structures J± on the Riemannian symmetric space G0 /K0 . In the ¯ because the complex sequel we denote the bounded domains by B and B, structures are conjugate. The latter case occurs, for example, when γ = B is a Hermitian bounded domain embedded as a G0 –orbit in its compact dual Z = G/P and the base cycle C0 consists of the unique K–fixed point in B. This is typical of the case where H = P , because, as we will see below, in this case the group theoretically defined cycle space is just the Hermitian bounded domain B. Thus, if H = P , we refer to γ as being of special Hermitian type. In the literature, in particular in [FHW], these orbits are said to be of holomorphic type. This notation is motivated by certain properties of the double fibration transform (see e.g. [HW4]) which is not discussed in the present article. ˜ where Proposition 4. If γ is not of special Hermitian type, then MZ = G/K ˜ K is a finite extension of K.

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If γ is not of special Hermitian type, then the closure of the affine homo˜ is a very interesting spherical variety. It would geneous space MZ = G/K be extremely interesting if the combinatorial theory of spherical embeddings could be applied to give precise descriptions of such varieties! Now let us turn to the spaces of cycles which at least historically have been the most important ones for applications in the representation theory of G0 . We first consider the case of an open orbit γ = D, where κ = cl(κ), and therefore κ itself can be regarded as a base point or base cycle C0 ∈ Cq (D). Using the classical notation which emphasizes that we are dealing with open orbits, we define MD := (G.C0 ∩ Cq (D))◦ . In other words we consider (the connected component) of the set of cycles which can be obtained from the base cycle by a transformation in the complex group G. Since C0 is compact in D, it is immediate that MD is an open subset of MZ . In the main case of interest, where D is not of special Hermitian type, MD is therefore a neighborhood of the totally real Riemannian ˜ symmetric space M0 = G0 /K0 in the affine symmetric space G/K. With the exception of the closed orbit γcl , in the case where γ is not open, we define the base cycle to be the closure C0 = cl(κ) in Z where κ is dual to γ. For γ = γcl where κ = κop we regard the base cycle as being the complement E = Z \ κop . As suggested in ([GM]) the cycle space should consist of the connected component of the set of perturbations of C0 so that the perturbed orbit g(κ) still intersects γ in a compact set. In order to insure that this condition is open we need the following fact which will be proved in the sequel. Proposition 5. If (γ, κ) is a dual pair, then cl(κ) ∩ cl(γ) = κ ∩ γ . With this in hand, we define C(γ) := {g(C0 ); g ∈ G and g(κ) ∩ γ is nonempty and compact}◦ . It follows from Proposition 5 that C(γ) is an open domain in MZ = G.C0 . In the case of γcl it is just the connected component containing E of those cycles in G.E which have empty intersection with γcl . Just as in the case of open orbits, we say that γ is of special Hermitian type whenever cl(κ) is P –invariant, where P is one of P± in the Hermitian case. In that case it is relatively straightforward to prove that C(γ) = B is just the associated bounded Hermitian symmetric domain. Otherwise we are in the main case of interest where C(γ) is a neighborhood of the Riemannian ˜ symmetric space M0 = G0 /K0 in the affine symmetric space MZ = G/K.


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We often refer to the cycle spaces MD and C(γ) as being group–theoretically defined. From the point of view of complex geometry there are other cycle spaces that are at least as interesting. An obvious example is the full cycle space component Cq (Z)◦ which is simply the irreducible component of Cq (Z) which contains C0 = cl(κ). In the case of open orbits, the full cycle space component Cq (D)◦ is also of particular interest. Part IV of ([FHW]) is devoted to detailed computations of the tangent spaces of these full cycle spaces (see §8 of the present paper). The study of their global analytic properties has not yet been attempted.

5 Schubert Incidence Varieties In this section we begin our work which, in particular, will lead to an exact description of the cycle spaces MD or more generally C(γ) for every γ ∈ OrbZ (G0 ). Of basic importance are Schubert varieties of Borel groups B which are as near to being defined over the reals as possible. Since these Schubert varieties are extremely important for our study of the cycles associated to the G0 –orbits in Z, we would like to expand a bit on the meaning of such Borel groups. The flag manifold Z = G/Q is said to be defined over the reals if the defining involution τ : G → G for G0 transcends to Z in the sense that an isotropy group Q = Gz0 is τ –invariant. Proposition 6. If Z is defined over the reals and Q = Gz0 is τ –invariant, then γcl = G0 .z0 is totally real and half-dimensional in Z. Proof. It only must be shown that the orbit G0 .z0 is closed. However, if this were not the the case, the closed G0 –orbit G0 .z1 would be on its boundary and would therefore be smaller than half–dimensional, violating the fact that G.z1 is open. Let p0 := m0 ⊕a0 ⊕n0 and p = m⊕a⊕n be the complexification p = p0 ⊕ip0 . If t = tm ⊕ a is a maximal torus of g containing a, then g = n− ⊕ p where n− is a direct sum of certain negative root spaces of g with respect to t. In particular, Z = G/P is a flag manifold which is defined over R. If B is a Borel subgroup of G which contains the AN of some Iwasawadecomposition G0 = K0 A0 N0 , then we refer to it as an Iwasawa-Borel subgroup. Proposition 7. A Borel subgroup B is Iwasawa–Borel if and only if it is the isotropy subgroup of a point in the closed orbit γcl in the flag manifold Z0 = G/B.

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Proof. Let G0 = K0 A0 N0 be an Iwasawa–decomposition. Note that A0 N0 is simply–connected and is acting real–algebraically on γcl . Thus its closed orbits there are fixed points. Hence, if B fixes a point in γcl , then some K0 conjugate of it contains A0 N0 and therefore B is Iwasawa–Borel. On the other hand, if B is an Iwasawa–Borel subgroup which contains A0 N0 and fixes z0 ∈ Z, then G0 .z0 = K0 .z0 is closed. Now let P be as above, Z = G/P and let π : Z0 → Z be the canonically defined G–equivariant map. If B is an Iwasawa–Borel subgroup, then, since it fixes a point in the closed G0 –orbit in Z0 , it likewise fixes a real point of Z. Hence, we may assume that it is contained in our special parabolic subgroup P = M AN . We may therefore write the above fibration as π : G/B → G/P . Now the fiber F := P/B is itself a compact flag manifold and therefore every maximal compact subgroup of P acts transitively on it, in particular M0 . Consequently, the fiber F0 = P0 /H0 = M0 /M0 ∩ H0 of the restricted fibration of closed orbits G0 /H0 → G0 /P0 agrees with F , and we make the following observation. Proposition 8. A Borel subgroup B is an Iwasawa–Borel subgroup if and only if it is contained in a parabolic P = M AN . Proof. Since F0 = F , it follows that there is only one G0 –orbit in Z0 over the closed orbit of real points G0 /P0 in Z = G/P . Therefore, a Borel subgroup B has a fixed point in the manifold G0 /P0 of real points of G/P if and only if it has a fixed point in the closed G0 –orbit in Z0 . It is also of interest that the closed orbit G0 /H0 in Z0 is Levi-flat with its complex foliation being given by the π–fibers over the totally real manifold G0 /P0 . This shows in particular that P is a minimal τ –invariant parabolic subgroup of G. 5.1 Schubert Varieties and Dual Pairs Here we fix an Iwasawa–decomposition G0 = K0 A0 N0 and an Iwasawa–Borel subgroup B of G which contains AN . Throughout this paragraph S denotes a B–Schubert variety. Occasionally we will remind the reader that B is an Iwasawa–Borel subgroup by referring to S as an Iwasawa–Schubert variety. The open B–orbit in S is always denoted by O and we write S = O ∪˙ Y where Y is the union of the (finitely many) B–orbits on the boundary of O. Given a dual pair (γ, κ), we let Sκ denote the set of B–Schubert varieties S of complementary dimension to that of κ, i.e., dim S = n − q, such that S ∩ cl(κ) = ∅. Note that since the set of all B–Schubert varieties generates the homology of Z, it follows that Sκ = ∅. Our main result in this context is the following existence theorem for Schubert slices Σ in an arbitrary G0 –orbit in Z = G/Q.


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Theorem 9. If (γ, κ) ∈ Orb(G0 ) × Orb(K) is a dual pair and S ∈ Sκ , then the following hold. 1. The intersection S ∩ cl(κ) is finite and contained in γ ∩ κ. If w ∈ S ∩ κ, then AN.w = B.w = O, in particular the intersection S ∩ cl(κ) takes place at smooth points of both varieties. Furthermore, this intersection is transversal at each of its points: Tw S ⊕ Tw κ = Tw Z. 2. The orbit Σ = Σ(γ, S, w) = (A0 N0 )(w) is open in S and is closed in γ with Σ ∩ κ = {w}. 3. The map K0 × cl(Σ) → cl(γ), given by (k0 , z) → k0 (z), is surjective. Proof. Assertion 1: Let w ∈ S ∩ cl(κ). Then Tw (AN.w) + Tw (K.w) = Tw Z

(1)

because g = a ⊕ n ⊕ k. Now w ∈ S = cl(O) and AN ⊂ B. Consequently dim AN.w ≤ dim B.w ≤ dim O = dim S. Furthermore w ∈ cl(κ). Thus dim K.w ≤ dim κ. If w were not in κ, this inequality would be strict, in violation of the above additivity (1) of the dimensions of the tangent spaces. Thus w ∈ κ and Tw Z = Tw S + Tw κ. Since dim S + dim κ = dim Z, this sum is direct, i.e., Tw Z = Tw S ⊕ Tw κ. In particular the intersection is transversal at each of its points and therefore finite. It also follows that dim AN.w = dim S. Thus AN.w is open in S. Now let B ∩ K =: BK so that B = BK AN , and observe that, since BK stabilizes both S and κ, it follows that BK ⊂ Bw and consequently AN.w = B.w = O. We have already seen that K.w is open in κ, forcing K.w = κ. It only remains to show that S ∩ κ is contained in the dual γ of κ. For this let γ ) = G0 (w). Since Σ = A0 N0 .w is transversal to κ in Z, it is also transversal ). If γ ) were not dual to κ, then γ ) ∩ κ ⊃ M as in Corollary 4, and to K0 .w in γ thus A0 N0 .w ∩ M would be positive–dimensional. This is of course contrary ) = γ is dual to κ. to A0 N0 .w being transversal to κ at w, and it follows that γ Assertion 2: The Iwasawa–decomposition G0 = K0 A0 N0 together with the transversality of the intersection O ∩ κ implies that Tw (K0 .w)+Tw (A0 N0 .w) = Tw γ. Furthermore, Tw (K.w) ⊕ Tw (AN.w) = Tw Z. Since by duality we have dim Z = dim K.w + dim γ − dim K0 .w, this implies that dim AN.w + dim K0 .w = dim γ. Therefore dim A0 N0 .w = dim AN.w is forced, and Σ := A0 N0 .w is open in S.

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If Σ were not closed in γ, there would be an orbit of lower dimension on its boundary. But since G0 = K0 A0 N0 , every A0 N0 –orbit in γ meets K0 .w. Furthermore, if p is such an intersection point, then Tp (γ) = Tp (K.p) + Tp (A0 N0 .p). If A0 N0 .p were lower–dimensional, this would be impossible, because A0 N0 .p would have smaller dimension than the orbit A0 N0 .w, which is transversal to K0 .w = K0 .p in γ. Finally, we must show that Σ ∩ κ = {w}. The essential ingredient for this is the following. Lemma 1. The map α : (K0 ∩Qw )×(A0 N0 ∩Qw ) → (G0 ∩Qw ), α(k0 , a0 n0 ) = k0 a0 n0 , is a diffeomorphism. Proof. The map K0 × A0 N0 → G0 , (k0 , a0 n0 ) → k0 a0 n0 , defined by the Iwasawa–decomposition is a diffeomorphism and thus α is a diffeomorphism onto its image. For surjectivity note that dim(K0 ∩ Qw ) + dim(A0 N0 ∩ Qw ) = dim(G0 ∩ Qw ), because dim K0 + dim(A0 N0 ) = dim G0 and dim K0 (w) + dim A0 N0 (w) = dim G0 (w). Thus the image Im(α) is open in G0 ∩Qw . Observe that the compact group K0 ∩ Qw acts freely and locally transitively on (G0 ∩ Qw )/(A0 N0 ∩ Qw ). Consequently, Im(α) is also closed in G0 ∩ Qw . Finally, since K0 .w is a strong deformation retract of γ, it follows that the quotient (G0 ∩ Qw )/(K0 ∩ Qw ) is connected. Thus the image of α has nonempty intersection with every component of (G0 ∩ Qw ) and therefore α is surjective. To complete the proof of Assertion 2 observe that if w ˜ is any point in Σ ∩ κ, then w ˜ = a0 n0 (w) = (k0 )−1 (w) for some a0 n0 ∈ A0 N0 and k0 ∈ K0 . It follows that k0 a0 n0 is in the G0 –isotropy at w and therefore by the above ˜ = w. Lemma so are k0 and a0 n0 , i.e., w Assertion 3: Since K0 is compact, the image K0 (cl(γ)) is closed in cl(γ). Since γ = K0 (Σ), this image is clearly dense in cl(γ). As a first example of how to use the Schubert slices we now prove that cl(κ) ∩ cl(γ) = κ ∩ γ for a dual pair (γ, κ) (Proposition 5): If cl(κ) ∩ cl(γ) contained a boundary point p of γ, then, by conjugation with an appropriate element of K0 , we could find a Schubert variety S and a Schubert slice Σ in S with p ∈ bd(Σ). In particular, p would be in the complement of γ in S. This is contrary to the fact that cl(κ) ∩ S is contained in γ. Similarly, if p ∈ bd(κ) ∩ γ, then by conjugating appropriately we could choose a Schubert slice S which contains it, and this is contrary to S ∩ cl(κ) being contained in κ ∩ γ. The following elementary application of Schubert slices indicates that the boundaries G0 –orbits in Z have a certain degree of pseudoconvexity. In the case of an open orbit D this is certainly related to the fact that, at least in the measurable case, D is q–convex in the sense of Andreotti–Grauert ([AnG]).


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To remind the reader of the numbers, q := dim κ and the Schubert slices in the dual γ are q–codimensional in Z. Proposition 9. For every γ ∈ OrbZ (G0 ) and p ∈ bd(γ) there exists an Iwasawa–Schubert variety S of dimension at least n–q–1 which contains p and which is contained in the complement of γ in Z. Proof. Such a point p is contained in a Schubert slice Σ in the orbit G0 .p. Therefore it is contained in Schubert varieties S of every dimension greater or equal to dim Σ. Now duality reverses the orbit ordering (Theorem 6) and ˜ the orbit γ˜ = G0 .p is in the boundary of the given orbit γ. Thus the dual κ has greater dimension than that of κ and this in turn implies that dim Σ ≤ n − q − 1. Since no A0 N0 –orbit of dimension less than n − q has nonempty intersection with γ, the desired reesult follows. 5.2 Schubert Envelopes Here we use Schubert incidence varieties to construct a domain in MZ (γ) which is an outer approximation of the cycle space C(γ). For this (γ, κ) is as usual the dual pair associated to γ and we consider Schubert varieties S ∈ Sκ of Iwasawa–Borel subgroups B which define Schubert slices Σ in γ. Recall that O denotes the open B–orbit in S and that S = O ∪˙ Y . Using the fact that O is the orbit of the unipotent radical of B, one shows that there is a B–equivariant embedding of S in a projective space so that Y is the intersection of S with a projective hyperplane. In the language of §4 we then have the following result. Proposition 10. The analytic subset Y of S has ample Cartier structure. If γ is not the closed G0 –orbit in Z, then the incidence variety IY is an analytic hypersurface in the cycle space MZ (γ) which is contained in the complement of C(γ). Proof. Given the results in §4 it is enough to prove that IY is contained in the complement of the cycle space C(γ). In the case of an open orbit D this means that every C in MD has empty intersection with Y and this is clear, because Y is contained in the complement of D. In the case of an orbit which is neither open nor closed this is a bit of a delicate point, because it is theoretically possible that g(cl(κ)) ∩ Y = ∅, but that g(cl(κ)) is nevertheless in C(γ). This would be possible when the intersection takes place on the boundary of g(κ). However, one can in fact show that in the case of such a nonempty intersection one can move g(cl(κ)) by transformations h ∈ G which are arbitrarily close to the identity and such that hg(κ) ∩ Y = ∅ (see [HN] and §12.2 and §12.3 of [FHW]). This then shows that g(cl(κ)) is not in the cycle space C(γ).

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Before discussing the case of the closed G0 –orbit we introduce the Schubert envelope ES (γ) defined by the Schubert variety S. For this we again let IY be the incidence variety in MZ and consider the union ∪ IY . Since IY is g∈G0

A0 N0 –invariant, this is the same thing as

∪ IY and is therefore a compact

k∈K0

subset of MZ . We define the envelope ES (γ) to be the connected component of its complement which contains the base cycle C0 in MZ . The above result can be reformulated as follows. Proposition 11. If γ is not the closed G0 –orbit, then the Schubert envelope ES (γ) contains the cycle space C(γ). The definition of the cycle space C(γcl ) of the closed orbit looks a bit different from the others. If E := Z \ κop is the complement of the open K–orbit, then C(γcl ) is the connected component of the set of cycles g(E) which have empty intersection with γcl . Thus, in order to unify the notation, we have regarded E as being the dual cycle to γcl . Now E is the union of its irreducible components Aj , j = 1, . . . , m, and each Aj is the closure of a K–orbit κj which is open in it. Here we would like to compare C(γcl ) with the cycle spaces C(γj ). It is inconvenient to do this directly; so we regard every cycle space as the preimage in G of the one defined above by the orbit map G → MZ (γ), g → g(cl(κ)). In the case of γcl this means by the map g → g(E). Since duality reverses orbit inclusion, the dual pairs (γj , κj ) are such that cl(γj ) = γj ∪˙ γcl , j = 1, . . . , m. Therefore, if g(cl(κj )) is on the boundary of C(γj ), then g(cl(κ)) ∩ γcl = ∅ and consequently g(κop ) does not contain γcl , i.e., bd(C(γj )) is contained in the complement of C(γcl ). Thus we have the following fact. Proposition 12. For all boundary components Aj of κop with dense open K– orbits κj and dual G0 –orbits γj it follows that C(γj ) ⊃ C(γcl ). In particular, C(γcl ) is contained in every Schubert envelope ES (γj ). 5.3 Orbits of Special Hermitian Type Using the Schubert incidence varieties introduced above, we are now able to characterize the cycle spaces in the case where the base cycle is of special Hermitian type. This paragraph is devoted to this characterization; therefore here G0 is assumed to be of Hermitian type. Recall that γ is said to be of special Hermitian type if cl(κ) is invariant with respect to one of the parabolics P± . Proposition 13. If P is one of the parabolic subgroups P± and the base cycle C0 is P –invariant, then C(γ) = G0 .C0 = B is the bounded Hermitian symmetric space in G/P with base point corresponding to P .


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Proof. Since C0 is P –invariant, it follows that MZ = G/P and C(γ) ⊃ B. Above we showed that C(γ) is contained in a Schubert envelope ES (γ). In this case the complement of the orbit of a Borel group in G/P is an irreducible hypersurface which has nonempty intersection with every positive–dimensional K–orbit. Thus the closure of every G0 –orbit γ with κ positive–dimensional is in the complement of C(γ). Unless G0 is of tube type where both B and B are open orbits, this already means that C(γ) ⊂ B. Furthermore, even in the tube type case the union B ∪ B is not connected. Thus in that case C(γ) ⊂ B as well. It should be remarked that the closed orbit γcl can be of special Hermitian type, i.e., the cycle E can be P –invariant.(The contrary was stated in in ([FHW], §12.5)) For example this is the case when Z = P2 and G0 = SU(2, 1). In that case the complement of κop is the union of a P –fixed point and a complementary P –stable projective line. For convenience of notation, as above we regard all cycle spaces, incidence varieties and Schubert envelopes as being in G. We say that an incidence variety IY is a lift from G/P if it is the preimage of a variety in G/P by the canonical map π : G → G/P . Proposition 14. Assume that γ is not closed so that C0 = cl(κ). Then C0 is P –invariant, if and only if every incidence variety IY is a lift from G/P . Proof. If C0 is P –invariant, then we know that C(γ) = B, and it is immediate that every incidence variety is a lift. Conversely, suppose that C0 is not P –invariant and let z0 ∈ Z be the base point so that κ = K.z0 . Since κ is not invariant, it is a proper subvariety of P.z0 . Now let S = O ∪ Y be an Iwasawa-Schubert variety in Sκ . We assume that it is a lift and will obtain a contradiction. We know that cl(κ) ∩ S ⊂ O and this intersection is transversal. Thus every component of P.z0 ∩ O is positive–dimensional and therefore has a point of Y in its closure. Thus, for every neighborhood U of the identity in G there exists h ∈ P and g ∈ U so that gh(z0 ) ∈ Y . Consequently gh ∈ IY . Since IY has been assumed to be a lift, it follows that g ∈ IY for all such g. But this is contrary to C(γ) being an open neighborhood of Id. Let us reformulate this in a weaker but more useful version. Corollary 5. If γ is not closed and cl(κ) is neither P+ − nor P− -invariant, then no Schubert incidence variety IY is a lift from either G/P+ or G/P− . We conclude this paragraph by indicating how the results here fit into the picture that will be described in the sequel. Even in the case of the closed orbit, if C0 is invariant by either P+ or P− , then we know that C(γ) = B. Thus we may assume that C0 is neither P+ – nor P− –invariant, and we know

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that no Schubert incidence variety is invariant by either of these groups. The desired description of the cycle space follows from the general results of §7, which in fact only require that some IY is neither P+ – nor P− –invariant to prove that C(γ) = U.

6 The Universal Domain One main goal of this survey is to explain in a relatively detailed way that a given orbit γ ∈ OrbZ (G0 ) is either of special Hermitian type, and its cycle space C(γ) is just the associated bounded Hermitian domain B, or C(γ) is naturally identifiable with a certain universal domain. This is a remarkable G0 – invariant neighborhood U of the Riemannian symmetric space M0 = G0 /K0 in its complexification X = G/K. In this section we introduce U from various viewpoints and indicate some of its properties. In particular we show how it is related to envelopes defined by invariant hypersurfaces of Iwasawa–Borel subgroups in X. 6.1 Definitions and Basic Properties Let us begin by defining U by means of a system of restricted roots. For this let a0 be a maximal commutative subspace of p0 as above and Σ = Σ(g0 , a0 ) be the resulting system of restricted roots, i.e., the nonzero weights of the restriction of adg0 to a0 . Let ω0 := {ξ ∈ a0 ; |λ(ξ)|
2d. 2. All Ql -vector spaces Héit (Y, Ql ) are finite dimensional. 3. The cohomology groups behave functorially with respect to morphisms of smooth algebraic varieties. ∼ 4. Poincaré duality holds, i.e. there is an isomorphism Hé2d t (Y, Ql ) = Ql and for each i ≤ d a perfect pairing ∼ Héit (Y, Ql ) × Hé2d−i (Y, Ql ) → Hé2d t (Y, Ql ) = Ql . t 5. The K¨ unneth decomposition theorem for products holds. 6. The Lefschetz trace formula holds: Let f : Y → Y be a morphism such that the set of fixed points Fix(f ) is finite and 1 − df is injective (here df denotes the differential of f ), then # Fix(f ) =

2d (−1)i tr(f ∗ | Héit (Y, Ql )). i=0

7. A comparison theorem with singular cohomology holds: if Y is smooth and projective over C, then there are isomorphisms of C-vector spaces Héit (Y, Ql ) ⊗Ql C ∼ = H i (Y, C). The crucial property for our purposes is the Lefschetz trace formula. The condition that 1 − df is injective means that the fixed points have multiplicity one. Assume now that X is a smooth projective variety defined over Q and that p is a prime of good reduction. Consider the geometric Frobenius morphism Frp : X p → X p given by sending the coordinates to their p-th power. Then X(Fpr ) = Fix(Frrp ). Now choose a different prime l = p (the choice of this prime will not matter). Then by the Lefschetz trace formula

Modularity of Calabi-Yau Varieties

Npr = #X(Fpr ) =

2d

275

(−1)i tr((Frrp )∗ | Héit (X p , Ql )).

i=0

We define polynomials Pi,pr (t) = det(1 − (Frpr )∗ t | Héit (X p , Ql )).

(1)

Using the identity

tr(f n | V )

n≥1

tn = − log det(1 − f t) n

for an endomorphism f of a finite-dimensional vector space V , it is then essentially an exercise in formal power series to show that Zpr (t) =

P1,pr (t) . . . P2d−1,pr (t) . P0,pr (t) . . . P2d,pr (t)

In particular, the polynomials Pi,pr (t) introduced in Sect. 2.1 coincide with the polynomials from equation (1). Deligne’s proof of the Riemann hypothesis for function fields of varieties defined over finite fields then follows from Theorem 2.2 (Deligne, [Del74], [Del80]). Let Y be a smooth projective variety defined over the field Fq , where q = pr is a prime power. For all i and l = p the eigenvalues of Fr∗q = (Frrp )∗ on Héit (Y p , Ql ) are algebraic integers of absolute value q i/2 . In the case of a curve X there is another, more explicit, construction of the étale cohomology. Let J(X) denote the Jacobian of X, then n Hé1t (X, Ql ) ∼ lim J(X)[l ] ⊗Zl Ql . = ←− n

If X is defined over Q then there is a natural action of Gal(Q/Q) on J(X)[ln ] which yields a Galois action on Hé1t (X, Ql ). For a general variety X defined over Q there is a natural action of Gal(Q/Q) on Héit (X, Q), but this nice description is missing. 2.3 L-Functions and Galois Representations We are now in a position to discuss L-functions. Let X be a model of a variety defined over Q. Whereas we have, so far, considered the reduction of a variety X defined over Q for a fixed prime p, we shall now vary the prime p. For any i ≥ 0 we define the L-function 1 , L(Héit (X, Ql ), s) = (∗) Pi,p (p−s ) p∈P

276

Klaus Hulek, Remke Kloosterman, and Matthias Sch¨ utt

where P is the set of primes of good reduction and (∗) denotes suitable Euler factors for the primes of bad reduction. In this survey article we shall not discuss the (difficult) question of how to define the Euler factors for bad primes in general, though we will explain this for the case of elliptic curves (see Sect. 4.1). Note that if X = P1 and i = 0, then we obtain exactly Riemann’s zeta function ζ(s), and if X is a smooth projective curve, one thus obtains the Hasse zeta function of X. In most cases, the interesting cohomology of a variety X is the middle cohomology. For this reason we define the L-series of a d-dimensional variety X as L(X, s) := L(Hédt (X, Ql ), s). ∼ H i (XC , C) as C-vector We have already remarked that Héit (X, Ql )⊗Ql C = spaces. Let p be a good prime. A further comparison theorem says that the reduction map gives an isomorphism ∼

ψ : Héit (X, Ql ) −→ Héit (X p , Ql ) as Ql -vector spaces. The isomorphism ψ is also compatible with the Galois module structure of these spaces in the following sense. The Galois group Gal(Fp /Fp ) acts on X p thus making Héit (X p , Ql ) a Gal(Fp /Fp )-module. Similarly, the absolute Galois group Gal(Q/Q) acts on X giving Héit (X, Ql ) a Gal(Q/Q)-module structure. This makes Poincaré duality a Galois-equivariant pairing. For every embedding Q → Qp one obtains an embedding Gal(Qp /Qp ) → Gal(Q/Q). p Two different embeddings yield conjugated subgroups of Gal(Q/Q). Let Fr i p on H (X p , Ql ) is conjugated to be a lift of Frp . Then the action of ψ ◦ Fr ´ et the action of Frp . Let ρi (X) : Gal(Q/Q) → GL(Héit (X, Ql )). be the natural Galois representations. Since any two lifts of Frp are conjugate they have the same trace and determinant. In this way we can still speak about the trace and determinant of Frobenius at a good prime p in connection with Gal(Q/Q)-representations.

3 Automorphic Origin of l-adic Representations 3.1 The Galois Representation Associated to a Modular Form To recall the notion of (elliptic) modular form, we recall first that the group SL(2, Z) acts on the complex upper half plane H by


277

aτ + b ab . : τ → cd cτ + d We can define H∗ by adding to H a copy of P1 (Q) in the following way: the point (x : 1) is identified with x ∈ Q ⊂ C and the point (1 : 0) is identified with the point at infinity along the complex axis, usually denoted by i∞. The action of SL(2, Z) extends to H∗ . For a given integer N we define the group ab Γ0 (N ) = ∈ SL(2, Z); c ≡ 0 mod N . cd The quotient Y0 (N ) = Γ0 (N )\H is a Riemann surface, which can be compactified to a projective curve X0 (N ) = Γ0 (N )\H∗ . The curve X0 (N ) is called the modular curve of level N and the points in X0 (N ) \ Y0 (N ) are called cusps. An elliptic modular form of weight k and level N is a holomorphic function f :H→C which satisfies the following properties: ab 1. For matrices M = ∈ Γ0 (N ) the function f transforms as cd aτ + b f = (cτ + d)k f (τ ). cτ + d 2. The function f is holomorphic at the cusps. To explain the latter condition, weconsider one of the cusps, namely the cusp 11 given by adding i∞. The matrix is an element of Γ0 (N ) and we thus 01 have f (τ + 1) = f (τ ). Writing q = exp2πiτ we can, therefore, consider the Fourier expansion bn q n . f (τ ) = f (q) = n

The cusp i∞ corresponds to q = 0 and f is holomorphic at the cusp i∞ if bn = 0 for n < 0. We say that f vanishes at this cusp, if in addition b0 = 0. A cusp form is a modular form which vanishes at all cusps. We shall also consider modular forms with a modN -Dirichlet character χ. In this case the transformation behaviour given in the definition of modular forms is aτ + b f = χ(d)(cτ + d)k f (τ ). cτ + d

278


The vector space of weight k cusp forms for Γ0 (N ) is denoted by Sk (Γ0 (N )) and, similarly, the vector space of weight k cusp forms with character χ is denoted by Sk (Γ0 (N ), χ). On Sk (Γ0 (N ), χ) we can define operators Tp for prime numbers p N . For a precise definition we refer to [Kna92, page 244]. The operators Tp are called Hecke operators. They generate a subalgebra of End(Sk (Γ0 (N ), χ) called the Hecke algebra T. Since Hecke operators Tq and Tq commute for distinct primes p, q not dividing N , we have simultaneous eigenspaces for all Tp . A form that is an eigenvector for all Tp is called a Hecke eigenform. Let g ∈ Sk (Γ0 (N ), χ) be a Hecke eigenform, so Tp (g) = cp g. Then there is a minimal N | N and a unique eigenform f= bn q n ∈ Sk (Γ0 (N ), χ) with the same system of eigenvalues such that bp = cp for all p N . The eigenform f is called a (normalised) newform. Let f be a normalised Hecke eigenform of some level N and weight k > 1. Let Kf,l the (finite) extension of Ql generated by the coefficients of f . We will now explain how one can associate to f a Galois representation ρf : Gal(Q/Q) → GL(2, Kf,l ) in the case that k > 2. The case k = 2 is treated in Remark 3.1. The following construction is due to Delinge [Del71]. Using the trivial inclusion Sk (Γ0 (N ), χ) ⊂ Sk (Γ0 (N M ), χ) we may assume that N > 4. In order to define ρf we start by considering the subgroup Γ1 (N ) ⊂ Γ0 (N ) of matrices M such that 1b M≡ mod N. 01 A modular form in Sk (Γ0 (N ), χ) is also a modular form for the group Γ1 (N ). Note, however, that the construction we present below uses the group Γ1 (N ) in an essential way. As above, we can define modular curves Y1 (N ) and X1 (N ). One can show that both curves have a natural model over Q. Since we assumed that N > 4 we have that Y1 (N ) is a fine moduli-space for pairs (E, P ) with E an elliptic curve and P a point of order N . Let π : E1 (N ) → Y1 (N ) be the associated universal family. We define define the l-adic sheaf Fk = Symk−2 R1 π∗ Ql on the étale site of Y1 (N ). Let j : Y1 (N ) → X1 (N ) be the natural morphism. Then the étale cohomology group H := Hé1t (X1 (N ), j∗ Fk )∨ has an action of the Hecke-algebra T ⊗Q Ql . It turns out that H is a direct sum of free rank two Hecke-algebras. Recall that we had fixed a normalised Hecke eigenform f . Let bp denote the eigenvalue of Tp . Consider the morphism T ⊗Q Ql → Kf,l defined by Tp → bp . Define


279

Vf := Hé1t (X1 (N ), j∗ Fk )∨ ⊗T⊗Q Ql Kf,l . It turns out that Vf is two-dimensional and that there is a natural action of Gal(Q/Q) on Vf . This gives our representation ρf : Gal(Q/Q) → GL2 (Kf,l ). Remark 3.1. If f has weight k = 2 then f dq q induces a holomorphic one-form on X1 (N ). Consider the two-dimensional subspace Vf := f

dq dq C ⊕ f C ⊂ H 1 (X1 (N ), C) q q

Using the comparison theory of étale cohomology and singular cohomology we obtain a two-dimensional subspace Vf of Hé1t (X1 (N ), Kf,l ). It turns out that Vf is Galois-invariant. One can show that this gives a definition of ρf . dqk−2 1 Similarly, if f has weight k > 2 then f dq q1 ∧ · · · ∧ qk−2 induces a holomorphic k − 2-form on Symk−2 E1 (N ). This leads to a two-dimension subrepresentation of H k−2 (E1 (N ), Kf,l ). This provides an alternative definition of ρf . Although this construction seems to be known to many experts, we could only find references for the special cases k = 3 [Shi72] and k = 4 [SY01]. To a normalised Hecke newform f we can associate an L-function L(f, s). One way of defining this is L(f, s) := L(ρf , s), i.e., as the L-series of the Galois representation ρ f . Equivalently, we can do the following: consider the Fourier expansion f = bn q n . One can associate to this its Mellin transform L(f, s) = bn n−s . n

If f is a normalised Hecke newform with respect to the group Γ0 (N ) and with character χ modulo N (possibly trivial), then the Fourier coefficients satisfy the properties bpr bp = bpr+1 + χ(p)pk−1 bpr−1 for p prime, bn bm = bnm if (n, m) = 1 where k is the weight of the form f . It follows from this that the series L(f, s) has a product expansion 1 L(f, s) = bn n−s = . (2) −s + χ(p)pk−1−2s 1 − b p p p n≥1

Note that χ(p) = 0 if p | N . 3.2 Fontaine-Mazur Conjecture Here we shall briefly discuss the Fontaine-Mazur conjecture. A more detailed discussion of this conjecture as well as many references for the facts mentioned below, can be found in Taylor’s survey paper [Tay04].

280


Consider a variety X/Q. We have already mentioned that the étale cohomology groups Héit (X, Ql ) have a natural Galois action, i.e., there is a continuous homomorphism ρ : Gal(Q/Q) → GL(Héit (X, Ql )). In order to discuss Galois representations we first have to list some subgroups of Gal(Q/Q). For each embedding Q → Qp we obtain Gal(Qp /Qp ) as a subgroup of Gal(Q/Q). Two such embeddings Q → Qp give subgroups of Gal(Q/Q) that differ only up to conjugacy. Fix for every prime p an embedding Q → Qp . The group Gal(Qp /Qp ) maps onto Gal(Fp /Fp ). The kernel of this map is called the inertia subgroup at p, denoted by Ip . Let V be a finite dimensional vector space over Ql . Let ρ : Gal(Q/Q) → GL(V ) be an irreducible subrepresentation of ρ. Then the representation ρ satisfies the following properties: 1. ρ is only ramified at finitely many primes. I.e., ρ (Ip ) = {1} for all but finitely many primes p. 2. The restriction of ρ to the subgroup Gal(Ql /Ql ) is of de Rham type. (For a definition of this see [FO94a], [FO94b]. Note that this is quite restrictive.). 3. For all but finitely many primes p we have that all roots α of the characteristic polynomial of ρ(Frp ) on V satisfy | α |= pi/2 , with | · | the complex absolute value. In general, an irreducible l-adic Galois representation ρ : Gal(Q/Q) → GL(V ) with V a finite dimensional Ql vector space, is called geometric if it satisfies the above properties 1 and 2. The i-th Tate twist of a Galois representation ρ is defined as the Galois representation obtain by tensoring ρ with µ⊗i l∞ if i > 0 and tensorif i < 0. Here µln denotes the ln -th roots of unity and ing ρ with µ∨⊗−i l∞ n lim µl∞ = ← −µl . Conjecture 3.2 (Fontaine-Mazur, [FM95, Conjecture 1]). Let V be a finite dimensional Ql -vector space and let ρ : Gal(Q/Q) → GL(V ) be an irreducible geometric l-adic representation. Then there exists a smooth projective variety X/Q and an integer i such that ρ is a Tate twist of an irreducible subrepresentation of the standard Galois representation on H i (X, Q ). In particular ρ satisfies property 3. Moreover, the L-function L(ρ, s) has a meromorphic continuation to C. A Tate twist of an irreducible subrepresentation of the standard Galois representation on H i (X, Ql ) is called a Galois representation coming from geometry.


281

In the two-dimensional case they make a stronger conjecture: Conjecture 3.3 (Fontaine-Mazur, [FM95, Conjecture 3c]). Let V be a two-dimensional Ql -vector space. Let ρ : Gal(Q/Q) → GL(V ) be an irreducible geometric Galois representation that is not a Tate twist of a finite representation. Then ρ is isomorphic to the Tate twist of a Galois representation associated to a modular form. Later on we will introduce rigid Calabi-Yau threefolds. A rigid Calabi-Yau threefold X has b3 (X) = 2. Suppose X is defined over Q. Then we have a two-dimensional Galois representation. The above conjecture states that the Galois representation on Hé3t (X, Ql ) is coming from a modular form. This conjecture is also stated in [SY01]. 3.3 Langlands’ Programme Here we shall only give a very brief outline. For a more extended introduction to this subject we refer to [Tay04]. In the previous section we have seen a relation between Galois representations, varieties over Q and L-series. In this section we introduce a fourth class of objects, namely automorphic representations. ˆ := limZ/N Z. Denote A∞ = Z ˆ ⊗ Q and A = A∞ × R. As a ring A∞ Let Z ←−

is the subring p Qp consisting of elements (xp ) such that xp ∈ Zp for all but finitely many primes p. Note that the topology of A∞ is different from the subspace topology of Qp . Let WQp be the Weil-group of Qp , that is, all σ ∈ Gal(Qp /Qp ) such that their image in Gal(Fp /Fp ) is a power of Frp . This is a dense subgroup of Gal(Qp /Qp ). Local class field theory gives a ‘natural’ isomorphism ∼

Artp : Q∗p −→ WQabp , where ∗ab means the abelianised group. Furthermore, global class field theory gives an isomorphism ∼

Art : A∗ /Q∗ R∗>0 −→ Gal(Q/Q)ab which is compatible with Artp . (A large part of) class field theory can be described in terms of the map Art. The Langlands’ programme can be considered as an attempt to generalise this to GL(n, Q)\ GL(n, A). Fix n ≥ 1. An automorphic form is a smooth function GL(n, Q)\ GL(n, A) → C ˆ satisfying certain conditions: for example, the translates of f under GL(n, Z)× O(n) (a maximal compact subgroup of GL(n, A)) form a finite-dimensional vector space. Other conditions concern a generalisation of the vanishing of

282


f at cusps and limit the growth of f . For reasons of space we will not give a complete definition, see [Tay04]. To an automorphic form one can associate an infinitesimal character H, where H is a multiset of n complex numbers (i.e., an element of Symn C). Fix such a multiset H. Let A := AoH (GL(n, Q)\ GL(n, A)) be the vector space of cuspidal automorphic forms with infinitesimal character H. We would like to consider A as a representation of GL(n, A). It turns out that GL(n, R) does not fix A. However, A has an action of GL(n, A∞ ) × O(n) and an action of the Lie-algebra gln . An irreducible constituent of A is called a cuspidal automorphic representation of GL(n, A). Consider n = 1. One easily describes Ao{s} (Q∗ \A∗ ), with s ∈ C, in terms of Ao{0} (Q∗ \A∗ ). It turns out that elements in this space are locally constant functions f on ∼ ˆ ∗ −→ A∗ /Q∗ R∗>0 . Z ˆ ∗ → C. Thus Ao{0} (Q∗ \A∗ ) is generated by continuous characters Z Consider n = 2. Write H = {s, t}. Then A = (0) only if s−t ∈ iR, s−t ∈ Z or s − t ∈ (−1, 1). If s − t ∈ Z>0 then to every element in A we can associate a weight 1 + s − t modular form and one can determine its level in terms of A. This shows that automorphic forms are generalisations of modular forms. Suppose n > 1. As in the case of modular forms we can associate local L-series Lp (π, s) to an automorphic form π. (Basically, we restrict the automorphic form π to πp : GL(n, Qp ) ⊂ GL(n, A) in order to define a local L-factor Lp (πp , s).) The Euler product p Lp converges on some half-plane, and has a meromorphic continuation to C. A weak form of the Langlands’ programme is to prove that every L-series of an automorphic form is also the L-series of an irreducible Galois-representation coming from geometry and that this correspondence is ‘natural’.

4 Two-dimensional Galois Representations Wiles’ proof of a special case of the Taniyama-Shimura-Weil conjecture was the essential ingredient in his proof of Fermat’s Last Theorem. In this section we briefly recall his result and discuss generalisations to higher dimension, in particular to rigid Calabi-Yau threefolds. 4.1 Elliptic Curves and Wiles’ Theorem We consider elliptic curves over Q, which we can assume to be in Weierstrass form y 2 = x3 + Ax + B, ∆ = 4A3 + 27B 2 = 0 where A, B ∈ Q. For good primes p the Lefschetz trace formula reads Np = #E(Fp ) = 1 − tr(Fr∗p | Hé1t (E p , Ql )) + p.


Writing

283

ap = tr(Fr∗p | Hé1t (E p , Ql ))

this formula simply becomes Np = 1 − ap + p. The Taniyama-Shimura-Weil conjecture (TSW), in one of its forms, predicts that the sequence Np , resp. ap , is given (for good primes) by the Fourier coefficients of a modular form. More precisely, Theorem 4.1 (Wiles-Taylor-Breuil-Conrad-Diamond, [BCDT01]). Let E be an elliptic curve defined over Q. Then E is modular, i.e., there exists a Hecke newform of weight two with Fourier expansion f (q) = n bn q n such that for all primes p of good reduction ap = 1 − Np + p = b p . Before commenting further on this theorem, we want to discuss an example. Let E be the elliptic curve given by the equation y 2 = x3 + 1. Then we find the following values for Np and ap : p 2 3 5 7 11 13 17 19 Np 3 4 6 12 12 12 18 12 ap 0 0 0 −4 0 2 0 8 In this case it is easy to describe the associated modular form explicitly. Recall that the Dedekind η-function is defined by 1 (1 − q n ). η(τ ) = η(q) = q 24 n≥1

Then the form f (τ ) = η(6τ )4 = q

(1 − q 6n )4 = 1 − 4q 7 + 2q 13 + 8q 19 + . . .

n≥1

is a weight two form of level 36 and one can easily verify the condition ap = bp for any primes within one’s computing capacity. One can provide a complete proof for this fact using Theorem 4.3 or the theory of CM-elliptic curves and CM-modular forms. Some comments regarding Theorem 4.1 are in order at this point. The Taniyama-Shimura-Weil conjecture also makes a prediction about the level of the modular form. If the equation of E is a global minimal Weierstrass equation (this is a Weierstrass equation for E whose discriminant ∆ is divisible by a minimal power of p for all primes p, see [Kna92, Chapter X.1]), then the

284


level of f is only divisible by the primes of bad reduction. More precisely, if p = 2, 3, then the level is at most divisible by p2 , for p = 3 it is at most divisible by p5 and for p = 2 by p8 (see [Ogg67]). In fact, the level equals the conductor of E (see the introduction of [BCDT01]. Also, the statement that E is modular, can be expressed in different, though equivalent, ways. One of these is to say that an elliptic curve E is modular, if there exists a non-constant morphism X0 (N ) → E defined over the rationals. This characterisation has no satisfactory known counterpart in the higher dimensional case. Yet another equivalent formulation, and one that can be generalised, is to say that the L-series of E is that of a weight two form f . In Sect. 2.3 we have already defined an L-function for varieties defined over the Q. In the case at hand we can make this explicit, including a definition of the Euler factors associated to the bad primes: we first assume that the elliptic curve E is given by a global minimal Weierstrass equation. Let ap = 1 − Np + p for good primes and define ⎧ ⎪ in the case of split multiplicative reduction ⎨1 ap = −1 in the case of non-split multiplicative reduction ⎪ ⎩ 0 in the case of additive reduction. Then we set L(E, s) =

p|∆

1 1 . −s 1 − ap p 1 − ap p−s + p1−2s p∆

In Sect. 3.1 we have also introduced the L-series of a normalised Hecke eigenform. Modularity of E can then be rephrased as an equality of L-series L(E, s) = L(f, s) for a suitable Hecke eigenform of weight two. 4.2 The Theorem of Dieulefait and Manoharmayum In view of Wiles’ result it is natural to ask for generalisations to other classes of varieties. Probably the most natural class to be considered for this question is that of Calabi-Yau varieties . A smooth projective variety is called a CalabiYau variety if the canonical bundle is trivial, i.e., ωX = OX

(3)

hi (X, OX ) = 0 for 0 < i < dim X.

(4)

and the following vanishing holds


285

For a curve, condition (4) is empty and we obtain the class of elliptic curves. Calabi-Yau varieties of dimension two are K3 surfaces (see Sect. 5.1 for a discussion of this case). Calabi-Yau threefolds have been investigated very thoroughly. A lot of this interest comes from the fact that they play an essential role in string theory. The easiest examples of Calabi-Yau threefolds are quintic hypersurfaces in P4 and complete intersections of type (2, 4) or (3, 3) in P5 . For a discussion of Calabi-Yau varieties, which appear as complete intersections in toric varieties, and their connection with mirror symmetry we refer the reader to [Bat94], [BB96]. If we look at Calabi-Yau vthreefolds from the point of view of modularity, then it is natural to start with examples whose middle cohomology is as simple as possible. By definition, we always have h3,0 (X) = h0,3 (X) = 1. A CalabiYau threefold X is called rigid if h2,1 (X) = h1,2 (X) = 0. The name comes from the fact that this condition implies H 1 (X, TX ) = 0, and hence that X has no infinitesimal complex deformations. In this case the middle cohomology H 3 (X) = H 3,0 (X) ⊕ H 0,3 (X) ∼ = C2 is two-dimensional. Now assume that X is a rigid Calabi-Yau threefold, which is defined over Q. Then one can show that the determinant of Frobenius on Hé3t (X, Ql ) is p3 . Hence P3,p (t) = det(1 − t Fr∗p | Hé3t (X, Ql )) = 1 − tr((Frp )∗ | Hé3t (X, Ql ))t + p3 t2 . We shall again set

ap = tr(Fr∗p | Hé3t (X, Ql )).

The L-series of (the middle cohomology of) X is then of the form L(X, s) = (∗)

p∈P

1 1 − ap

p−s

+ p3−2s

where the factor (∗) denotes the Euler factors associated to the bad primes, and P is the set of good primes. Comparing this to equation (2) leads one to expect a relationship with a modular form of weight four. Indeed, modularity of rigid Calabi-Yau threefolds defined over Q, had been conjectured for some time. This can be seen as a special form of the Fontaine-Mazur conjecture [FM95] (see Sect. 3.2) and has been stated explicitly by Saito and Yui [SY01]. In recent years, many examples of geometrically interesting rigid Calabi-Yau varieties have been found and in all known cases it has been possible to determine the associated weight four cusp form (see also Sect. 4.3). A general modularity result has been proved by Dieulefait and Manoharmayum [DM03], with further improvements provided later by Dieulefait [Di04b], [Di05]. Theorem 4.2. Let X be a rigid Calabi-Yau threefold defined over Q, and assume that one of the following conditions holds:

286


1. X has good reduction at 3 and 7 or 2. X has good reduction at 5 or 3. X has good reduction at 3 and the trace of Fr3 on Hé3t (X, Ql ) is not divisible by 3. Then X is modular. More precisely L(X, s) L(f, s) for some weight four modular form, where means equality up to finitely many Euler factors. Very recently Dieulefait has informed us that the above theroem can be further improved. More precisely, he can show that X is modular if it has good reduction at 3. The proof goes along the lines of [Di05]. This requires that a modularity lifting result of Diamond, Flach and Guo can be extended to the case of weight 4 and characteristic 3. Indeed, Kisin [Kis05] has recently announced such an extension. We remark that, as in the case of elliptic curves, one expects that the level of the modular form is only divisible by the primes of bad reduction. For the question of determining the level of f see also [Di04a] and Sect. 7. Very little is known about the Euler factors associated to the primes of bad reduction. The above result is purely an existence result and does not provide a method to determine the form f explicitly. There is, however, a method due to Faltings, Serre and Livné, which is very effective, if one wants to prove that a candidate modular form is indeed the right one. We shall discuss this method in the next section. Finally, we would like to remark that a number of non-rigid Calabi-Yau varieties have been found, for which modularity has been established. Most of the examples known up to date are of two types. They are either of Kummer type or they contain (many) elliptic ruled surfaces (or both). In both cases the geometry of the variety is such that the middle cohomology breaks up into two-dimensional pieces. The first type of example was found by Livné and Yui [LY03], for Calabi-Yau varieties containing elliptic ruled surfaces see [HV03] and [HV05a]. We shall discuss examples, mostly rigid, in more detail in Sect. 5. 4.3 The Method of Faltings-Serre-Livn´ e Given a rigid Calabi-Yau threefold, defined over Q, one often wants to determine the associated modular form explicitly. Generally, it is not hard to find a suitable candidate for such a form f : counting points, one can determine the first few Fourier coefficients of f and then compare these to existing lists of Hecke eigenforms such as [St]. In order to actually prove that the L-series of the variety and that of the modular form coincide, one can try to establish the equality of (the semi-simplifications of) the corresponding two-dimensional


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Galois representations (over Q2 ). In order to prove that two Galois representations ρ1 , ρ2 have isomorphic semi-simplifications, it suffices to prove that for all primes p in a certain finite set of primes T the traces of ρ1 (Frp ) and ρ2 (Frp ) are equal. This result is due to Faltings [Fal83]. In the sequel one of the representations is the Galois representation on the cohomology, the other representation is of the from ρf , with f a modular form. It is known that ρf is already simple, see [Rib77, Theorem 2.3]. If the representations have even traces then one can determine the set T effectively. This method is due to Serre and was recast by Livné [Liv87]. Otherwise one can use an approach of Serre [Ser85], which, however can only be made explicit in special cases. Theorem 4.3. Let ρ1 , ρ2 be two continuous two-dimensional two-adic representations of Gal(Q/Q), unramified outside a finite set S of prime numbers. Let QS be the compositium of all quadratic extensions of Q, which are unramified outside S and let T be a set of primes, disjoint from S, such that Gal(QS /Q) = {Frp |QS ; p ∈ T }. Suppose that 1. tr ρ1 (Frp ) = tr ρ2 (Frp ) for all p ∈ T , 2. det ρ1 (Frp ) = det ρ2 (Frp ) for all p ∈ T , 3. tr ρ1 ≡ tr ρ2 ≡ 0 mod 2 and det ρ1 ≡ det ρ2 mod 2. Then ρ1 and ρ2 have isomorphic semisimplifications, and hence L(ρ1 , s) L(ρ2 , s). In particular, the good Euler factors of ρ1 and ρ2 coincide. The gist of this theorem is, that, provided the parity of trace and determinant coincide, it is sufficient to check only the equality of the traces and determinants for a finite number of explicitly known primes, in order to establish that the semi-simplifications are isomorphic. To check the parity of trace and determinant is in practice often easy. The main point is to determine a suitable set T of primes. Given this, the calculation of the traces ap for p ∈ T is often straightforward. Of course, the set T depends on the bad primes of X. But in most concretely given examples, the set T is quite small. If the traces are not even, then one can apply Serre’s approach, provided that the mod 2-reductions ρ¯1 and ρ¯2 are absolutely irreducible and isomorphic. This means that their kernels cut out the same S3 -extension K of Q. In this situation, one has to compute all extensions L of K such that Gal(L/Q) is either S4 or S3 × C2 and which are unramified outside the ramification loci of the representations ρi . Let T be a set of primes such that for any such extension L there is a prime p ∈ T with Frp of maximal order in Gal(L/Q). Then, proving the isomorphism of the Galois representations amounts to checking equality of the traces at the primes of T . In practice, this method can only be used if there are only a few small bad primes, since the available tables of such number fields are limited (cf. [Jon]). We will sketch this approach for the Schoen quintic in Sect. 5.2. In the context

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of non-rigid Calabi threefolds, applications of this method can also be found in [Schu06].

5 Examples In this chapter we shall discuss various examples which have been found in recent years. For an extensive survey of modular Calabi-Yau threefolds we refer the reader to the book of Meyer [Mey05]. 5.1 K3 Surfaces In Sects. 4.1 and 4.2 we discussed elliptic curves and (rigid) Calabi-Yau threefolds. The reader may ask why we did not consider K3 surfaces before proceeding to dimension three. The reason is that in even dimension the middle cohomology will always contain algebraic classes, which changes the situation. If S is any K3 surface, then the middle cohomology has dimension 22. More precisely H 2 (S, Z) = 3U + 2E8 (−1) where U denotes the hyperbolic plane, i.e. the free lattice of rank 2, equipped 01 with the intersection form given by the matrix and E8 is the unique 10 positive definite, even, unimodular lattice of rank 8. The notation E8 (−1) indicates that we take the negative form. The Néron-Severi group of S is the group of divisors modulo numerical equivalence or, equivalently, NS(S) = H 1,1 (S) ∩ H 2 (S, Z). Its rank ρ(S) is called the Picard number of S. This can vary (in characteristic 0) from 0, in which case the K3 surface is not algebraic, to 20. K3 surfaces S with maximal Picard number ρ(S) = 20 are called singular K3 surfaces (a common but misleading terminology which does not mean that the surface has singularities). Singular K3 surfaces have no moduli, but they form an everywhere dense set (in the analytic topology) in the period domain of K3 surfaces. The orthogonal complement of NS(S) in H 2 (S, Z) is denoted by T (S) and is called the transcendental lattice of the surface S. If S is a singular K3 surface, this is a positive definite even lattice of rank 2. Shioda and Inose have proved that the map which associates to a singular K3 surface S the oriented lattice T (S) (where the orientation is given by the 2-form on S) defines a bijection between the sets of singular K3 surfaces modulo isomorphism on the one hand and positive definite, integral binary quadratic forms modulo SL(2, Z) on the other hand [SI77]. One also refers to the discriminant of the lattice T (S) as the discriminant of the surface S. The injectivity of the above map is a consequence of the Torelli theorem. To prove surjectivity Shioda and Inose proceed as follows. Fix a positive definite even lattice of rank 2 and


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discriminant d > 0. Recall that such a lattice determines uniquely a reduced quadratic form 2a b Q= b 2c where det(Q) = d and a > 0, −a < b ≤ a ≤ c (and b ≥ 0 if a = c). Set √ √ −b + −d b + −d and τ = . τ= 2a 2 By construction, the elliptic curves Eτ and Eτ have complex multiplication and are isogenous. The abelian surface A = Eτ × Eτ has Picard number ρ(A) = 4 and the intersection form on its transcendental lattice T (A) is given by Q, see [SM74]. For the Kummer surface Km(A) one has T (Km(A)) = T (A)(2), i.e. the form defined by Q, but multiplied by 2. In order to find a K3 surface with the form given by Q, Shioda and Inose exhibit an elliptic fibration on Km(A) and construct a suitable double cover, resulting in another K3 surface S for which T (S) = T (A). The surfaces S and Km(A) are related by a Nikulin involution. More precisely, S admits an involution ι with exactly 8 fixed points, leaving the non-degenerate 2-form on S invariant, such that Km(A) is the minimal resolution of the quotient S/ι. We shall now turn to the question of modularity of K3 surfaces. Note that we are now mostly considering varieties defined over a number field K rather than over Q. We can, however, still define an L-series for a variety X defined over K. In order to do this, we define for every prime ideal p ∈ Spec OK and each integer i the polynomial Pi,p (t) = det(1 − t Fr∗p | Héit (X p , Ql )). If the dimension of X is d, we set L(X/K, s) = (∗)

p∈P

1 , Pd,p ((Nm p)−s )

where Nm denotes the norm endomorphism from K to Q and P is the set of primes of good reduction. The symbol (∗) stands for suitably defined Euler factors at the bad primes. In the case K = Q this coincides with our previous definition. Since very little is known about the modularity of K3 surfaces which are not singular, we shall restrict ourselves to the singular case. Recall that a modular form f with Fourier expansion f = n an q n has complex multiplication by a Dirichlet character χ if f = f ⊗ χ where f ⊗χ= χ(n)an q n . n

The character χ is necessarily quadratic. We say that f has CM by the imaginary quadratic field K, if f has CM by the corresponding quadratic character.

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CM forms are closely related to Hecke Grössenscharacters: by a result of Ribet, every newform with CM comes from a Grössencharacter of an imaginary quadratic field K. For a treatment of the connection between CM forms and Hecke Grössencharacters see [Rib77]. √ Let S be a singular K3 surface of discriminant d and K0 = Q( −d). One can find a finite extension K of K0 , such that S has a model over K and that, moreover, the Néron-Severi group of S is generated by divisors which are defined over K. Shioda and Inose [SI77, Theorem 6] proved Theorem 5.1. Let S and K as above. Then 2

L(S/K, s) ζK (s − 1)20 L(s, ψ 2 )L(s, ψ ) where ζK (s) is the Dedekind zeta function of the field K and ψ is the Gr¨ ossencharacter of a model of the elliptic curve Eτ over K which is associated to S. Here denotes equality up to finitely many factors. We shall now turn to the more special case that the singular K3 surface S is defined over Q. Then the transcendental lattice T (S) is a two-dimensional Gal(Q/Q)-module, which defines an L-series L(T (S), s). This case was treated by Livné [Liv95, Theorem 1.3]. Theorem 5.2. Let S be a singular K3 surface of discriminant d, defined over Q. Then there √ exists a weight three modular form f3 with complex multiplication by Q( −d) such that L(T (S), s) L(f3 , s). If, moreover, the Néron-Severi group NS(S) is generated by divisors defined over Q, then one has in fact that L(S, s) ζ(s − 1)20 L(f3 , s). The level of the form f3 equals the discriminant of S up to squares composed of the bad primes [Schu05, Proposition 13.1]. An example where the latter situation occurs is the universal elliptic curve E1 (7) with a point of order 7. An affine model of this surface is given by y 2 + (1 + t − t2 )xy + (t2 − t3 )y = x3 + (t2 − t3 )x2 . In this case, NS(E1 (7)) is generated by divisors defined over Q, and the associated modular form is f3 = (η(τ )η(7τ ))3 , where η is the Dedekind η-function. For details of this example see [HV05b]. Another example of this kind was considered by Sch¨ utt and Top [ST05], who investigated the extremal elliptic fibration π : S → P1 with singular fibres I19 , I1 , I1 , I1 , I1 , I1 . At this point we would like to mention the following result, due to Shafarevich [Sha96].


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Theorem 5.3. Given an integer n, there is a finite set Sn of SL2 (Z)-equivalence classes of even, positive definite, binary quadratic forms, such that, if S is a singular K3 surface admitting a model over a number field of degree at most n then T (S) ∈ Sn . In particular, this result says that there exist only finitely many C-isomorphism classes of singular K3 surfaces which can be defined over Q. It is a natural question to ask which weight three newforms with rational coefficients can be realized by singular K3 surfaces defined over Q. In view of Shafarevich’s result, this number is finite (up to twisting). Motivated by this observation Sch¨ utt [Schu05] proved the following finiteness result for CM forms: Theorem 5.4. Assume the generalised Riemann hypothesis (GRH). Then, for given weight, there are (up to twisting) only finitely many CM newforms with rational coefficients. For weights 2, 3 and 4 this holds unconditionally. For weight two this is, of course, classical theory of CM elliptic curves. The fact that this theorem is true without the assumption of GRH for weights three and four follows from Weinberger’s result [Wein73] of the finiteness of imaginary quadratic fields with exponent 2 or 3. Sch¨ utt has determined lists of such forms for weights three and four. For weight three this list is complete under the assumption of GRH and without this assumption the list is complete up to at most one form of level at least 2 · 1011 . For weight four the list is complete up to level 1010 . So far, he has found singular K3 surfaces over Q for 24 out of the 65 weight three forms. 5.2 Rigid Calabi Yau Threefolds In this section we will introduce some of the rigid Calabi-Yau threefolds over Q for which modularity has been proven. For a complete treatment, the reader is referred to the book of Meyer [Mey05]. Nevertheless, we tried to include the most important and instructive examples. Note that for all of them modularity statements can also be derived from Theorem 4.2 (although this theorem does not give the explicit modular form). The Schoen Quintic To our knowledge the first rigid Calabi-Yau threefold over Q for which modularity was proved is the Schoen quintic X in P4 : X : x50 + x51 + x52 + x53 + x54 − 5x0 x1 x2 x3 x4 = 0. Note that X appears as a (singular) member of the one-parameter family M in Sect. 6.2. Schoen [Scho86] showed that X is rigid and derived the modularity of (a projective small resolution of) X explicitly:

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Theorem 5.5. X is modular and its L-series coincides with the Mellin transform of a normalised newform of weight four and level 25. Since the newform does not have even coefficients, the proof of the theorem requires Serre’s approach (cf. Sect. 4.3). In this case, this is easy, since by [Jon] (cf. [Liv87, Proposition 4.10]) there is a unique Galois extension K/Q with Galois group S3 or C3 which is unramified outside 2 and 5. By [Jon], there are only six extensions with Galois group S4 or S3 × C2 which have to be considered. One then finds that it suffices to check the primes in the set T = {3, 7, 11, 13}. Fibre Products of Rational Elliptic Surfaces A very useful method to construct Calabi-Yau threefolds goes back to Schoen. In [Scho88] he considers fibre products of two regular semi-stable rational elliptic surfaces with a section. Due to the semi-stability, such a fibre product has only ordinary double points as singularities, and hence a small resolution exists in the analytic category. Such a resolution has all properties of a CalabiYau variety, but may not be projective. Sometimes, it is possible to construct a small projective resolution by successively blowing up suitable subvarieties. In general, there are two major classes of fibre products where enough subvarieties are available: Proposition 5.6 (Schoen [Scho88, Lemma 3.1]). For i = 1, 2 let πi : Yi → P1 be a semi-stable rational elliptic surface with a section. Let X be the fibre product Y1 ×P1 Y2 . Then X has a projective small resoˆ if lution X 1. all singularities of X lie in fibres of type Im × In with m, n > 1, or 2. Y1 = Y2 and π1 = π2 . Schoen also investigated when a resolution of a fibre product is rigid. This is independent of its projectivity. He describes four cases where this happens. In this section, we will see examples for two of them. Let X be the self-fibre product of a semi-stable rational elliptic surface with the minimal number of four singular fibres. Then X has a projective ˆ by the proposition above. This is rigid due to [Scho88, small resolution X Proposition 7.1, (i)]. There are six such surfaces up to isomorphism over Q, as determined by Beauville [Be82]. All of them are modular in the sense of Shioda [Shi72], i.e., they are elliptic surfaces associated with a subgroup of SL(2, Z) of finite index. We give the corresponding congruence subgroups in Table 1, where we also include the types of the singular fibres. The fact that the elliptic surfaces can be defined over Q, and that X is a ˆ of X is defined over Q. self-fibre product imply that the small resolution X Hence, we expect it to be modular. There are three alternative proofs of modularity: one uses the EichlerShimura isomorphism and the fact that the respective spaces of newforms


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are one-dimensional. This has been worked out in [SY01]. The next approach consists of counting points and applying the Faltings-Serre-Livné method as sketched in Sect. 4.3. This was done by Verrill in [Yui03, Appendix]. We note that due to the self-fibre product structure, the number of points on the reductions and, consequently, the traces are always even, so Theorem 4.3 applies. Thirdly one can use Theorem 4.2 and then try to proceed as in [Di04a]. Take for every Q-isomorphism class the elliptic surface given by the equations listed in [Be82]. Table 1 gives the weight four newforms associated to the self-fibre products and their respective levels. We also remark that these newforms are the only ones of weight four which can be written as η-products (see [Mar96, Cor. 2]). Table 1. Beauville surfaces Γ Γ (3) Γ1 (4) ∩ Γ (2) Γ1 (5) Γ1 (6) Γ0 (8) ∩ Γ1 (4) Γ0 (9) ∩ Γ1 (3)

singular fibres weight 4 form level I3 , I3 , I3 , I3 η(3τ )8 9 I4 , I4 , I2 , I2 η(2τ )4 η(4τ )4 8 I5 , I5 , I1 , I1 η(τ )4 η(5τ )4 5 I6 , I3 , I2 , I1 η(τ )2 η(2τ )2 η(3τ )2 η(6τ )2 6 I8 , I2 , I1 , I1 η(4τ )16 η(2τ )−4 η(8τ )−4 16 I9 , I1 , I1 , I1 η(3τ )8 9

We shall give one further class of examples which was studied in [Schu04a]. As above, we shall work with the universal elliptic curve with a point of order 6, which we will denote by E1 (6). (This is the elliptic surface associated to Γ1 (6).) We take π2 = σπ1 with σ an automorphism of P1 permuting the three cusps with reducible fibres and not fixing the remaining cusp. ˆ This leads to case 1 of Proposition 5.6. The projective small resolution X is rigid by [Scho88, Proposition 7.1, (ii)]. There are five such automorphisms of P1 which are all defined over Q. The resulting rigid Calabi-Yau threefolds can be proved to be modular and one determines the corresponding newforms using Theorem 4.3. The associated newforms are forms of level 10, 17, 21, and twice the same form of level 73. Further Examples We shall briefly comment on other modular rigid Calabi-Yau threefolds. To our knowledge, the first examples which were published after the Schoen quintic were obtained by Werner and van Geemen [WvG90], based on work of Hirzebruch. Their paper pioneered an important method to determine the third Betti number b3 (X) for a Calabi-Yau threefold X over Q arithmetically. This method requires the knowledge of the Euler number e(X) and of a good prime p such that Frp operates trivially on the divisors of X. Then the Riemann hypothesis gives a bound in terms of b3 (X) on the trace of Frp on Hé3t (X, Ql ).

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This bound sometimes enables one to deduce the value of b3 (X) after counting points. Using this approach and Theorem 4.3, Werner and van Geemen proved modularity for four rigid Calabi-Yau threefolds over Q. They also determined the associated newforms, finding the forms of level 8 and 9 from above and two further forms of level 12 and 50 respectively. Further, we would like to mention two constructions which produced several modular Calabi-Yau threefolds. The first of them uses toroidal geometry involving root lattices. For the root lattices A31 and A3 , this was worked out by Verrill in [Ver00]. The corresponding newforms of level 6 and 8 appear in Table 1. For A4 , a family of Calabi-Yau threefolds was investigated by Hulek and Verrill in [HV03]. This family contains a number of (rigid and non-rigid) members for which they prove modularity. Another approach uses double octics, as studied by Cynk and Meyer in [CM05a], [CM05c]. In this case one starts with an octic surface S in P3 which has mild singularities. Many such double covers of P3 admit a desingularisation X, which is Calabi-Yau. In practice one uses for S highly reducible surfaces, such as a union of 8 planes (such configurations are also called octic arrangements). Such an X is called a double octic. Often it is possible to describe the deformations of a double octic X explicitly in terms of the surface S. Using this, Cynk and Meyer found further rigid Calabi-Yau threefolds over Q and the corresponding newforms in [CM05a]. One advantage of the constructions involving fibre products, root lattices or double octics is that they also produce non-rigid modular Calabi-Yau threefolds. Here, modularity is understood as the splitting of Hé3t (X, Ql ) into twodimensional Galois representations, all of which can be interpreted in terms of modular forms. In practice, this is often achieved by identifying enough elliptic ruled surfaces in X which contribute to Hé3t (X, Ql ). This method is explained in [HV05a]. For applications, the reader is also referred to [HV03], [Schu06] and [CM05c]. Apart from this strict notion of modularity, the Langlands’ programme hardly seems to be within reach in more generality at the moment. In [CS01], Consani-Scholten investigate a Calabi-Yau threefold X over Q with b3 (X) = 4. They prove that the Galois representation on√Hé3t (X, Ql ) is induced from a two-dimensional representation of Gal(Q/Q( 5)) and √ conjecture that this should come from a Hilbert modular form over Q( 5) of weight (2, 4) and level 5. Recently Yi [Yi05] has announced a proof of this conjecture. 5.3 Higher-dimensional Examples A number of examples of higher-dimensional modular Calabi-Yau manifolds were recently constructed in [CH05]. Most of these examples arise as quotients of products of lower dimensional Calabi-Yau manifolds by finite groups. The starting point is the Kummer construction, which was first used by Borcea [Bor92] and Voisin [Voi93], who constructed Calabi-Yau threefolds as (birational) quotients of a product of a K3 surface admitting an elliptic


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fibration and an elliptic curve, by the diagonal involution. This construction works in all dimensions: let Y be a projective manifold with H q (Y, OY ) = 0 for q > 0 and D ∈| −2KY | be a smooth divisor. Then D defines a double cover π : X → Y , branched along D with X a Calabi-Yau manifold. Now assume that we have two coverings πi : Xi → Yi as above. Then the group Z/2Z × Z/2Z acts on the product X1 × X2 . Proposition 5.7. Under the above assumptions, the quotient of the product X1 × X2 by the diagonal involution admits a crepant resolution X, which is Calabi-Yau. Moreover, there is a double cover X → Y , branched along a smooth divisor D, with H q (Y, OY ) = 0 for q > 0. The last remark of this proposition allows one to use the covering π : X → Y to repeat this process inductively. Starting with low-dimensional modular Calabi-Yau varieties, one can in this way easily construct many higher dimensional Calabi-Yau varieties. The middle cohomology of these varieties grows with the dimension, but splits in many cases into two-dimensional pieces, which allows to prove modularity. Here we give one example. As the first factor we choose the rigid Calabi-Yau threefold X3 , constructed as a resolution of singularities of the double covering of P3 , branched along the following arrangement of eight planes xt(x − z − t)(x − z + t)y(y + z − t)(y + z + t)(y + 2z) = 0. For details of this example see [Mey05, Octic Arr. No. 19]. As the second factor we take the K3 surface S which is obtained as a desingularisation of the double sextic branched along the following arrangement of six lines xy(x + y + z)(x + y − z)(x − y + z)(x − y − z) = 0. Performing the Kummer construction just described, we obtain a smooth Calabi-Yau fivefold X5 . The Hodge groups of X5 are the invariant part of the Hodge groups of X3 × S. From this it follows easily that b1 (X5 ) = b3 (X5 ) = 0 and b5 (X4 ) = 4. More precisely H 5 (X5 ) ∼ = H 3 (X3 ) ⊗ T (S) where T (S) is the transcendental lattice of S. It is known that L(T (S), s) L(g3 , s),

L(X3 , s) L(g4 , s)

where g3 and g4 are the unique weight three and weight four Hecke eigenforms of level 16 and 32 respectively, with complex multiplication. Both of these forms can be derived from the unique weight two level 32 newform g2 (q) = η(q 8 )2 η(q 4 )2 by taking the second, respectively, third power of the corresponding Gr¨ ossencharacter ψ of conductor 2 + 2i. The Gal(Q/Q)-module Hé5t (X, Ql ) is the

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tensor product of the Gal(Q/Q)-module Hé3t (X, Ql ) and of the transcendental lattice T (S) of S. Its semi-simplification splits into two two-dimensional pieces and one finds for the L-series L(X5 , s) L(g4 ⊗ g3 , s) = L(g6 , s)L(g2 , s − 2) where g2 is as above, and g6 is a level 32 cusp form of weight 6, which can be ossencharacter ψ. derived from g2 by taking the fifth power of the Gr¨ If one wants to find examples of higher dimensional Calabi-Yau manifolds which are the analogue of rigid Calabi-Yau varieties, i.e., have two-dimensional middle cohomology, then one has to divide by more complicated groups. One example of such a construction is the following. Consider the elliptic curve E : y 2 = x3 − D for some positive integer D. Then x → ρx, with ρ a third root of unity, defines an automorphism η of order 3 on E. Theorem 5.8. Let n > 1 be an integer and let X be the quotient of the cartesian product E n by the group {(η a1 × · · · × η an ) ∈ Aut(E n ); a1 + · · · + an ≡ 0 mod 3}. Then X has a smooth model X which is a Calabi-Yau manifold and the dimension of Hént (X, Q ) equals two if n is odd, respectively the dimenson of T (X) ⊗ Ql equals two if n is even, where T (X) is the transcendental part of the middle cohomology. Moreover, X is defined over Q and L(Hént (X, Ql ), s) L(gn+1 , s), respectively L(T (X) ⊗ Ql , s) L(gn+1 , s), √ where gn+1 is the weight n + 1 cusp form with complex multiplication in Q( −3), associated to the n-th power of the Gr¨ ossencharacter of E. One can also attempt to generalise the construction of double covers of Pn , branched along a configuration of hyperplanes, to higher dimension. In general, however, this becomes very difficult. The combinatorial problems in order to control the singularities of, say, a double cover of P5 branched along 12 hyperplanes become very hard. In [CH05] one example was studied, namely the double cover X of P5 given by w2 = x(x − u)(x − v)y(y − u)(y − v)z(z − u)(z − v)t(t − u)(t − v). This example was first studied by Ahlgren, who counted the number of points modulo p on the affine part of X, relating these numbers to a modular form. The following was proved in [CH05, Theorem 5.12]. Theorem 5.9. The variety X has a smooth model Z, defined over Q, which is a Calabi-Yau fivefold with Betti numbers b1 (Z) = b3 (Z) = 0, b5 (Z) = 2. More precisely, h5,0 = h0,5 = 1. One has L(Z, s) L(f6 , s) where f6 (q) = η 12 (q 2 ) is the unique normalised cusp form of level four and weight six.


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6 Zeta Functions and Mirror Symmetry 6.1 p-adic Cohomology In practice, it turns out to be hard to work with étale cohomology in an explicit way (at least in dimension two and higher). In this subsection we discuss another approach to calculate the zeta-function of a smooth variety over a finite field Fq . This gives enough information to obtain the characteristic polynomial of Frp on Héit (X, Ql ) for a prime of good reduction. We use cohomology with p-adic coefficients instead of l-adic coefficients. Cohomology theories with p-adic coefficients occured, for example, in Dwork’s proof [Dwo60] of the rationality of the zeta-function. There exist several p-adic cohomology theories which are rich enough to prove the Weil conjectures, and therefore are called ‘Weil cohomologies’. These theories have a Lefschetz trace formula. We discuss an easy variant, called Monsky-Washnitzer cohomology, which strictly speaking is not a Weil cohomology. For a survey on this theory we refer to [vdP86]. Let p be a prime number, r a positive integer. Set q = pr . Suppose V /Fq is a hypersurface in Pn and let X be the complement of V in Pn . Then X is an affine variety, say X = Spec R . Consider the ring Zq , the ring of integers of Qq , which is the unique unramified extension of Qp of degree r. Let π be the maximal ideal of Zq . The ring R can be ‘lifted’ to Zq , i.e., there is an algebra ˆ := Zq [X0 , . . . , Xm ]/(f1 , . . . , fk ) such that R/π ˆ R ˆ∼ R = R . ˆ Let R be the ‘overconvergent completion’ of R, that is the ring {g ∈ Zq [[X0 , . . . , Xm ]]; g converges on an open disc of radius more than 1} . (f1 , . . . , fk ) This definition enables us to have a good theory of differential forms on X: consider the complex of differential algebras 0 1 2 n 0 → ΩR → ΩR → ΩR → · · · → ΩR → 0. i Then we define HM W (X, Qq ) to be the i-th cohomology group of this complex tensored with Qq . It turns out that these groups are finite-dimensional. The reason to work with (overconvergent) p-adic algebras is that they admit lifts of Frobenius, i.e., there exists a morphism Frq : R → R such that the reduction Frq : R/πR → R/πR is the geometric Frobenius map on X. One can show that i (−1)i tr(q n Fr−1 q | HM W (X, Qq )) = #X(Fq ). i

The appearance of q n Fr−1 is comes from the fact that Monsky-Washnitzer q cohomology is the Poincaré dual of rigid cohomology with compact support.

298


The latter theory has a more usual Lefschetz trace formula. We note that R and Frq are far from being unique. Let (R1 , Frq,1 ) be another lift. Then there is a canonical isomorphism ψ between the cohomology groups associated to • • and the cohomology groups associated to ΩR such that ψ◦Frq = Frq,1 ◦ψ. ΩR 1 Example 6.1. Let V be the curve given by F = y 2 z − x3 − axz 2 − bz 3 for a, b ∈ Fq such that gcd(6, q) = 1 and 4a3 + 27b2 = 0. Let X = P2 \ V . Then 0 2 ∼ HM W (X, Qq ) = Qq and HM W (X, Qq ) is generated by xyz (xyz)2 Ω and Ω F F2 with Ω = dy ∧ dz − dx ∧ dz + dx ∧ dy. All other cohomology groups vanish. As a lift of Frobenius one can take [x, y, z] → [xp , y p , z p ]. Remark 6.2. Suppose E ⊂ P2C is a complex elliptic curve given by the zero-set of a cubic F as in the above example. Then one can show that the usual de Rham cohomology group H 2 (P2 \ E, C) is generated by ω1 :=

XY Z (XY Z)2 Ω and ω2 := Ω. F F2

A famous theorem of Griffiths [Gri69] applied to this case yields that ω2 generates F 2 H 2 (P2 \E, C) and F 1 H 2 (P2 \E, C)/F 2 H 2 (P2 \E, C) is generated by ω1 , where F • is the Hodge filtration. Example 6.3. Let g be a positive integer. Consider the algebra R := Fq [x, y, z]/(−y 2 + f2g+1 , yz − 1) with f2g+1 ∈ Fq [x] a polynomial of degree 2g + 1, with only simple roots. Then Spec R is a hyperelliptic curve of genus g minus the 2g + 2 Weierstrass points. Lift 0, 1, . . . , p − 1 to 0, 1, . . . , p − 1. This choice induces a unique lift σ : Qp → Qp of Frobenius. 1 A basis for HM W (X, Qq ) is xi dx for i = 0, . . . , 2g, and xi

dx for i = 0, . . . 2g − 1. y

There is a unique lift Frq of Frobenius extending σ, such that Frq (x) = xq , Frq (y) ≡ y q mod π and Frq (y)2 = Frq (f2g+1 ). Finally Frq (z) is defined by Frq (z) Frq (y) = 1. This description of the Monsky-Washnitzer cohomology enabled Kedlaya [Ked01] to present an efficient point counting algorithm on hyperelliptic curves over fields of small characteristic. We would like to point out that Frq (y) and Frq (z) are overconvergent power series which do not converge to a rational function. This is in contrast to the previous example.


299

The fact that we are working with a cohomology theory using differential forms enables us to use more analytic techniques. Consider the family Vt of elliptic curves X 3 + Y 3 + Z 3 + 3tXY Z where t is a parameter in the p-adic 2 unit disc ∆. Then HM W (Xt0 , Qq ) is two-dimensional for a general t0 ∈ ∆. One of the two eigenvalues of p2 Fr−1 p has p-adic valuation 0. This eigenvalue is called the unit root. Dwork showed that the unit root, as a function of t, satisfies the p-adic hypergeometric differential equation with parameters 1 1 2 , 2 ; 1. Specialising to values t0 on the boundary of the unit disc gives the 2 characteristic polynomial of Frobenius on HM W (Xt0 , Qq ) with t0 ≡ t mod π. n+1 N. Katz has generalised this idea: on the p-adic cohomology groups HM W (Xt ) one can introduce a Gauss-Manin connection, much in the same way as one usually does in the complex case. Theorem 6.4 (Katz, [Kat68]). Let Xt be a family of hypersurface complen ments in Pn and let (HM W (Xt , Qq ), ∇GM ) be the local system of cohomology groups. Let A(t) be a solution of the Picard-Fuchs equation (i.e., the differential equation associated to ∇GM .) Fix a basis e1,t , . . . , em,t for this local n system and let Fr(t) be the matrix of the Frobenius action on HM W (Xt , Qq ) with respect to e1,t , . . . , em,t . Let t0 ∈ Qq such that | t0 |q = 1. Then Fr(t0 ) = A(tq0 )−1 Fr(0)A(t0 ) provided that the reduction X t0 is smooth. 6.2 Mirror Symmetry and Zeta Functions The classical Picard-Fuchs equations of Calabi-Yau threefolds play an important role in Mirror Symmetry. Candelas, de la Ossa and Rodriguez-Villegas [COR00], [COR03] considered the p-adic Picard-Fuchs equation (cf. Theorem 6.4) of the following one-parameter family of quintics 4 : 5 M : Q(ψ) = xi − 5ψx0 x1 x2 x3 x4 = 0 . (5) i=0

For general ψ, i.e., ψ 5 = 1, ∞, this is a smooth quintic. They calculated explicit expressions for the number of points on Q(ψ), using tedious computations with so-called Dwork characters. Then they observed that these expression were solutions to the Picard-Fuchs equations. The above mentioned Theorem 6.4 gives a more mathematical explanation for this phenomenon. Using these calculations, Candelas et al. determined the structure of the zeta-function, more precisely, they showed that for any prime p = 5 and any ψ ∈ Fp , with ψ 5 = 0, 1, 20

30

R0 (t, ψ)RA (pρ tρ , ψ) ρ RB (pρ tρ , ψ) ρ ZM (t, ψ) = ZQ(ψ) (t) = (1 − t)(1 − pt)(1 − p2 t)(1 − p3 t)

(6)

300


where ρ = 1, 2 or 4 is the least integer such that pρ ≡ 1 mod 5. The R’s are quartic polynomials in t. We would like to point out that this factorisation can be obtained in a more direct way. As far as we know this strategy is not contained in the literature. Every member of the family M has many automorphisms. We distinguish three types. First of all, each σ ∈ S5 induces a permutation of coordinates xi → xσ(i) . Secondly, let ζ be a primitive 5-th root of unity. Then the group 3 4 G = diag(ζ n0 , ζ n1 , ζ n2 , ζ n3 , ζ n4 ); ni ≡ 0 mod 5 (7) acts on the family M. (Note that this action is defined over Fpρ .) Finally, the third type is the Frobenius automorphism Frpρ . Note that Frpρ commutes with all elements in G as well as all elements in S5 . It is easy to write down 4 4 an explicit basis for HM W (P \ Q(ψ), Qp ) and to calculate the action of the automorphisms of the first and second type on this basis. Since two commuting automorphisms respect each other’s eigenspaces, we can use the automorphisms coming from the S5 -action to identify all the 4 4 eigenspaces of Frobenius on HM W (P \ Q(ψ, Qp )). Then we can use the automorphisms of G to prove that many of the eigenvalues of Frobenius coincide. This gives a factorisation of ZP4 \Q(ψ) (t). Since ZP4 \Q(ψ) (t)ZM (t, ψ) = ZP4 (t) this factorisation can be translated in a factorisation of ZM (t, ψ) as in (6). The R’s are quartic polynomials, satisfying the functional equation 1 1 R , ψ = 6 4 R(t, ψ). p3 t p t In particular R0 (t, ψ) = 1 + a1 (ψ)t + b1 (ψ)pt2 + a1 (ψ)p3 t3 + p6 t4 where a1 (ψ), b1 (ψ) depend on the parameter ψ. The two factors RA and RB are closely related to genus 4 curves A = A(ψ) and B = B(ψ), namely ZA (u, ψ) =

RA (u, ψ)2 , (1 − u)(1 − pu)

with an analogous expression for RB . The curves A and B are the nonsingular models of y 5 = x2 (1 − x)β (1 − x/ψ 5 )5−β where β = 3 for A, and β = 4 for B. The geometric role of A and B does not seem to be well understood. There is no good notion of a mirror family over non-algebraically closed fields in positive characteristic. However, in this case we can mimic the mirror construction since the mirror family of (5), considered as a family of complex varieties, is given by a quotient construction using G: for a general element


301

Q(ψ) the quotient Q(ψ)/G has nodes and resolving these nodes gives CalabiYau threefolds Q (ψ), which form the mirror family W of M. In the complex case one has an equivalent construction, namely as family of hypersurfaces in the toric variety P∆ where ∆ is the simplex in R4 given by {e1 , . . . , e4 , −(e1 + · · · e4 )}. The mirror family is then given by crepant resolutions of the closure in P∆ of the affine family in the torus (C∗ )4 defined by the equation 1 x1 + · · · + x n + + ψ = 0. x1 · · · x4 It is known among experts how one can generalise this construction to p-adic toric varieties. Candelas et al. computed the zeta function of the mirror family for general ψ as ZW (t, ψ) = ZQ (ψ) (t) =

(1 − t)(1 −

R0 (t, ψ) 101 pt) (1 − p2 t)101 (1

− p3 t)

.

(8)

Using the explicit differential forms provided by Monsky-Washnitzer co4 4 G homology one can easily find an explicit basis for HM W (P \ Q(ψ)) . Along the same lines one shows that R0 (t, ψ) is the characteristic polynomial of 4 4 G Frobenius on HM W (P \ Q(ψ)) . This explains why R0 (t, ψ) is a factor of the numerator of the Zeta function of the mirror threefold. Comparing equations (6) and (8) shows that there is a close relationship between the zeta function of the family M and its mirror W. In particular, one obtains the congruence ZM (t, ψ) ≡

1 ≡ (1 − pt)100 (1 − p2 t)100 mod 52 . ZW (t, ψ)

(9)

In her thesis Kadir [Kad04] has investigated the case of octics in a certain weighted projective space, where she was able to prove analogous results. The question of studying the zeta functions of mirror families was also taken up by Wan [Wan04] and Fu and Wan [FW05]. Wan considered the family of hypersurfaces + · · · + xn+1 + ψx0 · · · xn = 0. Xψ : xn+1 n 0 The construction of the ‘mirror family’ Yψ is then completely analogous to the quintic case. He calls a pair (Xψ , Yψ ) with the same parameter ψ a strong mirror pair and proves the following generalisation of (9) for such pairs: #Xψ (Fqk ) ≡ #Yψ (Fqk ) mod q k .

(10)

Here q = pr is a prime power. He conjectures that such congruences should hold in greater generality. Such a generalisation for certain pairs (X, X/G) was then indeed obtained by Fu and Wan in [FW05]. As a consequence of their results they prove

302


Theorem 6.5. Let X be a smooth, geometrically connected variety X over the finite field Fq , q = pr . Assume that H i (X, OX ) = 0 for i > 0. Then for all integers k, the congruence #X(Fqk ) ≡ 1 mod q k holds. Moreover, let G be a finite group of Fq automorphisms acting on X. Then #(X/G)(Fqk ) ≡ #X(Fqk ) ≡ 1 mod q k . Remark 6.6. Fu and Wan proved a similar result when X is a Calabi-Yau variety. Namely, if G is a finite group of automorphisms of X fixing a nonzero n-from on X then #(X/G)(Fqk ) ≡ #X(Fqk ) mod q k . Their proof of these results uses crystalline cohomology and the MazurOgus theorem. Esnault [Esn05] has pointed out that these results can also be obtained using de Rham-Witt cohomology and rigid cohomology.

7 Open Questions We conclude this paper with a brief discussion of some open problems. These will mainly concern Calabi-Yau threefolds. We start with some questions related to modular forms. If f is a newform of weight two with field of coefficients F , then a classical construction associates to f an abelian variety A over Q of dimension [F : Q]. In particular, if f has rational coefficients, then this gives an elliptic curve E. For higher weight, however, no such construction is available. As remarked in Remark 3.1 one can always find some symmetric product of some universal curve for which part of the cohomology corresponds to f . One might ask whether any such form can be realised geometrically by a Calabi-Yau variety: Question 7.1. Let f be a newform of weight k > 2 with rational Fourier coefficients. Is there a Calabi-Yau variety X over Q such that L(f, s) occurs in the L-series of X? This question was formulated independently by B. Mazur and D. van Straten. Consider two non-isomorphic modular varieties defined over Q such that they are isomorphic over Q. In practice, it turns out that very often the associated modular forms differ by a twist. E.g., this can be seen in the case of elliptic fibrations in Weierstrass form or with double octics, where twists over quadratic extensions can easily be realised. For this reason the above question should be asked modulo twisting. By Theorem 5.4 (and assuming GRH) the number of available forms of given odd weight is finite up to twisting. In the case of singular K3 surfaces Shafarevich’s theorem 5.3 also tells us that


303

the number of available surfaces is finite, hence in the weight three case this reduces to a question of matching two finite sets. We have already mentioned that, so far, 24 of the 64 forms have been realised geometrically (see Sect. 5.1 or [Schu05, Section 15]). On the other hand, if the weight k is even, then there are infinitely many newforms with rational coefficients, even up to twisting. An answer to Question 7.1 would therefore require a general construction, similar to the weight two case of elliptic curves. In the case of weight four, we have seen a number of examples of modular Calabi-Yau threefolds in Sect. 5.2. In total, Calabi-Yau threefolds for around 80 newforms have been found so far (modulo twisting). We refer the reader to Meyer’s book [CM05a], which contains the most systematic collection of examples up to date. These include non-rigid and nodal varieties as well as some threefolds where the corresponding newform is only conjectural. Nevertheless, there are no known Calabi-Yau partners for the newforms of level 7 and 13. In general, it seems to be hard to find modular varieties with big bad primes. It would be interesting to find constructions which yield such examples. Another question concerns the level of the corresponding newform: Question 7.2. Let X be a modular variety over Q. Can one predict the level of the associated newform? In particular, this question concerns the bad primes of X. Once more, the difference with the special case of elliptic curves has to be emphasised. For an elliptic curve E, say over Q and hence modular, the theory provides a globally minimal model. This can be used to define the conductor of the curve, which then coincides with the level of the corresponding newform. For a general variety X over Q, no concept of arithmetic minimality is known. Nevertheless, we can define a conductor for any compatible system of étale Galois representations. Assuming modularity, this conductor coincides with the level N of the associated newform. This is a consequence of the compatibility with the local Langlands correspondence. In Sect. 4.1, we gave the classical bounds for the exponents ep of the conductor, if E is an elliptic curve. The same bounds also hold in many other cases. For a smooth projective variety X over Q of odd dimension n, these were proved by Serre in [Ser87] for the Galois representation of Hént (X, Ql ) provided this is two-dimensional and the Hodge decomposition is H n (XC , C) ∼ = H n,0 (XC ) ⊕ H 0,n (XC ), and assuming that the representation is modular. This was reproved by Dieulefait in [Di04a]. Dieulefait also remarks that his proof (combined with the potential modularity result of Taylor) is enough to conclude that the result still holds without the modularity assumption. In particular, these bounds apply to rigid Calabi-Yau threefolds over Q. For newforms of odd weight and rational coefficients (or in general CM-forms), the same bounds have been obtained

304


by Sch¨ utt in [Schu05, Cor. 12.2]. In practice, these bounds can be used to precisely determine the corresponding newform, if modularity is known (e.g., by Theorem 4.2). One then writes down all newforms with level composed of the bad primes and checks some coefficients after counting points. For more details we refer to [Di04a], where this was discussed for Calabi-Yau varieties. At the moment, this method only works if there are at most two very small bad primes or if the newform has CM. This takes us back to singular K3 surfaces and rigid Calabi-Yau threefolds over Q. If a bad prime p divides the level of the newform associated to X, then examples show the following: if the reduction Xp has a single A1 -singularity then ep = 1. Otherwise, ep = 2 (for p > 3) or ep ≥ 2 (if p = 2 or 3). In view of this, we would like to ask the following question: Question 7.3. Let X be a modular variety over Q with corresponding newform f . Under which conditions do the bad Euler factors of L(X, s) and L(f, s) coincide? Since the conductors have to be equal, one can hope for an unconditional answer. To our knowledge, this has not yet been investigated. We shall now mention one further tool which can sometimes be used to prove modularity. This consists in exhibiting a birational map (over Q) between a modular variety and one which is expected to be modular. In [HSGS01], two such varieties were called relatives. This paper gives a number of examples for such relatives. For example, birational maps over Q between the following varieties were given: the Barth-Nieto quintic, the self-fibre product of the universal elliptic curve E1 (6) and Verrill’s threefold associated to the root lattice A3 . Recently, Cynk and Meyer published a study of rigid Calabi-Yau threefolds of level 8 where the question of relatives was treated systematically [CM05b]. This issue can also be discussed for the self-fibre products from Sect. 5.2. Here, each permutation of four cusps leads to isogenous elliptic fibrations. Then it is a matter of the field of definition of the isogeny which newform occurs. If the isogeny is not defined over Q, then the newform might be twisted. For instance, the modular elliptic surfaces associated to Γ1 (4) ∩ Γ (2) and Γ0 (8) ∩ Γ1 (4) are isogenous. If one works with the equations from [Be82], this isogeny is only defined over Q(i). The newforms of level 8 and 16 are twists by the corresponding quadratic character. In general, this problem is addressed by the Tate conjecture. Let ρ1 , ρ2 be isomorphic two-dimensional l-adic Galois representations coming from geometry. In particular, these Galois representations occur in the étale cohomology of some smooth projective varieties X1 , X2 over Q. Then the Tate conjecture predicts a correspondence, i.e., an algebraic cycle in the product X1 × X2 , which is defined over Q and which induces an isomorphism between ρ1 and ρ2 . For details on known correspondences, the reader is referred to the book of Meyer [Mey05]. Here we shall only mention one particular example. In


305

[vGN95], van Geemen and Nygaard study three Siegel modular threefolds over Q. One of them is the first non-rigid Calabi-Yau threefold for which modularity has been established. Denoting this variety by X, van Geemen and Nygaard show that h1,2 (X) = 1. An automorphism of X is used to split the four-dimensional Galois representation of Hé3t (X, Ql ) into a sum of twodimensional representations. Then the authors deduce the equality L(X, s) = L(f4 , s) L(f2 , s − 1) with newforms of level 32 and weight four and two respectively. Here, f2 comes from the Gr¨ ossencharacter ψ of Q(i) with conductor (2 + 2i) and ∞-type 1. The other newform, f4 is derived from ψ 3 . On the other hand, ψ is related to the elliptic curve with CM by Z[i]: E : y 2 = x3 − x as

L(E, s) = L(ψ, s).

As a consequence, L(X, s) = L(Sym3 Hé1t (E, Ql ), s), and hence the Tate conjecture predicts the existence of a correspondence between X and E × E × E which induces an isomorphism between the two representations corresponding to ψ 3 . This is established by means of a curve of genus 5 over Q which admits rational maps over Q to both varieties, X and E × E × E. However, there are modular varieties X and Y with the same associated form, for which there is no known correspondence. Question 7.4. Let X and Y be modular varieties which have an associated form in common, find a correspondence between them. We would like to conclude this section with a question related to arithmetic mirror symmetry. In Sect. 6.2 we discussed the zeta function of the family M of quintics in P4 (See (6)). This expression involves genus 4 curves A and B whose geometric meaning has, so far, remained a mystery. Question 7.5. What is the geometric meaning of the genus 4 curves which appear in the motive of the one-parameter quintic family M? Acknowledgements We cordially thank the DFG who has generously supported our research in the framework of the Schwerpunktprogramm 1094 ‘Globale Methoden in der komplexen Geometrie’. In our project on the modularity of Calabi-Yau varieties we closely cooperated with the research group in Mainz led by D. van Straten. We thank him and C. Meyer for numerous discussions. We are grateful to S. Cynk, J. Top and H. Verrill who are coauthors of some of the papers discussed here and with whom we have collaborated in many ways. Last but not least we thank S. Kadir and N. Yui for many stimulating discussions.

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Some Recent Developments in the Classification Theory of Higher Dimensional Manifolds Priska Jahnke1 , Thomas Peternell2 and Ivo Radloff3 1 2 3

Mathematisches Institut der Universit¨ at Bayreuth, D-95440 Bayreuth, Germany [email protected], Mathematisches Institut der Universit¨ at Bayreuth, D-95440 Bayreuth, Germany [email protected] Mathematisches Institut der Universit¨ at Bayreuth, D-95440 Bayreuth, Germany [email protected]

Introduction This article describes some recent developments in the classification theory of projective varieties and compact K¨ ahler manifolds. There are two mains themes: • pseudo-effective and nef line bundles and the geometry of projective varieties and K¨ ahler manifolds, • singular Fano threefolds and smooth threefold which are nearly Fano. In Sect. 2 we discuss general properties of pseudo-effective line bundles, various cones and duality theorems. We also discuss algebraicity problems. Section 3 gives applications to uniruledness and rational connectivity, a strong generic nefness of the cotangent bundle of a non-uniruled variety and to abundance. In Sect. 4 wee describe the status of Mori theory on non-algebraic K¨ ahler threefolds, whereas Sect. 5 deals with approximation of K¨ ahler by projective manifolds. In Sect. 6 we use the strong nefness from Sect. 3 to study subsheaves of sheaves of p−forms, a refined Kodaira dimension and the universal cover. Then we are turning to threefolds X with nef anticanonical bundle −KX . Section 7 gives structure results in case −KX is not big. Sections 8 and 9 deal with smooth threefolds with −KX big and nef (but not ample) and b2 (X) = 2 and with singular Gorenstein Fano threefolds, in particular their anticanonical linear systems. The final section is devoted to smoothings of Gorenstein Fano threefolds.

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1 Preliminaries Throughout this paper we are dealing with complex varieties. We recall some basic notions. A projective manifold X – or more generally, a normal projective Q−Gorenstein variety (see below) is called Fano if the anticanonical divisor −KX is ample. A projective manifold X is uniruled if there exists a covering family of rational curves on X, or, equivalently, if through the general point x of X, there is a rational curve, which is to say that there is a non- constant holomorphic map f : P1 → X whose image contains x. X is rationally connected if two general points can be joined by a chain of rational curves. Fano manifolds are rationally connected. We recall here some definitions around the singularities of the minimal model program. We ignore the log case even though it implicitely plays a role below. Let X be a normal projective Cohen Macaulay variety and let f : X −→ X be a resolution of singularities. Let ωX denote the dualizing sheaf and KX a canonical Weil divisor on X. We assume that X is Q−Gorenstein, i.e., some multiple mKX is Cartier. Denote the exceptional divisors of f by Ei . Then we have an equation of Q−divisors KX = f ∗ KX + ai Ei , ai ∈ Q. (1.0.1) The coefficient ai ∈ Q of some Ei is called the discrepancy of f at Ei . Now X is said to have (1) canonical singularities if ai ≥ 0 for all i; (2) terminal singularities if ai > 0 for all i. Terminal singularities in dimension 3 are points, canonical singularities may occur in codimension two (e.g. the product P1 × S with S a surface with rational double points). Canonical singularities are known to be rational, i.e., Ri f∗ OX = 0.

2 Cones, Nef and Pseudo-effective Line Bundles We fix a compact Kähler manifold X. Some notations first: we set HRq,q (X) = H q,q (X) ∩ H 2q (X, R), q where as usual H q,q (X) = H q (ΩX ). We also set

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N S(X) = N 1 (X) = (H 1,1 (X) ∩ H 2 (X, Q)) ⊗ R and N1 (X) = (HRn−1,n−1 (X) ∩ H 2n−2 (X, Z)) ⊗Z R. The letters N S stand for Néron-Severi. In Mori theory however the notation N 1 (X) is standard. Recall that the vector spaces N 1 (X) and N1 (X) are finite-dimensional and dual to each other in a natural way. 2.1 Definition. (1) The K¨ ahler cone K = K(X) is the (open) cone in HR1,1 (X) generated by the classes of K¨ ahler forms. (2) The ample cone A is the open cone in N 1 (X) generated by the classes of ample line bundles, its closure is the nef cone. (3) The pseudo-effective cone E is the closed cone in HR1,1 (X, X) of classes of positive closed (1, 1)−currents. The intersection EN S = E ∩ N S(X) is also called pseudo-effective cone. Notice that A = K ∩ N S(X). Moreover, EN S is the closure of the cone of effective divisors, [De90,92]. A line bundle L is nef , if c1 (L) ∈ A. It is classical in case X is projective that this is equivalent to the condition L·C ≥0 for all irreducible curves C ⊂ X. Moreover L is pseudo-effective if c1 (L) ∈ EN S . Again by [De90], this is the same as to say that L carries a singular metric with (semi-)positive curvature current. A singular hermitian metric h is given locally (in a chart) by the weight function e−φ with φ locally L1 , and its curvature current is i ∂∂φ. π So if v ∈ Lx and λ : LU → U × C the local trivialization, then h(v) = |λ(v)|e−φ(x) . In other words, L is pseudo-effective, if there exists a singular metric whose local weight function are pluri-subharmonic. Often we say that (L, h) is pseudoeffective. For example, if L = OX (D) with D an effective divisor given by a section s of L, we define a singular hermitian metric as follows. If locally in a chart s is given by a holomorphic function f , then the weight function φ = log|f |. If the metric h can be chosen C ∞ , then L is called hermitian semi-positive.

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2.2 Definition. (1) Let φ be a plurisubharmonic function on a complex manifold X. Then the multiplier ideal sheaf I(φ) is the ideal sheaf of germs of holomorphic functions f such that |f |2 e−2φ is locally integrable. (2) Now let (L, h) be pseudo-effective on the compact K¨ ahler manifold X. Then the multiplier ideal I(h) is defined as follows. The metric h is locally given by the weight function e−φ , and we set locally I(h) = I(φ). One checks easily that this does not depend on the choice of the chart of L and thus defines a global ideal sheaf I(h). Basic references are [De01] and [La04]. If the metric h is smooth, then I(h) = OX . It was a fundamental question whether any nef line bundle on a projective manifold is hermitian semipositive. This turned out not to be the case, and the most conceptual way to see this is via the Hard Lefschetz Theorem for pseudo-effective line bundles wich we state now. 2.3 Theorem. Let (X, ω) be a compact K¨ ahler manifold, (L, h) pseudoeffective with multiplier ideal I(h). Then for all q, the canonical map ∧ω q : H 0 (X, Ω q ⊗ L ⊗ I(h)) → H q (X, KX ⊗ L ⊗ I(h)) is surjective. For the proof see [DPS01]. In order to check that a given nef line bundle L is not hermitian semi-positive, it suffices to show that Hard Lefschetz does not hold when the multiplier ideal sheaf is omitted. To get a specific example, consider an elliptic curve E and a non-split extension 0 → OE → V → OE → 0. Let X = P(V ) (in the sense of Grothendieck) and L = OP(V ) (1). Let ω be any K¨ ahler form on X. Then one can check that 1 ∧ω : H 0 (ΩX ⊗ L3 ) → H 1 (KX ⊗ L3 )

vanishes. Hence L3 is not hermitian semi-positive and so does L as well as −KX = L2 . For details see again [DPS01]. One even sees that no multiple Lm is hermitian semi-positive. However a nef line bundle is not too far from being hermitian-semi-positive [De90]: 2.4 Theorem. Let X be compact K¨ ahler; fix some K¨ ahler form ω. Then a line bundle L on X is nef if and only if for all > 0 there exists a smooth metric h on L such that ΘL,h ≥ −ω.

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We now turn to the dual situation. 2.5 Definition. (1) The cone of classes of positive closed (n − 1, n − 1)−currents on X is denoted by N ⊂ HRn−1,n−1 (X). (2) The movable cone M ⊂ HRn−1,n−1 (X) is the closed cone generated by classes of currents of the form µ∗ (˜ ω1 ∧ . . . ∧ ω ˜ n−1 ). ˜ → X is any modification from a compact K¨ ˜ Here µ : X ahler manifold X ˜ and ω ˜ i are K¨ ahler forms on X. (3) NN S := N ∩ N1 (X); and MN S := M ∩ N1 (X). (4) N E(X) ⊂ N1 (X) is the closed cone generated by the classes of irreducible curves. (5) If X is projective, we say that an irreducible curve C ⊂ X is strongly movable, if ˜1 ∩ . . . ∩ H ˜ n−1 ) C = µ∗ (H ˜ j very ample on X. ˜ Then the closed cone generated by the classes with H of strongly movable curves is denoted by SM E(X) ⊂ N E(X). (6) An irreducible curve C is called movable , if C = Ct0 is a member of a family (Ct ) of curves such that X = t Ct . The closed cone generated by the classes of movable curves is denoted by M E(X). (7) M E nef is the closed cone generated by curves C ⊂ X whose normal sheaf NC := Hom(IC /IC2 , OC ) is nef. There are three basic duality statements. • First, by the fundamental results of Demailly-Paun [DP04], the cone N is ahler cone. dual to the closure K of the K¨ • Second, a classical result says that in case X is projective, N E(X) is dual to the nef cone (we already mentioned that a line bundle L is nef if and only if L · C ≥ 0 for all curves C). • Third, the paper [BDPP04] answers the question for the dual of the pseudo-effective cone EN S . To motivate the third item, consider the blow-up π : X → Y of a submanifold A in the projective manifold Y and let E denote the exceptional divisor. The line bundle OX (E) is pseudo-effective but not nef: E · C < 0 for all curves C ⊂ E contracted by π. But if C is a curve, possibly inside E, which deforms in a family Ct covering X, then E · C = E · Ct ≥ 0. This example is typical: 2.6 Theorem. Let X be a projective manifold. Then the cones EN S and SM E are dual. In particular, a line bundle L is pseudo-effective if and only if L · C ≥ 0 for all strongly movable curves C ⊂ X.

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This theorem allows to identify various cones of curves. In fact, it is not difficult to establish the following sequence of inclusions SM E ⊂ M E ⊂ M E nef ⊂ (EN S )∗ and we also know that M E ⊂ MN S ⊂ (EN S )∗ . Thus Theorem 2.6 gives 2.7 Corollary. Let X be a projective manifold. Then SM E = M E = M E nef = MN S = (EN S )∗ For any details we refer to [BDPP04]; the fact that (EN S )∗ contains the cone of curves with nef normal bundles is proved in [DPS96]. 2.8 Corollary. Let X be a projective manifold and L a line bundle on X. (1) L is pseudo-effective if and only if L · C ≥ 0 for all curves C with nef normal sheaves NC = (IC /IC2 )∗ . (2) If L is big, then L · C > 0 for all curves with nef normal sheaves. One can rephrase the chararacterization of pseudo-effective line bundles also by saying that L is pseudo-effective if and only if L · C ≥ 0 for all very general curves C. This is to say that there is a countable union of algebraic subvarieties A such that L · C ≥ 0 for all C ⊂ A. Also the characterization carries over to normal Moishezon spaces. It is an open problem to relate the cone of curves with ample normal sheaves to the other cones; possibly this cone could be smaller. Concerning the dual of E one has 2.9 Conjecture. Let X be a compact K¨ ahler manifold. Then the cones E and M are dual. It is clear that on a non-projective compact Kähler manifold nefness of a line bundle L cannot be rephrased by saying that L · C ≥ 0; there are simply too few curves. However for the canonical bundle this should be true – and is in fact in dimension at most 3. We will come back to that point in Sect. 3. If K contains a rational point a, then a famous theorem of Kodaira says that X is projective. In fact, some multiple ma is in H 2 (X, Z) and thus it is represented by (the Chern class of) an ample line bundle. It is interesting to ask the dual question: 2.10 Problem. Let X be compact K¨ ahler and a an interior rational point in the dual N of the K¨ ahler cone. How algebraic is X? I.e., what can be said on the algebraic dimension of X? Somehow weaker: assume that a is an interior rational point of M. What does this imply for X?

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This problem is treated in [OP04]. Whereas in the case of surfaces there is a complete answer [Hu99],[OP00], the situation in higher dimensions is very much open. Here is what is known so far. 2.11 Theorem. Let X be a compact K¨ ahler threefold, C ⊂ X an effective curves such that [C] is an interior point of N . Then the algebraic dimension a(X) ≥ 1 unless X is both simple and not Kummer. We say that a K¨ ahler manifold is simple if there is no proper compact subvariety through the very general point of X. Moreover X is Kummer if X is bimeromorphic to A/G with A a torus and G a finite group. We will come back to this class of manifolds later and just say here that it is expected that a simple threefold is always Kummer. The proof of Theorem 2.9 uses very much arguments from dimension 3 and does not generalize immediately to higher dimensions. Also it is very unclear whether examples with a(X) = 2 really exist. The case that [C] is in the interior of M might be slightly easier and is certainly also very interesting. The condition that [C] is an interior point of N is somehow related to positivity of the normal sheaf. So consider a smooth curve C ⊂ X with ample normal bundle in the compact K¨ ahler manifold X. Again X should be “rather” algebraic. One has ([OP04]): 2.12 Theorem. Let X be a compact K¨ ahler manifold, C ⊂ X a smooth curve of genus g. Suppose that NC is ample and dim X = 3 for the statements (2)-(4). (1) (2) (3) (4)

If g ≤ 1, then X is projective; κ(X) ≤ 0, then X is projective or simple non-Kummer; a(X) = 1; If C moves in a family that covers X, then X is projective.

A comparison with the previous theorem suggests that [C] should be an interior point of N . Here is a problem related to (2.10) dealing with higher dimensional subvarieties. 2.13 Problem. Let X be a compact K¨ ahler manifold (or a manifold bimeromorphic to a compact K¨ ahler manifold). Let Y ⊂ X be a compact submanifold whose normal bundle is ample. Is a(X) ≥ dim Y + 1? Using methods from cycle space theory, Barlet and Magnusson [BM04] show the following ahler manifold. Suppose there is an 2.14 Theorem. Let Xn be a compact K¨ equidimensional map π : X → S to a normal compact complex space Sm . Let Y ⊂ X be a compact submanifold of dimension m − 1 with (Griffiths) positive normal bundle. Then S Moishezon, in particular a(X) ≥ dim Y + 1.

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Griffiths-positivity is a differential-geometric analogon to ampleness and potentially (?) equivalent to ampleness, see [De01]. For the proof of Theorem 2.14 and more general results we refer to [BM01]. A key point is to consider the set Σ = {s ∈ S|Xs ∩ Y = ∅}. Since Y is projective, one knows that Σ is Moishezon. Also notice that the positivity assumption on NY implies that Σ has codimension 1 in S and one want to conclude that the normal sheaf of Σ has positivity properties in order to conclude that S is Moishezon. This approach also works in a more general context, namely when – instead of assuming the existence of the map π – one only has a covering family (Xs ) of compact cycles of dimension n − m such that for general s, the cycle Xs meets Y in a finite set (plus some technical condition). Finally we mention the important work of Boucksom [Bo02a,02b,04], who constructs a “Zariski decomposition” of a pseudo-effective class into some “exceptional” class and a class which is nef in codimension 1. We do not go into any detail here.

3 Uniruledness and Abundance In this section we apply the results of the last section to the canonical bundle. The main result is 3.1 Theorem. Let X be a projective manifold that KX is not pseudoeffective. Then X is uniruled. The converse is easy: if X is uniruled and C a general covering rational curve, then KX · C < 0. Proof. Since KX is not pseudo-effective, we find by 2.6 a covering family (Ct ) of curves such that KX · Ct < 0. Hence by Miyaoka-Mori [MM86] we also find a covering family (Cs ) of rational curves (with KX · Cs < 0). Hence X is uniruled. In some sense Theorem 3.1 can be generalized to the cotangent bundle. The starting point is Miyaoka’s 3.2 Theorem. Let Xn be a projective manifold or normal projective variety 1 is with only canonical singularities. Assume that X is not uniruled. Then ΩX 1 generically nef , i.e., ΩX |C is nef, where C is a sufficiently general curve cut out by hyperplane sections of sufficiently large degree. The curves C – often called general complete intersection curves – in the theorem are of the form

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C = m1 H1 ∩ . . . ∩ mn−1 Hn−1 , where Hj are ample divisiors and mj ' 0. A vector bundle E such that E|C is nef for these general complete intersection curves is said to generically nef. For the proof of Theorem 3.2 we refer to [Mi87] or [SB92]. In [CP04], the following generalization was etablished: 3.3 Theorem. Let X be a projective manifold. Suppose that X is not unir1 ⊗m ) : uled. Let Q be a torsion free quotient of some tensor power (ΩX 1 ⊗m (ΩX ) → Q → 0.

Then det Q := Q∗∗ is pseudo-effective. Actually one might expect more to be true: ˜ → X a birational 3.4 Conjecture. Let X be a projective manifold, π : X ˜ If X is not uniruled, π ∗ (Ω 1 ) is generically map from a projective manifold X. X nef. Of course, Theorem 3.3 would be a consequence of Conjecture 3.4. In some 1 is pseudo-effective in some weak sense unless X is sense one can say that ΩX uniruled. The proof of Theorem 3.3 uses 2.6 in essential way. One might suspect a stronger form of pseudo-effectivity for the cotangent 1 is nef on almost all curves of a non-uniruled variety. bundle, namely that ΩX But this is not true: in [BDPP04] it is shown that on a K3-surface X or a Calabi-Yau threefold, there exists a covering family (Ct ) of curves such that 1 |Ct is not nef for general t. Of course, Ct cannot be a general complete ΩX 1 |Ct is not nef, then one can find a torsion free quotient intersection curve. If ΩX Qt living on Ct such that det Qt is not nef. However such a Qt does not extend to X. It would by the theorem of Mehta-Ramanathan [MR82], if Ct would be a general complete intersection curve. Here is another application of generic nefness, reproving and potentially extending an old result of Rosenlicht. 3.5 Corollary. Let X be a projective manifold and v a holomorphic vector field on X. Suppose that v has a zero. Then X is uniruled. ˜ → X be the blow In fact, consider p ∈ X such that v(p) = 0. Let π : X up at p with exceptional divisor E. We obtain a positive number k and an 1 inclusion kE ⊂ TX˜ . This clearly contradicts the generic nefness of ΩX ˜ if X would not be uniruled. If Conjecture 3.4 holds, then one can generalize Corollary 3.5 even to sections 1 ) would be generically in (TX )⊗m . In fact, if X were not uniruled, then π ∗ (ΩX ˜ Let C = nef, hence nef on the general complete intersection curve C˜ ⊂ X. ˜ Then (Ω 1 )⊗m |C is nef. Since C˜ ∩ π −1 (p) = ∅, the point p lies on C. π(C). X Hence (TX )⊗m |C has a non-zero section with a zero. This is in conflict with

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the previous statement and produces a contradiction. The original argument of Rosenlicht of course uses group actions. Theorem 3.1 should also hold in the K¨ ahler case. In dimension > 3 this is completely open; in dimension 3 it follows from classification results [Fu83] unless X is simple non-Kummer. This case – together with a general proof not using classification - was settled recently by Brunella [Br05]: 3.6 Theorem. Let X be a smooth non-algebraic compact K¨ ahler threefold with KX not pseudo-effective. Then X is uniruled. Brunella uses the fact that a non-algebraic K¨ ahler threefold X carries a holomorphic 2-form. This 2−form induces an injective map −KX → TX and therefore defines a foliation F ⊂ TX by curves. If D is the divisorial part of the zeroes of the 2-form, then the canonical bundle KF = F ∗ of the foliation is of the form KF = KX − D, hence not pseudo-effective. Now the claim of the theorem follows from another result of Brunella in the same paper: 3.7 Theorem. Let X be a smooth compact K¨ ahler manifold, F ⊂ TX a foliation by curves. If KF is not pseudo-effective, then F is a foliation by rational curves. Important ingredients here are a uniformization theorem of 1-dimensional foliations and again Theorem 2.6. We will now have a look at rationally connected varieties. It is important to find good characterizations of this notion. By [Pe05] and [KST05], the following holds: 3.8 Theorem. Let X be a projective manifold and C ⊂ X an irreducible curve. If TX |C is ample, then X is rationally connected. Notice that the ampleness of TX |C forces [C] ∈ M E(X). This result is a consequence of a weak version of Mumford’s conjectural characterization of rationally connected manifolds: 3.9 Conjecture. Let X be a projective manifold. If 1 ⊗m ) )) = 0 H 0 ((ΩX

for all positive integers m, then X is rationally connected. Here is an asymptotic answer [Pe05]: 3.10 Theorem. Let X be a projective manifold and L be a big line bundle on X. If 1 ⊗m ) ⊗ L)⊗N ) = 0 (∗) H 0 (X, ((ΩX for all m, N ' 0, then X is rationally connected.

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In the proof Theorem 2.6 again plays an important role. The following foliated version of Theorem 3.8 is very important. 3.11 Theorem. Let X be a projective manifold, F ⊂ TX a foliation and C ⊂ X an irreducible curve. Assume that F is a subbundle near C and that FC is ample. Then all leaves of F meeting C are algebraic and their closures are rationally chain connected. We refer to [BM01],[KST05] and the article [KS05] in this volume. Notice that by applying Theorem 3.11 to F = TX , one gets another proof of Theorem 3.8. We now approach the second main theme of this section, abundance. There are several versions/parts of the Abundance Conjecture: 3.12 Conjecture. (1) Let X be a normal projective (or K¨ ahler) variety with only terminal singularities. If KX is nef , then κ(X) ≥ 0. (2) X as above, KX nef with κ(X) ≥ 0. Then some multiple mKX is spanned. (3) X a projective (K¨ ahler) manifold, KX pseudo-effective , then κ(X) ≥ 0. (4) X a projective (K¨ ahler) manifold, KX pseudo-effective with κ(X) ≥ 0, then κ(X) = ν(X), the numerical Kodaira dimension of X. 3.13 Remark. (3) is birational version of (1) and (4) is a birational version of (2); therefore it suffices in (3) and (4) to consider manifolds instead of singular varieties. (1) is known in the projective case in dimension 3 by [Mi87,88a] (see also [SB92]); in the K¨ ahler threefold case it holds by [DP03]. (2) holds for projective threefolds by [Mi88b], [Ka92], [Ko92]. In the K¨ ahler case in dimension 3 it holds by [Pe01] unless possibly X is simple non-Kummer (with κ(X) = 0). (3) holds for projective threefolds X as a consequence of (1): take a minimal ahler threefolds, the existence of model X of X and apply (1) to X . For K¨ minimal models is unkown, see sect. 4. However by classification it is easy to see that (3) except possibly for simple non-Kummer threefolds. (4) Here we have to explain the meaning of ν(X). If L is a nef line bundle, then ν(L) is the largest integer l such that Ll ≡ 0. In case L is only pseudoeffective, the usual intersection product has to be replaced by the “movable intersection product” < > introduced by Boucksom [Bo02], and then ν(L) is the largest integer l such that < Ll >= 0. Of course, ν(X) := ν(KX ). Also in case KX is nef, abundance can be formulated as κ(X) = ν(X) because of a result of Kawamata [Ka88], saying that if κ(X) = ν(X) and if KX is nef, then some multiple mKX is spanned. In dimension at least 4, abundance, in any form, is widely open. Let us consider a projective manifold Xn with KX pseudo-effective. One should have in mind that pseudo-effectivity is a numerical property, depending only on

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the numerical equivalence class of a line bundle. So it might happen that a pseudo-effective line bundle is numerically equivalent to an effective divisor, but no multiple of L has a section; of course such line bundles exist (the cheapest example is a topologically trivial line bundle which is not torsion). For L = KX however, such a phenomenon does not occur: 3.14 Theorem. Let X be a projective manifold, L a numerically trivial line bundle. Then κ(KX + L) ≤ κ(X). For the proof see [CP05]. It makes essential use of a result of Simpson [Si93]. Namely, consider S = {G ∈ Pic0 (X)|h0 (KX + G) ≥ r}. Then S is a union of translates of subtori (Green-Lazarsfeld) by torsion elements. Let us come back to the projective manifold X with KX pseudo-effective. Of course, we may assume that KX is on the boundary of E; otherwise KX is big and we are done. Thus we find α ∈ SM E(X), necessarily also on the boundary, such that KX · α = 0. If α = [Ct ], with (Ct ) a covering family of curves, then we are in a good situation. In fact, we consider the quotient f : X Y defined by the family (Ct ) and analyze this rational map. To be more precise, we say that the family (Ct ) is connecting , if two sufficiently general points x, y ∈ X can be joined by a chain of Ct ’s. We also say that X is (Ct )−connected. Campana’s reduction theory (see e.g. [Ca04]) gives: 3.15 Proposition. Let X be a projective manifold, L a line bundle on X and (Ct ) a covering family with L · Ct = 0. Then there exists an almost holomorphic dominant meromorphic map f : X Y with connected fibers such that the general compact fiber is (Ct )−connected. We say that f is the partial nef reduction of L with respect to (Ct ). Recall that a meromorphic map f : X Y is almost holomorphic, if there exists an open non-empty set U ⊂ X such that f |U is holomorphic and ˜ → X is a modification such that the induced proper. In other words, if π : X ˜ ˜ of π does not map X → Y is holomorphic, then the exceptional locus E ⊂ X project onto Y. Hence f has nice compact fibers. There is also the notion of a “full” nef reduction:

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3.16 Theorem. Let L be a nef line bundle on a normal projective variety X. Then there exists an almost holomorphic dominant meromorphic map f : X Y with connected fibers such • L is numerically trivial on all compact fibers of f of dimension dim X − dim Y ; • for every general x ∈ X and every irreducible curve C through x with dim f (C) > 0, one has L · C > 0. This theorem was first postulated by Tsuji, for a proof see [workshop]. For further results in the pseudo-effective case we refer to [Ec05]. We now apply the reduction theory to abundance for 4−folds and refer again to [BDPP04]. 3.17 Theorem. Let X be a smooth projective 4-fold with KX pseudoeffective. Let (Ct ) a covering family of curves such that KX · Ct = 0. Then κ(X) ≥ 0. The proof proceeds in several steps. Let f : X Y be the almost holomorphic partial nef reduction with respect to (Ct ) (we take Y smooth). Of course, it may happen that dim Y = 0, a case which needs special treatment. First we assume that dim Y > 0. Since the general fiber F of f cannot be uniruled, we have κ(F ) ≥ 0. If now κ(Y ) ≥ 0, we are going to use the so-called conjecture Cn,m which says that κ(X) ≥ κ(F ) + κ(Y ) for a surjective map f ; Xn → Ym with connected fibers. Now in our cases Cn,m is known to be true (see [Mo87] for further references), hence we may conclude κ(X) ≥ 0 as long as κ(Y ) ≥ 0. By taking into account the Albanese map, we may assume that q(Y ) = 0. Therefore we are reduced to Y = P1 resp. Y rational in dimension ≤ 2. If dim Y = 3, then Y is at least uniruled. By considering the rational quotient of Y , which must P1 or a rational surface, if it is not trivial, and by applying [CT86] or [GHS03], we conclude that Y is rationally connected. If now dim Y = 1, 2, 3, a direct analysis of KX leads to the conclusion that κ(X) ≤ 0. Let us demonstrate this in case dim Y = 3. Here the general fiber F is an elliptic curve. We choose a holomorphic birational model f : X → Y (with X and Y smooth), such that • f is smooth over Y0 and Y \ Y0 is a divisor with simple normal crossings only; • the j−function extends to a holomorphic map J : Y → P1 . By the first property, f∗ (KX ) is locally free [Ko86], and we obtain the wellknown formula of Q−divisors (∗)

KX = f ∗ (KY + ∆) + E

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Here E is an effective divisor such that f∗ (OX (E)) = OY . Moreover ∆ = ∆1 + ∆2 with ∆1 =

1 (1 − )Fi + ak mi

and

1 ∗ J (O(1)). 12 Here Fi are the components over which we have multiple fibers and Dk are 1 N the other divisor components over which there singular fibers. The ak ∈ 12 according to Kodaira’s list. Then by a general choice of the divisor ∆2 , the pair (Y, ∆1 + ∆2 ) is klt. Now KY + ∆ is pseudo-effective. In fact, by 2.6 it suffices to show that KY + ∆ · Ct ≥ 0 for every covering family of curves. But this is checked easily. Hence the log Minimal Model Program [Ko92] in dimension 3 implies that KY + ∆ is effective. Hence κ(X) ≥ 0 by (*). ∆2

The final case is dim Y = 0. This is settled in any dimension by 3.18 Theorem. Let X be a projective manifold with KX pseudo-effective, (Ct ) a connecting family of curves. Assume KX · Ct = 0. Then κ(X) = 0. Actually one has 3.19 Theorem. Let X be a projective manifold, L a pseudo-effective line bundle on X. Let (Ct ) be a connecting family of curves such that L · Ct = 0. Then there exists a line bundle L ≡ L such that κ(L ) = 0. Theorem 3.18 is a consequence of Theorem 3.19 by virtue of 3.14. The proof of 3.19 is rather technical; the essential case deals with an (n−1)−dimensional subfamily (Ct )t∈T and analyzes carefully the graph of this family. As a prototype, assume that dim X = 2 and let (Ct )t∈T be a 1-dimensional connecting family. We may assume T smooth; let p : C → X be the graph with projection q : C → T. Let us also assume that p is not finite, i.e. there is a point x0 ∈ X such that all Ct contain x0 . Then p∗ (L) · q −1 (t) = 0 for all t (and numerically trivial on the general q −1 (t)). Moreover there is a multi-section C0 of q such that p∗ (L)|C0 ≡ 0. Then it is not difficult to conclude p∗ (L) is numerically equivalent to an effective divisor by analyzing q ∗ q∗ (p∗ (L) ⊗ L ) with L a suitable numerically trivial line bundle (on C a priori). Of course also the case that p is finite has to be considered. For any details we refer to [BDPP04]. In order to complete the Abundance Conjecture 3.6(3) in dimension 4, it remains to consider the case that KX · Ct > 0 for all covering families. Here we expect KX to be big, of course. A priori, it seems that this case should be easier, since we have more positivity. However we have no tool to deal with this case. The difficulties become clearer when we rephrase the problem in

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the following form. Since KX is assumed pseudo-effective, there exists a class α ∈ ∂M E(X) such that KX · α = 0. The problem is to prove that a suitable α can be represented (up to a scalar) by a movable curve. Of course, this is not true for a general pseudo-effective line bundle but must be a special property of the canonical bundle. Even if KX is nef on the 4−fold X, this is wide open. The reason why in dimension 3 things are tractable is the following. Suppose X is a threefold with KX pseudo-effective. Let X be a minimal model, so KX is nef. One may assume q(X ) = 0, otherwise C3,2 and C3,1 yield the claim. Thus we may assume that χ(OX ) ≥ 1; here of course dim X = 3 is essential. Now let π : Y → X be a desingularization. Since X is not uniruled, 1 is generically nef and as an important consequence ΩX c2 (Y ) · π ∗ (D) ≥ 0 for all nef divisors D on X . Then Riemann-Roch gives χ(Y, π ∗ (nKX )) ≥ χ(OY ) = χ(OX ) ≥ 1 fo all sufficiently divisible positive integers n. Here again the dimension assumption is crucial. It follows h2 (nKX ) = 0 and this leads finally to a contradiction by a careful analysis of KX , distinguishing the various possible numerical dimensions of KX . There is no argument at the moment to prove directly – without going to a minimal model – that the original threefold X has non-negative Kodaira dimension. We finish this section by proving that the “Rational Connectedness Conjecture” 3.9 is a consequence of a part of Abundance, 3.12(3). 3.20 Theorem. Suppose that Conjecture 3.12(3) holds, i.e., given any projective manifold X with KX pseudo-effective, then κ(X) ≥ 0. Then Conjecture 3.9 holds, i.e. given a projective manifold X with 1 ⊗m ) )=0 H 0 (X, (ΩX

(∗)

for all positive integers m, then X is rationally connected. Proof. First observe that (*) implies of course κ(X) = −∞. Then by 3.12(3), KX is not pseudo-effective. Hence X is uniruled by (3.1). Let f : X S be the rational quotient. We may assume S smooth and f holomorphic; S is not uniruled [GHS03]. If X were not rationally connected, then dim S > 0. Since clearly H 0 (S, (ΩS1 )⊗m ) = 0 for all m ≥ 1, S is rationally connected (uniruledness suffices) by induction on the dimension. This is a contradiction.

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We see that in order to establish 3.9, it suffices to show that KX pseudoeffective is incompatible with the vanishing 1 ⊗m H 0 (X, (ΩX ) )=0

for all m.

4 Mori Theory on K¨ ahler Manifolds Mori theory should work on compact K¨ ahler manifolds in the same way it does – or should do – on projective manifolds/varieties. Of course, terminal (canonical) singularities have to be allowed also here. We recall the notion of a normal K¨ ahler variety. we require the existence of a K¨ ahler form ω on the regular part on X with the following additional property. Every singular point x of X admits an open neighborhood U such that U can be embedded as closed subvariety of an open set V in some CN and such that ω|U is the restriction of a K¨ ahler form on all of V. 4.1 Problem. Let X be a compact K¨ ahler manifold or a Q−factorial K¨ ahler variety with only terminal singularities. Then X has a bimeromorphic model ahler variety with only terminal singularX , which is again a Q−factorial K¨ ities and either X admits a Fano fibration with relative Picard number 1, or KX is nef. More specifically, X can be found be a sequence of contractions and flips. To be more precise, a contraction, or extremal contraction, is a map φ : X → Y with the following properties: (1) −KX is φ−ample, (2) ρ(X/Y ) = 1, (3) Y is again K¨ ahler. If additionally dim Y < dim X, then the general fiber of φ is Fano and we say that φ is a Fano fibration. In the projective setting φ is the contraction of an extremal ray of N E(X). However a cone theorem is unknown in the K¨ ahler case and all projective methods used in Mori theory. In the following we will completely restrict ourselves to dimension 3. Suppose that KX is not nef. A priori this does not mean that there is a curve C with KX · C < 0; it just says that KX is not in the closure of the K¨ ahler cone. ahler. If KX is not nef , then there exists 4.2 Theorem. Let X3 be smooth K¨ an irreducible curve such that KX · C < 0. The curve C can even be taken rational.

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If X is not both simple non-Kummer , the proof is given in [Pe01]; it uses heavily structure results on non-algebraic K¨ ahler threefolds. Moreover the structure of surfaces plays an important role. E.g., when KX = OX (D) with an effective divisor D, then the non-nefness of KX implies that D|D is not nef and by adjunction this says that KD is not nef. If X is not pseudo-effective, we know by Brunella’s result 3.6 that X is uniruled. In the remaining case where KX is pseudo-effective (and X simple non-Kummer, but this is not really needed), one can apply Theorem 4.3 to obtain a curve C with KX · C < 0 and similar methods as in [Pe01] should also give a rational curve. Theorem 4.2 should also hold in the singular case with similar proofs, but this has not yet been carried out. In the case of higher dimensions, definitely completely new ideas will be needed to construct KX −negative curves. If KX is pseudo-effective and even X singular with isolated singularities (but still of dimension 3), one can directly establish a curve C with KX · C < 0, see [DPS01,5.4]. Namely, consider a singular metric with positive curvature current on X. If φ is a local weight function for the metric with Lelong number ν(x, φ), then one defines for c > 0 the globally defined analytic sets Ec = {x ∈ X|ν(x, φ) ≥ c}. By [De92,6.4], dim Ec ≥ 1 for a sufficiently small c, otherwise KX would be nef. Using a result of Paun and Siu’s decomposition theorem of currents, one deduces the existence of a curve or a surface S ⊂ Ec such that KX |S is not nef; even more, if dim S = 2, then for some positive a, the line bundle KX − aS|S is still not nef. From that the existence of a curve C with KX · C < 0 is easily derived. Hence: 4.3 Theorem. Let X be a normal compact K¨ ahler space with isolated singularities. Suppose that X is Q−factorial and that KX is pseudo-effective. If KX is not nef, then there exists a curve C with KX · C < 0. In general one can say that L is nef if and only if L|Ec is nef for all c > 0. Now suppose that X is a non-algebraic smooth K¨ ahler threefold carrying a rational curve C with KX ·C < 0. Then by Ein’s deformation lemma [Ko95], C moves in a positive-dimensional family – this does not use the rationality of C, but smoothness – or Gorenstein – is essential. Now a special member of this family could decompose into several components. One of these components must again be KX −negative, and one can start the game again. Using the K¨ ahler form one sees easily that after finitely many steps, we obtain a nonsplitting family of rational curves. This is the starting point of [CP94], and [Ko91] in the algebraic case. If the family covers X, then this family forms a P1 −bundle over a non-algebraic surface and a contraction is found. So we may assume that the family fills up a divisor D. This divisor can be classified including its normal bundle: we obtain

328

• • • • •


(P2 , O(−1)), (P2 , O(−2)), (Q2 , O(−1)), where Q2 = P1 × P1 or a 2-dimensional quadric cone, D is a ruled surface with D · l = −1 for the ruling line l, or D is a non-normal surface; there exists a conic bundle f : X → S such that D consists of reducible conics.

In the first three cases D can be contracted to a point, in the fourth D can be contracted along the ruling. We obtain a bimeromorphic contraction φ : X → Y, and the list is exactly the same as in the projective setting. However there is a important point: Y might not be K¨ ahler again. This cannot happen when dim φ(D) = 0 as shown in [Pe98], however it is very well possible when φ(D) = B, the smooth curve blown up by φ, even when X is projective. In the algebraic setting, in order to force X to be projective, the ruling line l ⊂ D must be ahler case we can at least prove geometrically extremal in N E(X). In the K¨ [Pe98]: Y is K¨ ahler if and only if [l] is geometrically extremal in N . If [l] is not geometrically extremal, it could e.g. happen that B is homologous to 0. In that case one needs to find a better non-split family, and it is completely unclear how to do this. The study of the singular case was begun by Kronenthaler [Kr05]. He considered a normal non-projective K¨ ahler threefold which is Q−factorial and has at most terminal singularities. Let (Ct )t∈T be a non-split family of rational curves with −KX · Ct > 0 (dim T ≥ 1). If the Ct cover X, then again X is a P1 −bundle and therefore smooth; so we may assume that Ct S= t∈T

is an irreducible surface S. Then in [Kr05] the following is proved. 4.4 Theorem. (1) Suppose that dim T = 1 and all Ct pass through a fixed point or that dim T ≥ 2. Then there exists a normal compact variety Y , Q−factorial with at most terminal singularities and a bimeromorphic map φ : X → Y contracting S to a point and nothing else. (2) Suppose that X is Gorenstein, that dim T = 1 and T is maximal, i.e. a component in the cycle space. Suppose furthermore that the Ct do not have a point in common. Then we are in one of the following two cases. (a) S · Ct < 0, and there exists φ as in (1), contracting S to a curve; S is generically a P1 bundle and S · Ct = −1; (b) S · Ct ≥ 0, X is uniruled, and X admits some divisorial contraction or a conic bundle structure and the Ct are components of conics.

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In the non-Gorenstein there is a new difficulty. Namely, suppose we have a rational curve C with KX · C < 0. Then C might be rigid. In that case one might hope that C could be flipped. Also it is much more difficult to handle a surface filled up by a non-splitting family of rational curves, since S is no longer Cartier on X. In contrast, if X is Gorenstein, then automatically every Weil divisor on X is Cartier. Once the existence of contractions is settled (including the K¨ ahler property of the image), the whole Mori program in dimension 3 runs through. In particular, there is no problem with the existence of flips, which works in the analytic setting; actually in Mori’s extistence theorem, open neighborhoods in the usual topology are considered.

5 Approximation of K¨ ahler Manifolds In [Ko63] Kodaira showed that every compact K¨ ahler surface is the limit of projective surfaces and asked whether this is also true in higher dimensions. To be more precise we formulate the following 5.1 Problem. Let X be a compact K¨ ahler manifold. Does there exist a complex manifold X with a proper surjective holomorphic submersion π : X → ∆ to the unit disc in some CN such that • X X0 ; • there exists a sequence (tk ) converging to 0 such that Xtk is projective for all k? Here of course Xt = π −1 (t). We say for short that X can be algebraically approximated. In [Vo04a,b] C.Voisin showed that the answer to this problem is in general “no”. She proved actually much more: 5.2 Theorem. In any dimension n ≥ 4 there exist compact K¨ ahler manifolds which do not have the homotopy type of a projective manifold. Even more, in any even dimension n ≥ 8 there exist compact K¨ ahler manifolds with the property that no smooth bimeromorphic model has the homotopy type of a projective manifold. This seems to suggest that there are compact Kähler manifolds which are – even bimeromorphically – far away from projective manifolds. However Mori theory suggests that – instead of merely smooth bimeromorphic models – one should rather take models with terminal singularities into account. Therefore we reformulate Problem 5.1:

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5.3 Problem. Let X be a compact K¨ ahler manifold. Does there exist a normal complex space X with proper surjective holomorphic π : X → ∆ to some disc ∆ such that all Xt = π −1 (t) are Q−factorial with only terminal singularities and such that • X is bimeromorphic to X0 ; • there exists a sequence (tk ) converging to 0 such that Xtk is projective for all k? Instead of terminal singularities one might ask for canonical singularities, only. Furthermore one could ask whether the minimal models in the sense of Mori theory can be approximated algebraically, or, under which circumstances Fano fibrations can be approximated. In other words, if Problem 5.3 has a positive answer, which are the models that can be approximated? We are now discussing the approach of [DEP05] in this direction. Consider a 3−dimensional torus A with Picard number ρ(A) ≥ 3 and three linearly independent line bundles Li over A. Let U be an open neighborhood of [A] in the universal deformation space of A, which has dimension 9. Using the description of line bundles on tori, one sees that every Li defines a subspace Vi ⊂ U of codimension 3 such that Li – considered as topological line bundle on A – is still holomorphic on A if and only if [A ] ∈ Vi . We consider Y = P(OA ⊕ L1 ) ×A P(OA ⊕ L2 ) ×A P(OA ⊕ L3 ). Observe that Y is P31 -bundle over A with projection π : Y → A. In each subspace P(OA ⊕ Li ) there is a section Zi at infinity given by the direct summand OA . This gives a section Z of π by choosing over every a ∈ A the point (x1 , x2 , x3 ), where {xi } = Zi ∩ π −1 (a). 5.4 Proposition. Suppose that [A] is an isolated point of V1 ∩ V2 ∩ V3 . Then the blow up σ : X → Y of Z ⊂ Y is rigid, even infinitesimally. “Unfortunately” one can prove that X must be projective, i.e., the condition that [A] is an isolated point of V1 ∩ V2 ∩ V3 cannot be fulfilled in the non-projective setting. More is true [DEP05]: 5.5 Theorem. The 6−fold Y can always be algebraically approximated. Finally we want to show how a positive solution of Problem 5.3 could be used to carry over theorems from the projective to the K¨ ahler case. We are going to use 5.6 Proposition. Let π : X → ∆ be a family of normal compact K¨ ahler varieties with at most terminal or canonical singularities. Let α ∈ HR1,1 (X ) such that αt := α|Xt is big for a sequence (tk ) converging to 0. Then α0 is big.

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Proof. This is a consequence of results of Boucksom [Bo02]. The results there are proved in the smooth case, but they carry over to our situation. Let v(α) denote the “volume” of αt . Since αtk is big, v(αtk ) > 0 by [Bo02,3.1.31]. Hence v(α0 ) > 0 by ibid., 3.1.29. Hence α0 is big, again by 3.1.31. In case α is c1 (L) for a line bundle L, the bigness of L0 of course follows from upper semi-continuity of h0 (Lt ). 5.7 Theorem. Let X be a compact K¨ ahler variety with at most canonical singularities. Suppose that X can be approximated algebraically (by varieties with only canonical singularities). If KX is not pseudo-effective, then X is uniruled. Proof. Let π : X → ∆ be a family of normal compact K¨ ahler varieties with at most terminal or canonical singularities such that X0 X and such that there is a sequence (tk ) converging to 0 such that Xtk is projective for all k. Since KX is not pseudo-effective, Proposition 5.6 tells us that KXtk is not pseudo-effective for infinitely many k. Hence Xtk is uniruled for those k and therefore X0 is uniruled.

6 Cotangent Bundle and Universal Cover In Sect. 2 we saw that the cotangent bundle of a non-uniruled projective manifold X has the following positivity property: given a positive integer m and a torsion free quotient 1 ⊗m (ΩX ) → Q → 0,

then det Q is pseudo-effective. This has the following consequence [CP05]. p 6.1 Theorem. Let Xn be a projective manifold. Suppose that some ΩX contains a subsheaf F such that κ(det F ) = n. Then κ(X) = n.

In fact, X cannot be uniruled, because then we would find rational curves C with TX |C. On the other hand det F |C will be ample. This yields a contradiction ,hence X is uniruled. Let Q = Ω p /F ; of course we may assume F saturated, so that Q is torsion free. Then we conclude that det Q is pseudoeffective. Since mKX = det F + det Q for a suitable positive integer m, it follows that KX is big as a sum of a big and of a pseudo-effective divisor. To put things into perspective, we recall Campana’s notion of a “refined Kodaira dimension” [Ca95]. This notion measures the geometric positivity of the cotangent bundle, and not only that one of the canonical bundle.

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6.2 Definition. Let X be a compact (or projective) manifold. Then κ+ (X) p for 1 ≤ p ≤ dim X is a is the maximal number κ(det F ), where F ⊂ ΩX (saturated) coherent subsheaf. Obviously we have κ+ (X) ≥ κ(X) for any X. Assuming the standard conjectures of the Minimal Model Program, one can easily describe κ+ (X) as follows (see [Ca95] for details). 6.3 Conjecture. Let X be a projective manifold. If X is not uniruled (or if κ(X) ≥ 0), then κ+ (X) = κ(X). Theorem 6.1 confirms Conjecture 6.3 in case κ+ (X) = dim X. When X is uniruled, one has κ+ (X) = κ+ (R(X)), where R(X) is the rational quotient of X. This rational quotient is not uniruled, and so should be either one point or have κ+ (R(X)) = κ(R(X)) ≥ 0. Thus if X is uniruled, one has κ(X) = −∞ but κ+ (X) ≥ 0, unless R(X) is one point, which means that X is rationally connected. In this latter case κ+ (X) = −∞. Conversely: 6.4 Conjecture. If κ+ (X) = −∞, then X is rationally connected. p ) = 0 for p > 0. Notice that χ(OX ) = 1 if κ+ (X) = −∞, because h0 (X, ΩX Moreover it is shown in [Ca95] that X is simply connected if κ+ (X) = −∞, giving support to the above conjecture.

This conjecture can be seen as geometric version of the “stability of the cotangent bundle” of X when X is not uniruled. In order to obtain more positive results concerning Conjecture 6.3, we introduce 6.5 Conjecture. Let X be a projective manifold. Assume that N KX = A+B for some positive integer N with A effective and B pseudo-effective. Then κ(X) ≥ κ(A). It is easily seen that Conjecture 6.4 implies Conjecture 6.3 for non-uniruled varieties. Now one proceeds as follows. Suppose that κ(X) ≥ 0 and consider the Iitaka fibration with general fiber G of Kodaira dimension κ(G) = 0. Put d = n − κ(X). If Conjecture 6.4 holds for all manifolds G with dim G = d and κ(G) = 0, then (6.4) and a fortiori (6.3) holds for X. So we are reduced to varieties G with κ(G) = 0. Here we show that if G has a good minimal model, i.e. a model G with KG torsion, then (6.4) holds for G. Since good minimal models exist in dimension up to 3, we obtain 6.6 Theorem. Let Xn be a projective manifold, κ(X) ≥ 0. If κ+ (X) ≥ n−3, then κ+ (X) = κ(X). In particular Conjecture 6.3 holds in dimension 4.

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One can also show that (6.4) holds if κ(A) = n − 1, and possibly also for n − 2 and n − 3. An interesting special case of Conjecture 6.4 is when B is numerically trivial. This case is also settled in [CP04], as already said in sect. 2 (see Theorem 3.8). One of the important properties of κ+ is the relation to the geometry of the universal cover [Ca95], [Ko93]. To explain that, let ˜ →X π:X denote the universal cover of the compact Kähler manifold X. By identifying ˜ which can be joined by a compact connected analytic set, one points in X obtains an almost holomorphic meromorphic map ˜ % Γ (X). ˜ X ˜ is holomorphically convex (which is expected to be always true by the soIf X called Shafarevitch conjecture), then this map is holomorphic and is just the usual (holomorphic) Remmert reduction. In any case it induces the so-called Shafarevich map ˜ ˜ γX : X % Γ (X) = Γ (X)/π 1 (X). 6.7 Definition. γd(X) = dim Γ (X) is the Γ −dimension of X. Notice that • γd(X) = 0 iff π1 (X) is finite, and that ˜ there is no positive di• γd(X) = dim X iff through the general point of X ˜ mensional compact subvariety, i.e. X geometrically seems as a modification of a Stein space. The following result [Ca95,(4.1)] gives a relation between κ+ (X) and γd(X). 6.8 Theorem. Let X be a compact K¨ ahler manifold. If χ(X, OX ) = 0, then either (1) κ+ (X) ≥ γd(X), or (2) κ+ (X) = −∞, and so X is simply connected. By (6.6) we obtain 6.9 Corollary. Let Xn be a projective manifold, κ(X) ≥ 0. Suppose γd(X) ≥ n − 3 and χ(OX ) = 0. Then κ(X) ≥ γd(X) ≥ n − 3. In particular, if n = 4, κ(X) ≥ 0, π1 (X) is infinite and χ(OX ) = 0, then κ(X) ≥ 1. In other words, if X is a projective 4-fold with κ(X) = 0 and π1 (X) is not finite, then χ(OX ) = 0; so there is either a holomorphic 1-form, or a holomorphic 3-form.

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From Theorem 6.8 we also deduce 6.10 Corollary. Let X be a projective manifold and suppose that its universal cover is not covered by its positive-dimensional compact subvarieties. Then either X is of general type, or χ(OX ) = 0. This corollary also holds for normal projective varieties with at most rational singularities. 6.11 Corollary. Let Xn be a projective manifold (or a normal projective variety with at most terminal singularities) whose universal cover is Stein (or has no positive-dimensional subvariety). Then either KX is ample or χ(OX ) = n = 0. 0, KX is nef and KX This is immediate from (6.10) by observing that X does not have any rational curve, so that KX must be nef by Mori theory. Moreover if KX is big, then KX is ample by Kawamata [Ka92]. Corollary 6.11 leads us to ask for the structure of projective manifolds Xn n whose universal cover is Stein and with KX = 0. ˜ 6.12 Conjecture. Let Xn be a projective manifold whose universal cover X n = 0. Then up to finite étale cover of X, the manifold is Stein. Assume KX X has a torus submersion over a projective manifold Y with KY ample and universal cover again Stein. A birational version of (6.12) has been proved by [Ko93,5.8]. In our situation we have 6.13 Proposition. Conjecture 6.12 holds if κ(X) ≥ n − 3. For the proof we again refer to [CP04].

7 Threefolds with Nef and Non-big Anticanonical Bundles In this section we describe, following [BP04], the structure of smooth projective threefolds X whose anticanonical bundle −KX is nef but not big, i.e., (−KX )3 = 0. We shall also assume that KX is not numerically trivial, which comes down to say that κ(X) = −∞, or X is uniruled. In some sense X has some positive and some flat directions and the problem is to describe and separate them. We start by describing some general results which hold in all dimensions. Let us first have a look to the special case that −KX is hermitian semi-positive [DPS96]. 7.1 Theorem. Let X be a compact K¨ ahler manifold with −KX hermitian semi-positive. Then

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˜ admits a holomorphic and isometric splitting (1) The universal covering X ˜ Cq × X Xi with Xi either a Calabi-Yau manifold or a symplectic manifold or a com1 ⊗m ) ) = 0 for all m > 0. pact manifold with H 0 (Xi , (ΩX i ˆ ˆ has (2) There exists a finite étale cover X → X such that the Albanese Aˆ of X ˆ ˆ dimension q and the Albanese map α ˆ : X → A is a locally trivial hlomorphic fiber bundle whose fibers are products of simply connected manifolds Xi as in (1). ˆ Z2q , in particular π1 (X) is almost abelian. (3) π1 (X) The proof uses methods from differential geometry (holonomy etc.) and does not carry over to the nef case. Recall that the compact K¨ ahler manifolds 1 ⊗m ) ) = 0 are expected to be rationally connected. In the Xi with H 0 (Xi , (ΩX i nef case we have the following result of Zhang [Zh96]: 7.2 Theorem. Let X be a projective manifold with −KX nef. Then the Albanese map X → Alb(X) is surjective. More generally, if f : X → S is surjective map to a projective manifold S, then −KS is generically nef and κ(S) ≤ 0. The proof uses methods from characteristic p, namely Miyoaka’s result that the relative anticanonical bundle of a fibration X → Y with X and Y smooth and dim Y > 0 can never be ample. Theorem 7.2 should actually be true in the K¨ ahler case, too, and much more should; as suggested by Theorem 7.1: 7.3 Conjecture. Let X be a compact K¨ ahler manifold with −KX nef. Then the Albanese map is a surjective submersion. For projective threefolds this is proved in [PS98] using Mori theory; for non-algebraic K¨ ahler threefolds in [DP01] and for 4−folds in [CPZ03]. See also [Pa01]. It is also expected that the fundamental group is almost abelian for first results in this direction see [DPS93]. Paun proved this in the projective setting [Pa97]: 7.4 Theorem. Let X be a compact K¨ ahler manifold with −KX nef.Then π1 (X) has polynomial growth. If X is projective, then π1 (X) is almost abelian, ˜ such that the Albanese map gives an and there exists a finite étale cover X ˜ ˜ isomorphism π1 (X) → π1 (Alb(X)). For non-algebraic K¨ ahler threefolds 7.4 holds by [DPS01]. We now describe the structure of smooth projective threefolds X with −KX nef in detail. First we introduce a new invariant. Given a nef line bundle L on a projective manifold X, we define the nef dimension n(L) by n(L) = dim B,

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where B is the base of the nef reduction of L (3.16). Since B is well-defined up to a birational equivalence, n(L) is indeed an invariant of L. In particular we will be interested in n(−KX ). In general it is likely that a nef reduction cannot be chosen to be holomorphic, although no explicite example is known. For the anticanonical bundle things are much better: 7.5 Theorem. Let X be a projective threefold with −KX nef. Then there exists a holomorphic map f : X → B to a normal projective variety B such that (1) −KX is numerically trivial on all fibers of f (2) for x ∈ X general and every irreducible curve C passing through x such that dim f (C) > 0, we have −KX · C > 0. In case X is rationally connected and n(−KX ) = 1, 2 then even some multiple −mKX is spanned by global sections, so that we can take f to be (the Stein factorisation of ) the map defined by the sections of −mKX . The proof is a little bit involved; we refer to [BP04]. The starting point is the investigation of the (completed) family of (general) fibers of f. We now have three different methods to our disposal: • the Albanese, which allows us to reduce ourselves to simply connected varieties; • the nef reduction , which fibers out the flat directions; • Mori theory, which takes care of the positive directions. One might say that Mori theory and the nef reduction are in a sense transverse. Notice also that we will only be interested into a classification up to finite étale cover and that a smooth projective threefold with no holomorphic 1−forms even after finite étale covers must be simply connected by Theorem 7.3. The main classification results are the following. 7.6 Theorem. Let X be a smooth projective threefold with −KX nef, KX ≡ 0. Then one of the following statements holds. (1) X is a product after finite étale cover. (2) q(X) > 0 after some finite étale cover and n(−KX ) = 3. (3) X is rationally connected. In particular π1 (X) is almost abelian. Recall that n(−KX ) = 3 means that −KX · C > 0 for all movable curves. These threefolds are somehow unpleasant and possibly cannot be classified easily, however one can prove that they still form a bounded family. Hence one can state: 7.7 Theorem. Smooth projective threefolds with −KX nef and which are not rationally connected form a bounded family.

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An interesting special feature is whether −mKX is spanned for some m, at least after finite étale cover. Here we have 7.8 Theorem. Let X be a smooth projective threefold, −KX nef. Then −mKX˜ is spanned for some positive m and some finite étale cover, or one is one of the following cases. ˜ = C × S with C elliptic and S a surface with −KS (1) n(−KX ) = 2 and X nef but not multiple spanned. ˜ is a special P1 −bundle over an abelian surface. (2) n(−KX ) = 3 and X (3) n(−KX ) = 3 and X is rationally connected. We are now reduced to the case that X is rationally connected. These threefolds are the generalization of P2 blown up in at most 9 points in general position, so are in some sense the most interesting ones, especially when n(−KX ) = 3. In the cases n(−KX ) = 1, 2 we study the nef reduction in connection with a Mori contraction ϕ : X → Y . When dim Y ≤ 2, we get very precise structure theorems. If φ is birational, we show that φ must be the blowup of a smooth curve in the smooth threefold Y – up to some rather precise cases – and we would like to argue by induction. However one unpleasant thing happens: −KY might no be nef anymore. This happens exactly when C = P1 with normal bundle NC = O(−1) ⊕ O(−2) resp. O(−2) ⊕ O(−2). The first type can be handled by some birational transformations but the second is really unpleasant. The way out is probably to follow [PS98] and to enlarge the category, namely to consider threefolds such that −KX is nef up to finitely many rational curves and prove everything for these threefolds. This has not yet been done, so that we have to rule the possibility of having (−2, −2)-curves. Here is a boundedness result: 7.9 Theorem. Smooth rationally connected threefolds subject to the following conditions are bounded modulo boundedness of smooth threefolds Y with −KY nef and n(−KY ) = 3 : • −KX is nef • X does not admit a contraction of type (−2, −2) • 1 ≤ n(−KX ) ≤ 2 It remains to consider the case n(−KX ) = 3. The cheapest example is P1 × S where S is P2 blown in 9 points in such a position that S does not admit an elliptic fibration. 7.10 Proposition. Let X be a smooth rationally connected projective three2 = 0. The anticanonical fold with −KX nef and n(−KX ) = 3. Then KX bundle induces a holomorphic map f : X → P1 . The general fiber F of f has nef anticanonical bundle with KF2 = 0 and n(−KF ) = 2.

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Thus the general fiber F is either P2 blown up in 9 points without elliptic fibration; the anticanonical system contains either of one elliptic curve or of rational curves. Or F is ruled over an elliptic with n(−KF ). The first part of the first case is further studied in [BP04], we refer to that paper for details. One method used there is to run the Minimal Model Program relative to f. The other cases still need to be studied. At the end of the theory we expect in particular a boundedness theorem for smooth projective threefolds X with −KX nef, not numerically trivial and 3 = 0. Also threefolds with KX ≡ 0 are expected to be bounded, but this KX is a completely area.

8 Almost Fano Threefolds Recall that a projective manifold X is called Fano if the anticanonical divisor −KX is ample, hence some multiple of −KX defines an embedding into some projective space. For any fixed dimension the number of deformation families of Fanos is finite ([Ma70], [Na91], [KMM92]). This makes it in principle possible to write down complete lists. In the singular case much less is known. If we allow canonical Gorenstein singularities, then in the case of surfaces (allowing DuVal singularities) only certain degenerations of smooth del Pezzo surfaces come into the picture. In the threedimensional case the list is again finite ([Bo01]) but unknown. Easy examples are the following: 8.1 Example. (1) Let S be a del Pezzo surface with at most DuVal singularities. Then X = S × P1 is a Fano threefold with at most canonical singularities. A particular example would be the following. Take the blowup of P2 in 3 points lying on a line l. Then the strict transform of l is a (−2)–curve, its contraction yields a del Pezzo surface with one A1 point. (2) Let X = Q ⊂ P4 be a (double-) cone over a smooth conic in P2 or a smooth quadric in P3 . Then X is a Gorenstein Fano threefold with ρ(X) = 1. If the vertex is a point, then the singularity of X is terminal; if it is a line, then X has canonical singularities. The finer classification of canonical and terminal singularities is mainly due to M. Reid (see [Re87]). We restrict again to the Gorenstein case. If X has a canonical Gorenstein singularity in a point p, then a general hyperplane section Hp through p either has a DuVal singularity or an elliptic singularity. In the former case p is called a compound DuVal singularity (cDV for short). Here X is locally given by f (x, y, z) + tg(x, y, z, t) = 0

in C4 ,

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where f is a DuVal singularity. The precise definition of elliptic surface singularities is not important here. Instead let us consider an example: 8.2 Example. Let S ⊂ PN be a smooth surface. The restriction of OPN (1) to S defines a polarization OS (1). Let Sˆ be the cone over S in PN +1 with vertex v. Blowing up v we get ˆ = P(OS ⊕ OS (1)) Blv (S)

|O(1)|

/ Sˆ .

The exceptional divisor E S of the blowup is in |O(1) ⊗ π ∗ OS (−1)|, where π denotes the projection onto S and O(1) the tautological line bundle. Considering the discrepancy of the blowup at E as defined in (1.0.1) we find that v is a terminal singularity if and only if −KS = OS (a) for some 1 < a ∈ N; the singularity is canonical iff a = 1. But Sˆ is usually not factorial: ˆ lying over disjoint curves in S. Their assume we find two divisors in Blv (S) ˆ images in S will then be (Weil)–divisors intersecting in a single point. ˆ be a very general hyperplane section of Sˆ passing through v and Let H ˆ Restricting to E ˆ in Blv (S). S we denote by H the strict transform of H find H ∩ E ∈ |OS (1)|. If v is a terminal point, then S = P1 × P1 or P2 and H ∩ E is a rational curve (the diagonal (1, 1) or a line or a conic in P2 ). If v is canonical point, which is not terminal, then OS (1) = −KS and H ∩ E is an elliptic curve. In fact isolated cDV points are exactly the terminal points. If X has canonical singularities along a curve, then analytically near any general point on that curve X (DuVal) × C. (8.2.1) As in the surface case, the idea is to resolve the singularities by so called crepant blowups. A birational morphism ψ : Y −→ X between canonical Gorenstein threefolds is called crepant if ψ ∗ K X = KY . Crepant blowups reduce the singularities of any X to cDV points. For curves in Xsing we have the generic description (8.2.1) and there is a general theory of how to simultaneously resolve the DuVal locus of a family of surfaces (Brieskorn). This reduces Xsing up to a number of isolated cDV points. 8.3 Theorem ([Ka88b], [Re83]). Let X be a Gorenstein threefold with at most canonical singularities. Then there exists a partial crepant resolution ψ : Y −→ X, where Y is a Gorenstein threefold with at most terminal and Q–factorial singularities, such that KY = ψ ∗ KX .

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We compose here the resolution with the so called Q–factorialization due to Kawamata. Note that a partial crepant resolution is in general not unique. It may be thought of as a relative minimal model and such are usually not unique in dimension three. For example, in the case of a cone as above over a del Pezzo ˆ over surface S we may flop any anticanonically trivial curve in Y = Blv (S) some (−1)–curve in S in order to get a different partial crepant resolution Y ˆ of S. 8.4 Definition. A normal projective variety Y , such that the anticanonical divisor −KY is a big and nef Cartier divisor, is called almost Fano. In the case where X is a Gorenstein Fano threefold, the partial resolution Y is almost Fano. By the Kawamata–Viehweg vanishing theorem, H i (Y, OY ) = H i (X, OX ) = H i (X, −KX ) = 0,

for i > 0.

(8.4.1)

Moreover, all integral cohomology groups H m (X, Z) are torsion free. Almost Fano threefolds are always simply connected and covered by rational curves, even rationally connected. By the Riemann–Roch theorem, χ(X, −KX ) = h0 (X, −KX ) =

(−KX )3 + 3 = 2g − 2. 2

Here g is genus of a general curve section, that is the complete intersection S1 ∩ S2 of two general elements S1 , S2 ∈ |−KX | (in the case |−KX | is base point free), also called the genus of X. The highest power (−KX )3 is called degree of X. Let on the other hand Y be an almost Fano threefold with at most canonical singularities. By the base point free theorem, |−mKY | is spanned for m ' 0, defining a morphism ψ : Y −→ X onto some Fano threefold X, the so called anticanonical model of Y . The anticanonical map ψ is again crepant, and X has at most canonical singularities. 8.5 Example. (1) Back to the threefold X = S × P1 in Example 8.1, let ψS : S˜ −→ S be a resolution of singularities of S. Then KS˜ = ψS∗ KS , since S has only DuVal singularities. Then the product map ψ : S˜ × P1 −→ X is a partial crepant resolution of X: the product Y = S˜ × P1 is smooth and almost Fano, and ψ is crepant. (2) Consider again the quadric cone X = Q ⊂ P4 and assume first the vertex ˆ 2 is the cone over the two dimensional quadric is a point p, i.e., X = Q Q2 P1 × P1 ⊂ P3 . Hence π : P(OP1 ×P1 ⊕ OP1 ×P1 (1, 1)) −→ X is a resolution of singularities. But this map is not crepant: denote the exceptional divisor of π by E P1 × P1 . Then −KP = π ∗ (−KX ) − E.

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This shows that p is a terminal singularity of X. Blowing down E first along one of its rulings, we obtain a small resolution of X: ψ : Y = P(OP1 ⊕ OP1 (1) ⊕ OP1 (1)) −→ X. Here the tautological bundle O(1) is base point free and contracts a single section onto the vertex of the cone X. This shows in particular, that X is not Q–factorial. (3) If the vertex of the cone is a line, then X is the double cone over a conic in P2 , i.e., we get a a crepant resolution ψ : Y = P(OP1 ⊕ OP1 ⊕ OP1 (2)) −→ X, contracting the P1 × P1 corrresponding to the first two trivial summands onto the vertex. It is clear, that this is degeneration of the last case: the bundle OP1 (1)⊕OP1 (1) may be viewed as the non–split extension of OP1 (2) by OP1 (Euler sequence), which degenerates to the splitting one. The classification of singular Fano threefolds is still an ongoing project. There are several partial results, so for example lists of hypersurfaces or complete intersections in weighted projective spaces, which are Fano by Reid, Iano-Fletcher, Johnson and Koll´ ar ([Re79], [IF89], [JK01]). Following the line of arguments of the classification in the smooth case, the first steps towards a classification of Gorenstein Fano threefolds with canonical singularities, or almost Fano threefolds, respectively, are already complete. In 2003, Prokhorov proved an effective bound for the degree: 8.6 Theorem ([Pr05]). Let X be a Gorenstein Fano threefold with at most canonical singularities. Then (−KX )3 ≤ 72 and equality holds iff X is one of the weighted projective spaces P(13 , 3) or P(12 , 4, 6). The next step is a classification of all those X, which are not “Gino Fano”, i.e., where the anticanonical system is not very ample. Consider the rational map ψ : X _ _ _/ W ⊂ Pg+1 defined by the anticanonical system. Then two things may happen. Either ψ is not even a morphism, i.e., |−KX | is not base point free. In that case W is a surface. There exist two families of smooth Fano threefolds with this property, in the singular case the list enlarges, but is completely given in [JR06a], the results are content of Sect. 9 below. Secondly, |−KX | may be base point free, but still not very ample. In that case, ψ is a double covering and W is a variety of minimal degree in Pg+1 , i.e., deg(W ) = codim(W ) + 1. Since a general curve section of X is a hyperelliptic curve, these threefolds are called hyperelliptic. Hyperelliptic Fano threefolds are classified by Cheltsov, Shramov and Przyjalkowski ([CSP05]).

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A last “non–general type” of Fano threefolds are del Pezzo threefolds. We define the Fano index of a Fano variety (in any dimension n) to be the maximal number rX , such that −KX = rH for some H ∈ Pic(X). A Fano variety X is then called “del Pezzo”, if rX = n − 1. In our case this means rX = 2. These varieties are treated in a different context in the paper [CJR06]. Having classified all “ugly” cases, we started in [JPR05a] a complete classification of all smooth almost Fano threefolds Y . Here the first step is to assume the Picard number as small as possible. This means ρ(Y ) = 2, since Y should not be Fano. Then an anticanonical model X of Y is a Fano threefold with at most canonical singularities and ρ(X) = 1. So in a sense we classify all Fanos X, such that already the first step of a partial crepant resolution becomes smooth. This means that the singularities of X are “as easy as possible”. There are two different cases: the anticanonical map ψ : Y −→ X may either contract an irreducible divisor (= divisorial, content of [JPR05a]), or a finite number of rational curves (= small, considered in [JPR05b]). In the latter case the target X is not Q–factorial, but has only terminal singularities. In that case there exists by [Na97] a smoothing of X, i.e., X moves in family X −→ ∆, such that X0 = X and the general fiber Xt is a smooth Fano threefold. In [JR06b] we study such families, the results can be found in Sect. 10 below. We obtain that there are no “new” cases in this situation, meaning X is always a degeneration of some smooth Fano threefold with ρ = 1 from Iskovskikh’s list. Maybe more interesting from this point of view is the second case, which we considered in [JPR05a]. From now on we assume that ψ is divisorial, contracting an irreducible divisor D either to a curve B in X, or to a point. In this case, the singularities of X are Q–factorial and canonical, but non– terminal. In particular, the existence of a smoothing of X is not known, and may indeed not exist. We obtain 8.7 Theorem ([JPR05a]). There are 68 families of smooth almost Fano threefolds Y with ρ(Y ) = 2, such that the anticanonical map ψ is divisorial. Almost all of these families can be explicitly described. For two numerically possible families a construction is still missing, so the number 68 could still drop to 66 or 67. The missing ones should be described ) for d ∈ as follows: consider the projective bundle P = P(OP1 (2) ⊕ OP⊕d 1 {3, 4, 5, 6, 8}. Now find a smooth almost Fano threefold Y ⊂ P with ρ(Y ) = 2,

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such that the induced map Y −→ P1 is a del Pezzo fibration with general fiber a del Pezzo surface of degree d. This can easily be done for d = 3 and 4 by simply taking complete intersections: for d = 3 for example let Y ∈ |O(3)| be a general member, etc. But it seems to be a difficult problem to determine whether d = 5 and 6 are possible. To prove Theorem 8.7, we use the minimal model program. The assumptions imply that KY is not nef, hence there exists an elementary extremal contraction φ : Y −→ Z to some (normal) variety Z, such that −KY is φ– ample and ρ(Z) = ρ(Y ) − 1 = 1. These contractions are classified by Mori ([Mo82]). We obtain the following diagram ψ

Y

/X

φ

Z

Dividing the project into parts corresponding to the type of (Z, φ), we obtain the following • If dim Z = 1, then Z P1 and φ is a del Pezzo fibration. We obtain 15 families. • If dim Z = 2, then φ is a conic bundle over P2 . Here we get 8 proper conic bundles and 4 families of P1 –bundles. • Assume dim Z = 3 and Y is the blowup of Z along a smooth curve. Then Z is a smooth Fano threefold with ρ(Z) = 1. There are 24 families, where the excetional divisor D of ψ is also contracted to a curve, and 10, where D maps to a point. • Assume dim Z = 3 and Y is the blowup of Z in a point p. In this case Z may have a terminal singularity in p. We get 7 families. We consider again the special cases, where the anticanonical system of X is not very ample and obtain here: 8.8 Proposition. Let Y be a smooth almost Fano threefold with ρ(Y ) = 2, such that the anticanonical map ψ is divisorial. Then |−KY | is base point free. For the proof just note that the anticanonical model X is a Q–factorial Gorenstein Fano threefold with canonical, non–terminal singularities and Picard number one. Now the result follows from the list in Theorem 9.1 below. 8.9 Proposition. Let Y be a smooth almost Fano threefold with ρ(Y ) = 2, 2:1 such that the anticanonical map ψ : Y −→ X is divisorial. If X −→ W is hyperelliptic, then we are in one of the following cases (1) (−KX )3 = 2, W = P3 and X → W is ramified along a sextic; (2) (−KX )3 = 4, W ⊂ P4 is a quadric and X → W is ramified along a quartic;

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(3) (−KX )3 = 8, W is the cone in P6 over the Veronese surface in P5 and X = X6 ⊂ P(13 , 2, 3), i.e., X → W is ramified along a cubic. All of these threefolds are the expected degenerations of Iskovskikh’s list. For small genus we find in our situation (see [IP99], Proposition 4.1.12. for the smooth case): 8.10 Proposition. Let Y be a smooth almost Fano threefold with ρ(Y ) = 2, such that the anticanonical map ψ : Y → X is divisorial. Assume X is not hyperelliptic. (1) If g = 3, then X4 ⊂ P4 is a quartic. (2) If g = 4, then X2,3 ⊂ P5 is a complete intersection of a quadric and a cubic. (3) If g = 5, then X2,2,2 ⊂ P6 is a complete intersection of three quadrics. Since the canonical curve section C ⊂ X is a smooth canonical curve of genus g, (1) and (2) are easily obtained. Assume g = 5. We have two possible cases: either X is cut out by quadrics or it is trigonal. Since X is already a complete intersection in the first case, assume the latter one. Then by [CSP05], X is the anticanonical model of an almost Fano threefold V with canonical singularities, where V is a divisor in |O(3) + π ∗ OP1 (−1)| on the projective bundle π : P(⊕4i=1 OP1 (di )) → P1 , where either d1 = d2 = d3 = 1, d4 = 0 or d1 = 2, d2 = 1, d3 = d4 = 0. Neither case is possible in our situation: in the first case the map V → X is small and in the second case the Picard number of X is larger than one. By assumption, the anticanonical map ψ is a birational contraction. The assumption ρ(Y ) = 2 guarantees that ψ is primitive, i.e., it does not factor. The structure of the exceptional locus of such contractions is studied by Wilson, Paoletti and Minagawa. We have in our situation 8.11 Proposition ([Wi92], [Wi97], [Pa98], [Mi03]). Let Y be a smooth almost Fano threefold with ρ(Y ) = 2, such that the anticanonical map ψ : Y → X is divisorial, contracting the divisor D to a curve B. Let lψ be a general exceptional fiber. Then (1) B is a smooth curve of cDV singularities, Y = BlB (X); (2) D is a conic bundle over B; each fiber is either isomorphic to a smooth conic or to a line pair. In particular, D.lψ = −2.

9 Base Points of the Anticanonical System In the classification of Fano varieties, those where the anticanonical system is not base point free, are usually annoying. In his classification of smooth Fano threefolds Iskovskikh classified these threefolds first and found two deformation families ([Is80]):

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(1) the blowup of the Fano threefold V2,1 (see below) along a complete intersection curve of genus 1, and (2) S1 × P1 , where S1 is a del Pezzo surface of degree 1. Some more details: (1) Here V = V2,1 denotes a Fano threefold of index rV = 2 and degree (−KV )3 = rV3 · 1 = 8, i.e., there exists an ample H ∈ Pic(V ), such that −KV = 2H and H 3 = 1. We may describe V as a double cover of the Veronese cone W → P5 , ramified along a smooth cubic not passing through the vertex. The system |H| on V is not base point free, the base locus being a single point p. Still two general members of |H| cut out a smooth elliptic curve through p. Blowing up this curve, we obtain a Fano threefold X, π : X −→ V. Denote the exceptional divisor of π by E. Then −KX = π ∗ (−KV ) − E, i.e., rX = 1 and (−KX )3 = 4. The base locus of |−KX | is a P1 , namely the fiber of E over p. (2) The second example is easier: the anticanonical divisor −KS1 of a del Pezzo surface of degree 1 is not base point free. Indeed, write S1 = Blp1 ,...,p8 (P2 ) the blowup of P2 in 8 (general) points. Then |−KS1 | is the system of all cubics in P2 passing through the 8 points. There are only two, and these meet in a ninth point, the base point p. The base locus of |−KX | is hence again a P1 , namely the fiber over p in the product. Degenerating the two cases above, we easily obtain Gorenstein Fano threefolds X with canonical singularities and base points in the anticanonical system. Move for example the 8 points blown up to obtain S1 = Blp1 ,...,p8 (P2 ) in a more special position, such that three of them lie on a line l. Then the blowup becomes almost Fano, containing the (−2)–curve ˆl over l. Contracting ˆl, we obtain a del Pezzo surface of degree 1 with an A1 –singularity. The product with P1 gives X. The natural question arises whether Iskovskikh’s list remains complete in this sense allowing canonical or terminal singularities. The answer (for both cases) is “no”, there are more examples. For terminal singularities there is only one further, which appeared several times in the literature in different contexts ([Me99], [Mo88], [IT01]). Allowing canonical singularities, the list enlarges maybe a little bit unexpectedly by a whole sequence of varieties X2m−2 , with 3 ≤ m ≤ 12: 9.1 Theorem ([JR06a]). Let X be a Gorenstein Fano threefold with at most canonical singularities and Bs|−KX | = ∅. Then we are in one of the following cases. (1) dim Bs|−KX | = 0. In this case X is a complete intersection in P(14 , 2, 3) of a quadric Q, defined in the first four linear variables, and a sextic F6 ;

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(2) dim Bs|−KX | = 1. Then Bs|−KX | P1 and either (i) X is the blowup of a sextic in P(13 , 2, 3) along a complete intersection curve of arithmetic genus 1, or (ii) X S1 × P1 , where S1 is a del Pezzo surface of degree 1 with at worst DuVal singularities, or (iii) X = X2m−2 is an anticanonical model of the blowup of the variety Um (see below) along a smooth, rational complete intersection curve Γ0 ⊂ Um,reg for 3 ≤ m ≤ 12. Here Um denotes a double cover of P(OP1 (m)⊕OP1 (m−4)⊕OP1 ) with at worst canonical singularities, such that −KUm is the pullback of the tautological line bundle O(1). For m ≥ 4, this is a hyperelliptic Gorenstein almost Fano threefold of degree 4m − 8. The curve Γ0 lies over the complete intersection of some general element in |O(1)| and the “minimal surface” B ∈ |O(1) − mF |, where |F | denotes the pencil (note that Γ0 is always contained in the ramification locus). If m = 3, then Γ0 is the only curve, on which −KU3 is not nef. For details of the construction see below. In the following we present the ideas of the proof. Main tools are on the on hand the partial crepant resolution of X as in Theorem 8.3, on the other hand the existence of a general elephant, that is a “good” member of |−KX |. The existence of a K3–elephant was proved by Shokurov for the smooth case in [Sh80a], and by Reid in [Re83] in general. 9.2 Theorem ([Sh80a], [Re83]). Let X be a Gorenstein Fano threefold with at most canonical singularities. Then |−KX | contains an irreducible surface S with at worst DuVal singularities, called general elephant. Note that if X is smooth, then the general elephant S is a smooth K3 surface. We fix such a surface S. By the Kawamata–Viehweg vanishing theorem, H 0 (X, −KX ) −→ H 0 (S, −KX |S ) is surjective, implying Bs|−KX | = Bs|−KX |S |. A resolution of singularities ν : S˜ −→ S will be a smooth K3 surface, and pulling back −KX , we obtain a linear system, which is nef and big, but not base point free. Such systems have been studied first by Saint–Donat ([SD74], for the ample case), and later Shin generalized his results in [Sh89]. We obtain: ν ∗ |−KX | = |Γ + mf |, where m ≥ 2 and (1) |f | is an elliptic pencil and (2) Γ = Bs|Γ + mf | P1 is a section. Hence depending on whether Γ is ν–exceptional or not, we have two cases (this is part of a general result in [Sh89]):

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• Bs|−KX | is a point in Xsing . In that case m = 2 and we get (1) in the Theorem. • Bs|−KX | = Γ P1 is contained in Xreg . This leads to all the other cases. Assume from now on Bs|−KX | = Γ of the rational map given by |−KX |

P1 . We start considering the image

X _ _ _/ W ⊂ Pg+1 .

(9.2.1)

Due to Reid ([Re83]), W is a (not necessarily smooth) surface, and one easily verifies that it is a surface of minimal degree, i.e., deg(W ) = codim(W ) + 1. Varieties with this property have been classified by del Pezzo in dimension two, and by Bertini in general ([dP85], [Be07]). The list (with some repetitions) is as follows: (1) (2) (3) (4)

Pn ; the n–dimensional quadric Qn ⊂ Pn+1 ; (a cone over) the Veronese surface; (a cone over) a rational scroll.

The cone over a (rational) scroll, denoted F(d1 , . . . , dn ), is the image of F(d1 , . . . , dn ) = P(OP1 (d1 ) ⊕ · · · ⊕ OP1 (dn )),

d1 ≥ · · · ≥ d n ≥ 0

in Pd1 +···+dn +n−1 under the map associated to the tautological line bundle which will be denoted O(1). Note that for dn ≥ 1, F(d1 , . . . , dn ) and F(d1 , . . . , dn ) are isomorphic. In our case, W is either a ruled surface Fe , or the cone Cˆd over a rational normal curve of degree d. Blowing up the base locus Γ ⊂ X, we obtain a morphism onto W ϕ /W XΓ {= { { { X ∗ ) of the blowup is a rational ruled The exceptional divisor EΓ = P(NΓ/X surface which is either mapped isomorphically onto W (case W Fe ), or ϕ contracts the minimal section (case W Cˆd ). We have Fe ; (i), (ii) in Theorem 9.1 W Cˆd ; (iii) in Theorem 9.1.

The “new cases” hence occur if W is not smooth. Very roughly speaking the idea of the proof is now to find some contractible divisor Z in XΓ (actually not

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directly in XΓ , but inside some modification) and to study the corresponding morphism. The divisor Z will be a component of the pullback of the vertex of W . The main difficulty (and the reason why these cases do not occur in the smooth case) is, that XΓ is not necessarily Q–factorial, we first have to choose a suitable partial crepant resolution. The idea how to construct the remaining examples in the list is as follows. Assume we have found some Gorenstein threefold U (which will of course be Um ) with at most canonical singularities, such that |−KU | contains a smooth K3 surface S with −KU |S = 2Γ0 + mf for some m ≥ 3. Moreover, assume P1 Γ0 ⊂ Ureg and |f | is an elliptic pencil as above. Note that this already implies that U is a hyperelliptic almost Fano threefold (if m ≥ 4). Let Y = BlΓ0 (U ) be the blowup of U in Γ0 . The strict transform of S is a smooth K3 surface in |−KY | which we denote by S as well. We have −KY |S = Γ0 + mf, implying Bs|−KY | = Γ0 P1 . An anticanonical model X of Y is a Gorenstein Fano threefold with canonical singularities for which Bs|−KX | P1 . 9.3 Example. We want to construct U = Um with the properties above. We have already seen that U has to be almost Fano and the anticanonical map associated to −KU sends U to a variety of minimal degree U −→ V ⊂ P2m−2 (this follows since U is hyperelliptic, compare Sect. 8). Here S is sent to F4 , the fourth Hirzebruch surface. The idea is therefore to construct U as a ramified twosheeted covering of some variety of minimal degree, for which a general hyperplane section is isomorphic to F4 and verify the correct splitting of −KU on the surface S over F4 . Define Um as double cover of the projective bundle V = F(m, m − 4, 0),

m ≥ 3,

which we may view as a resolution of singularities of the cone over F4 = F(m, m − 4). We find a smooth surface isomorphic to F4 in |OV (1)|, corresponding to the projection onto the first two summands. Moreover, there exists a unique section B ∈ |OV (1) − mF | meeting the surface F4 in its minimal section C0 (as usual, the pencil of V is denoted by |F |). We claim that we may choose a D ∈ |OV (4) − (4m − 12)F |,

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such that the square root of D yields a threefold Um with at worst canonical singularities. Then 2:1

µ : Um −→ F(m, m − 4, 0)

and −KUm = µ∗ OV (1).

The section C0 = B ∩ F4 should not meet the singular locus of D. Its reduced inverse image in Um will then be the curve Γ0 and X2m−2 is by definition an anticanonical model of BlΓ0 (Um ). We define the surface S = µ−1 (F4 ), the induced double covering of F4 . This shall be our general elephant. It remains to show that for D general enough Um has the correct type of singularities, and that −KUm |S = 2Γ0 + mf with an elliptic pencil |f |. Since F4 comes from a splitting sequence, D ∩ F4 is a general member of |4C0 + 12f|, with f P1 a fiber of F4 . A general member of |4C0 + 12f| splits as C0 + C with C ∈ |3C0 + 12f| smooth and disjoint from C0 . This can be seen as follows. Write |4C0 + 12f| = |O(4) − (4m − 12)f|. If x1 , x2 denote homogeneous coordinates of P1 corresponding to the summands of our vector bundle, then the monomial xe11 xe22 with e1 + e2 = 4 has as coefficient a function taken from H 0 (P1 , OP1 (me1 + (m − 4)e2 − (4m − 12))). Now C0 corresponds fiberwise to x1 = 0. It therefore suffices to prove that the coefficient function of x42 vanishes. This is supposed to be a section of OP1 (−4), so the claim follows. The double covering of F4 yields a smooth K3 surface S ∈ |−KUm | = |µ∗ OV (1)| with µS : S −→ F4 ramified along C0 and C. The pullback of f gives an elliptic pencil |f | on S with the section Γ0 lying over C0 and −KUm |S = µ∗S O(1) = 2Γ0 + mf as claimed. Concerning the singularities of Um note that for m = 3, Um is smooth, and for m ≥ 4, we always have D =B+R with R ∈ |OV (3) − (3m − 12)F |. Fiberwise D ∩ F consists of a line together with some cubic. For 4 ≤ m ≤ 12 we can take R to be irreducible, i.e., D ∩ F consists of a line and an irreducible cubic. For m = 4, the cubic is smooth, meeting the line transversally in three points. For m ≥ 5, the line and the cubic intersect in one point, i.e., in a flex if the cubic is smooth. This gives an A–D–E singularity in the fiber, implying that Um indeed has at worst canonical singularities for 3 ≤ m ≤ 12. For m ≥ 13 on the other hand, R = R1 + R2 + R3 with Ri ∈ |OV (1) − (m − 4)F |, so D ∩ F consists of four lines through a point. This means that over F we will not have DuVal singularities, implying that Um is not canonical for m ≥ 13. 9.4 Remark. The construction works for m = 2 as well. Here Bs|−KX2 | = {p} and we get a special case of (1) in Theorem 9.1 with Q the quadric cone.

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10 Smoothings Let S be any Gorenstein del Pezzo surface with DuVal singularities. Then a ˜ These resolution of singularities is crepant, yielding an almost Fano surface S. are easily classified: it is either P1 ×P1 , F2 , or the blowup of P2 in r ≤ 8 points in “almost general” position. Assume for simplicity that the points are indeed different. By moving them into general position we see that S˜ comes with a family S˜ −→ ∆, where S˜t is a smooth del Pezzo surface for t = 0. The anticanonical image yields a smoothing S −→ ∆ of S = S0 with St S˜t for t = 0. Starting from the smoothing S, the family S˜ is nothing but an example of a simultaneous resolution. 10.1 Definition. Let X be a Gorenstein (almost) Fano threefold. A smoothing of X is a (not necessarily smooth) complex space X together with a flat proper morphism π : X −→ ∆, such that X0 X and Xt is a smooth (almost) Fano threefold for t = 0. The general fiber Xt is also referred to as a smoothing of X. Smoothings already appeared in Sect. 8. We will see in this section, under which conditions a smoothing is known to exist, and show some properties. We will also see that a smoothing does not always exist, contrarily to the surface case. Motivated by the above example we say that a smoothing of X has a simultaneous resolution if there exists a diagram Y@ @@ @@ @@

∆

/X ~ ~ ~~ ~~ ~ ~

(10.1.1)

such that Yt −→ Xt is a desingularisation for all t. Similarly we define a simultaneous partial crepant resolution. Such resolutions rarely exist (blowing up subvarieties usually produces a reducible central fiber). Smoothings are known to exist in the following cases: • for Gorenstein Fano threefolds with terminal singularities ([Na97]); • for Gorenstein almost Fano threefolds with terminal and Q–factorial singularities ([Mi01]). The idea of the proofs is always the same: one studies the Kuranishi space of X and shows that there are “good directions” smoothing X.

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10.2 Example. 1.) A smoothing of a terminal Fano threefold: let p1 = [1:0:0:0] and p2 = [0:1:0:0] in P3 = Proj C[w, x, y, z] and π

Y = Blp1 ,p2 (P3 ) −→ P3 with exceptional divisors E1 and E2 . Then Y is almost Fano with one (−KY )– trivial curve corresponding to the line through p1 and p2 . On Y , −KY = π ∗ O(4) − 2E1 − 2E2 . The half anticanonical system maps Y to a terminal Fano threefold X in P7 of degree 6. The map is given by the eight quadrics through p1 and p2 . The quadrics cutting out X are the 2 × 2 minors of the matrix on the right hand side of ⎛ ⎞ ⎞ ⎛ wx wy wz t0 t1 t2 ⎝ xz yz z 2 ⎠ = ⎝ t3 t4 t5 ⎠ xy y 2 yz t6 t7 t4 The minors of the general 3 × 3 matrix with linear entries on P7 cut out P(TP2 ), a smooth Fano threefold. Indeed, embed P2 × P2 into P8 by Segre. Then P2 × P2 is cut out by the 2 × 2 minors of the matrix (ui,j ), where ui,j , 1 ≤ i, j ≤ 3, are homogeneous coordinates of P8 . A general hyperplane section of P2 × P2 is P(TP2 ). 2.) The results of Namikawa and Minagawa together do not give a simultaneous partial crepant resolution. Consider ˆ 2 ⊂ P4 , Y = P(OP1 ⊕ OP1 (1)⊕2 ) −→ X = Q the small resolution of the quadric cone as in Example 8.5. Here X is a degeneration of a smooth quadric in P4 , but Y is rigid. We could take Y = Y × ∆ but we do not get a diagram as in (10.1.1). Terminal Gorenstein singularities are in a sense as close to smooth points as one can whish. In fact, with the existence of a smoothing one can show that the classification of terminal Gorenstein Fano threefolds is in fact the same as the smooth classification: 10.3 Theorem ([JR06b]). Let X be a Gorenstein Fano threefold with at most terminal singularities and let π : X −→ ∆ be a factorial smoothing of X. Then Pic(X ) Pic(Xt ) Pic(X). In the end of this section we will see that the same is not true in the case of canonical singularities. To give the idea of the proof let X be a Gorenstein Fano threefold with at most terminal singularities. Let X −→ ∆ be a smoothing of X. We collect some properties of X :

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(1) Xsing ⊂ Xsing , X is Cohen Macaulay, Gorenstein and normal; (2) X has at most terminal singularities (“inversion of adjunction”). Terminal Gorenstein singularities in dimension 3 are isolated hypersurface singularities (Sects. 1 and 8). (3) X has at most isolated hypersurface singularities; (4) X is factorial (Grothendieck’s proof of Samuel’s conjecture, [Gr68]); (5) the Leray spectral sequence and the exponential sequence imply Pic(X ) H 2 (X , Z), Pic(X) H 2 (X, Z), and Pic(Xt ) H 2 (Xt , Z); (6) X is a deformation retract of X , implying H i (X , Z) H i (X, Z) for all i. The primitive part of H 2 defines a polarized variation of Hodge structures of pure type (1, 1) by (8.4.1) in Sect. 8. The isometries of a lattice form a finite group. (7) The monodromy ρ : π1 (∆∗ ) −→ H 2 (Xt , Q) is finite. After a base change we may assume that the monodromy is trivial. The following proof shows that there is in fact no monodromy. The only thing that remains open is Pic(X) Pic(Xt ). We have to show that we can extend any line bundle Lt ∈ Pic(Xt ) to the whole of X . The monodromy being trivial we may assume Lt is defined on X \X. Let ϕ : Y −→ X be a log–resolution of (X , X). On Y, the maps H m (Y, Q) −→ H m (Yt , Q)inv are surjective. This is the local invariance cycle theorem ([Cl77]). In other words, any monodromy invariant cohomology class on the general fiber Yt comes by restriction from a class on Y. Therefore ϕ∗ Lt extends to some LY ∈ Pic(Y). This shows that we can extend Lt to X \Xsing . As X is factorial, we get an extension to the whole of X . 10.4 Corollary. In the case of a smoothing of terminal Gorenstein singularities a simultaneous resolution does not exist. This is in fact a Corollary to point (4) above which says that the total space X is factorial. If we assume moreover the central fiber X to be Q–factorial, then even all elementary extremal contractions of the central and the general fiber coincide. This means terminal and Q–factorial Fano threefolds are degenerations of smooth ones even concerning their Mori fiber structures. This agrees with Mukai’s classification of Fano threefolds with terminal singularities and ρ = 1 in [Mu95]. We already mentioned that the above results do not hold for canonical singularities. Here much less is known.

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For the rest of this section let X be a Gorenstein Fano threefold with canonical singularities. First thing to note is that a smoothing need not exist: consider Y = P(OP2 ⊕ OP2 (−KP2 )) −→ X = P(13 , 3) ⊂ P10 . Here Y is the blowup of the cone over P2 → P9 embedded by |−KP2 |. It is easy to see that Y is almost Fano and (−KX )3 = (−KY )3 = 72. But there does not exist a smooth Fano threefold V with (−KV )3 = 72 by Iskovskikh’s classification. The degree is of course constant in a smoothing. We conclude that X is not smoothable. Note, however, that X has an isolated non–cDV point. It is an open question whether canonical Gorenstein Fano threefolds with at most cDV singularities are always smoothable. Secondly, there might exist different smoothings: consider the cone over the del Pezzo surface S of degree 6. This is a Gorenstein Fano threefold with canonical singularities and ρ = 1. Here we have two smoothings, one with general fiber P(TP2 ) and one with fiber P1 × P1 × P1 . Indeed, S is the general hyperplane section in the half anticanonical system of P(TP2 ) and of P1 × P1 × P1 . Any smooth projective manifold flatly degenerates to the cone over its general hyperplane section.

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[Hu99] [IF89] [IT01] [Is80] [IP99] [JPR05a] [JPR05b] [JR06a] [JR06b] [JK01] [Ka88a] [Ka88b]

[Ka92a] [Ka92b] [Ka92c] [KS05] [KST05]

[KO70] [Ko63] [Ko86] [Ko92] [Ko93]

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Grothendieck, A.: Cohomologie locale des faisceaux cohéhents et Théorèmes de Lefschetz locaux et globaux (SGA 2). Adv. Stud. in Pure Math. 2 (1968) Huybrechts, D.: Compact hyperk¨ ahler manifolds: Basic results. Invent. Math. 135, 63-113 (1999). Iano-Fletcher, A.R.: Working with weighted complete intersections. Bonn preprint MPI/89-35 (1989) Ishii, S., Tomari, M.: Hypersurface non-rational singularities which look canonical from their Newton boundaries. Math. Z. 237, 125–147 (2001) Iskovskikh, V.A.: Anticanonical models of three-dimensional algebraic varieties. J. Soviet Math. 13, 745–814 (1980) Iskovskikh, V.A., Prokhorov, Yu.G.: Algebraic Geometry V: Fano varieties. Springer 1999 Jahnke, P., Peternell, Th., Radloff, I.: Threefolds with big and nef anticanonical bundles I. Math. Ann. 333, 569–631 (2005) Jahnke, P., Peternell, Th., Radloff, I.: Threefolds with big and nef anticanonical bundles II. In preparation. Jahnke, P., Radloff, I.: Gorenstein Fano threefolds with base points in the anticanonical system. Comp. Math. 142, 422–432 (2006) Jahnke, P., Radloff, I.: Terminal Fano threefolds and their smoothings. math.AG/0601769 Johnson, J. M., Koll´ ar, J.: Fano hypersurfaces in weighted projective 4-spaces. Exp. Math. 10, 151–158 (2001) Kawamata, Y.: Pluricanonical systems on minimal algebraic varieties. Invent. Math. 79, 567-588 (1985). Kawamata, Y.: Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces. Ann. Math. 127, 93–163 (1988) Kawamata, Y.: Moderate degenerations of algebraic surfaces. In: Lecture Notes in Math. 1507, 113-133. Springer 1992 Kawamata, Y.: Abundance Theorem for minimal threefolds. Inv. Math. 108, 229-246 (1992) Kawamata, Y.: Boundedness of Q–Fano threefolds. In: Proc. Int. Conf. Algebra. Contemp. Math. 131, 439–445 (1992) Kebekus, S., Sol´ a Conde, L.: Existence of rational curves on algebraic varieties, minimal rational tangents, and applications. This volume Kebekus, S., Sol´ a Conde, L., Toma, M.: Rationally connected foliations after Bogomolov and McQuillan. To appear in J. Alg. Geom.; math.AG/0505222 Kobayashi, S., Ochiai, T.: On complex manifolds with positive tangent bundles. J. Math. Soc. Japan 22, 499–525 (1970) Kodaira, K.: On compact analytic surfaces III. Ann. Math. 78, 1–40 (1963) Koll´ ar, J.: Higher direct images of dualizing sheaves. I. Ann. Math. 123, 11-42 (1986) Koll´ ar, J.: Flips and abundance for algebraic threefolds. Astérisque 211 (1992) Koll´ ar, J.: Shafarevich maps and plurigenera of algebraic varieties. Inv. math. 113, 177-215 (1993)

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[Na91] [Na97] [OP00] [OP04] [Pa98]

Koll´ ar, J.: Rational curves on algebraic varieties. Erg. d. Math. vol 32. Springer, Heidelberg (1996) Koll´ ar, J., Miyaoka, Y., Mori, S.: Rational connectedness and boundedness of Fano manifolds. J. Diff. Geom. 36, 765–769 (1992) Koll´ ar, J., Mori, S.: Birational Geometry of Algebraic Varieties. Camb. Univ. Press 1998 Kronenthaler, W.: Zur Moritheorie auf K¨ ahlerdreifaltigkeiten mit h¨ ochstens terminalen Singularit¨ aten. Ph.D. Thesis, Bayreuth, (2005) Lazarsfeld,R.: Positivity in algebraic geometry I, II. Erg. d. Math. vol. 49,50. Springer 2004 Matsusaka, T.: On canonically polarised varieties (II). Amer. J. Math. 92, 283–292 (1970) Mella, M.: Existence of good divisors on Mukai varieties. J. Alg. Geom. 8, 197–206 (1999) Mehta,V.B.; Ramanathan,A.: Semistable sheaves on projective varieties and their restriction to curves. Math. Ann. 258, 213–224 (1982) Minagawa, T.: Deformations of weak Fano 3-folds with only terminal singularities. Osaka J. Math. 38, 533–540 (2001) Minagawa, T.: Global smoothing of singular weak Fano 3-folds. J. Math. Soc. Japan 55, 695-711 (2003). Miyaoka,Y.: Deformations of a morphism along a foliation.In: Proc. Symp. Pure Math. 46, 245–268 (1987) Miyaoka,Y.: On the Kodaira dimension of minimal threefolds. Math. Ann. 288, 325–332 (1988) Miyaoka,Y.: Abundance conjecture for threefolds: ν = 1 case. Comp. Math. 68, 203–20 (1988) Miyaoka,Y; Mori,S.: A numerical criterion for uniruledness. Ann. Math. 124, 65–69 (1986). Mori, S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math. 116, 133–176 (1982) Mori, S.: Classification of higher-dimensional varieties. Proc. Symp. Pure Math. 46, 269–331 (1987). Mori, S.: Lectures on 3-dimensional algebraic varieties. Utah University, Fall 1988 Mori, S., Mukai, S.: Classification of Fano threefolds with b2 ≥ 2. Manus. Math. 36, 147–162 (1981) Mukai, S.: New developments in the theory of Fano threefolds: vector bundle method and moduli problems. S¯ ugaku 47, 125–144 (1995); Transl. in Sugaku Exp. 15, 125–150 (2002) Nadel, A.M.: The boundedness of degree of Fano varieties with Picard number one. J. AMS 4, 681–692 (1991) Namikawa, Y.: Smoothing Fano 3-folds. J. Alg. Geom. 6, 307–324 (1997) Oguiso, K.; Peternell, Th.: Projectivity via the dual Kähler coneHuybrechts’ criterion. Asian J. Math. 4, 213–220 (2000). Oguiso, K.; Peternell, Th.: The dual Kähler cone of compact K¨ ahler threefolds. Commun. Anal. Geom. 12, 1131–1154 (2004). Paoletti, R.: The K¨ ahler cone in families of quasi-Fano threefolds. Math. Z. 227, 45–68 (1998)

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Paun,M.: Sur le groupe fondamental des variétés k¨ ahleriennes compactes a classe de Ricci numériquement effective. C. R. Acad. Sci., Paris, Sér. ` I, 324, No.11, 1249–1254 (1997). [Pa01] Paun, M.: On the Albanese map of compact K¨ ahler manifolds with numerically effective Ricci curvature. Commun. Anal. Geom. 9, 35–60 (2001). [Pe01] Peternell,Th.: Towards a Mori theory on compact K¨ ahler threefolds, III. Bull.Soc. Math. France 129, 339–356 (2001) [Pe05] Peternell,Th.: Kodaira dimension of subvarieties II. To appear in Intl. J. Math.; math.AG/0504266 [Pr05] Prokhorov, Yu.G.: The degree of Fano threefolds with canonical Gorenstein singularities. Mat. Sb. 196, 81–122 (2005), in Russian. English translation: Sb. Math. 196, 77–114 (2005) [Re79] Reid, M.: Canonical 3-folds. Proc. Alg. Geom. Anger 1979. Sijthoff and Nordhoff, 273–310 (1980) [Re83] Reid, M.: Projective morphisms according to Kawamata. Unpublished manuscript (1983) [Re87] Reid, M.: Young person’s guide to canonical singularities. In Algebraic Geometry Bowdoin 1985, Proc. Symp. Pure Math. 46 (1987) [SD74] Saint-Donat, B.: Projective models of K3 surfaces. Am. J. Math. 96, 602–639 (1974) [Sh89] Shin, K.-H.: 3-dimensional Fano Varieties with Canonical Singularities. Tokyo J. Math. 12, 375–385 (1989) [SB92] Shepherd-Barron,N.: Miyaoka’s theorem on the seminegativity of TX . In: Flips an abundance for algebraic threefolds; Astérisque 211, 103– 114 (1992) [Sh80a] Shokurov, V.V.: Smoothness of a general anticanonical divisor on a Fano variety. Math. USSR, Izv 14, 395–405 (1980) [Sh80b] Shokurov, V.V.: The existence of a straight line on Fano 3-folds. Math. USSR Izv. 15, 173–209 (1980) [Si93] Simpson,C.: Subspaces of Moduli Spaces of Rank One Local Systems. Ann.Sc. ENS. 26, 361–401 (1993). [Vo04a] Voisin,C.: On the homotopy type of compact K¨ ahler and complex projective manifolds. Inv. Math. 157, 329–343 (2004) [Vo04b] Voisin,C.: On the homotopy type of K¨ ahler manifolds and the birational Kodaira problem. math.AG/0410040 [Wi92] Wilson, P.M.H.: The K¨ ahler cone on Calabi-Yau threefolds. Inv. Math. 107, 561–583 (1992); Erratum: 114, 231–233 (1993) [Wi97] Wilson, P.M.H.: Symplectic deformations of Calabi-Yau threefolds. J. Diff. Geom. 45, 611–637 (1997) [workshop] Bauer,T. et al.: A reduction map for nef line bundles. In: Festband zu Ehren von H.Grauert; 27–36. Springer 2002 [Zh96] Zhang, Q.: On projective manifolds with nef anticanonical bundles. J. Reine Angew. Math. 478, 57–60 (1996)

Existence of Rational Curves on Algebraic Varieties, Minimal Rational Tangents, and Applications Stefan Kebekus1 and Luis Sol´ a Conde2 1 2

Mathematisches Institut, Universit¨ at zu K¨ oln, Weyertal 86–90, 50931 K¨ oln, Germany. [email protected] Mathematisches Institut, Universit¨ at zu K¨ oln, Weyertal 86–90, 50931 K¨ oln, Germany. [email protected]

1 Introduction Over the last two decades, the study of rational curves on algebraic varieties has met with considerable interest. Starting with Mori’s landmark works [Mor79, Mor82] it has become clear that many of the varieties met daily by the algebraic geometer contain rational curves, and that a variety at hand can often be studied by looking at the rational curves it contains. Today, methods coming from the study of rational curves on algebraic varieties are applied to a broad spectrum of problems in higher-dimensional algebraic geometry, ranging from uniqueness of complex contact structures to deformation rigidity of Hermitian symmetric manifolds. In this survey we would like to give an overview of some of the recent progress in the field, with emphasis on methods developed in and around the DFG Schwerpunkt “Globale Methoden in der komplexen Geometrie”. Accordingly, there is a large body of important work that we could not cover here. Among the most prominent results of the last years is the breakthrough work of Graber, Harris and Starr, [GHS03], where it is shown that the base of the rationally connected fibration is itself not covered by rational curves3 . Another result not touched in this survey is the recent progress toward the abundance conjecture, by Boucksom, Demailly, P˘ aun and Peternell, [BDPP04] —see the article of Jahnke-Peternell-Radloff in this volume instead. 1.1 Outline of the Paper We start this survey in Sect. 2 by reviewing criteria that can be used to show that a given variety is covered by rational curves. After mentioning 3

see [Ara05] for a good introduction

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Mori’s results, we discuss foliated varieties in some detail and present a recent criterion that contains Miyaoka’s fundamental characterization of uniruledness, [Miy87], as a special case. Its proof is rather elementary, and a number of known results follow as simple corollaries. Keel-McKernan’s work [KMcK99] on rational curves on quasi-projective varieties will be briefly discussed. We will then, in Sect. 3 discuss the geometry of (higher-dimensional) varieties that are covered by rational curves, and present some ideas how minimal degree rational curves can be used to study these spaces. The hero of these sections is the “variety of minimal rational tangents”, or VMRT. In a nutshell, if X is a projective variety covered by rational curves, and x ∈ X a general point, then the VMRT is the subvariety of P(TX |∨ x ) which contains the tangent directions to minimal degree rational curves that pass through x. If X is covered by lines, this is a very classical object that has been studied by Cartan and Fubini in the past. In the general case, the VMRT is an important variety, similar perhaps to a conformal structure, whose projective geometry as a subset of P(TX |∨ x ) encodes much of the information on the underlying space X and determines X to a large extent. In Sects. 4–6 we will apply the methods and results of Sects. 2 and 3 in three different settings. To start, we discuss the moduli space of stable ranktwo vector bundles on a curve in Sect. 4. There exists a classical construction of rational curves on these spaces, the so-called “Hecke-curves”. These have been used to answer a large number of questions about moduli of vector bundles. We name a few of the applications and sketch a proof for a result that helps to give an upper bound for the multiplicities of divisors at general points of the moduli space; the result bears perhaps some resemblance with the classical Riemann singularity theorem. The existence results of Sect. 2 can also be used to study varieties for which it is known a priori that they are not covered by rational curves. We conclude this paper by giving two examples in Sects. 5 and 6. Section 5 deals with deformations of surjective morphisms f : X → Y , where the target is not covered by rational curves, or at least not rationally connected. It turns out that there exists a natural refinement of Stein factorization which factors f via an intermediate variety Z, and that the existence results of Sect. 2 can be used to show that the associated component of the deformation space Hom(X, Y ) is essentially the automorphism group of Z. In Sect. 6 we apply the results of Sect. 2 to the study of families of canonically polarized varieties. Generalizing Shafarevich’s hyperbolicity conjecture, it has been conjectured by Viehweg that the base of a smooth family of canonically polarized varieties is of log general type if the family is of maximal variation. Using Keel-McKernan’s existence results for rational curves on quasiprojective varieties, we relate the variation of a family to the logarithmic Kodaira dimension of the base and sketch a proof for an affirmative answer to Viehweg’s conjecture for families over surfaces.

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Unless explicitly mentioned, we always work over the complex number field. 1.2 Other References The reader who is interested in a broader perspective will obviously want to consult the standard reference books [Kol96] and [Deb01]. Hwang’s important survey [Hwa01] explains by way of examples how rational curves can be employed to study Fano manifolds of Picard number one. The article [AK03] contains an excellent introduction to the deformation theory of rational curves and rational connectivity, also in the more general setting of varieties defined over non-closed fields.

2 Existence of Rational Curves The modern interest in rational curves on projective varieties started with Shigefumi Mori’s fundamental work [Mor79]. In his proof of the HartshorneFrankel conjecture , he devised a new method to prove the existence of rational curves on manifolds whose tangent bundle possess certain positivity properties. Though not explicitly formulated like this, the following results appear in his papers. Theorem 1 ([Mor79, Mor82]). Let X be a complex-projective manifold. If X is Fano, i.e., if −KX is ample, then X is uniruled, i.e., covered by rational curves. More precisely, if x ∈ X is any point, then there exists a rational curve ⊂ X, such that x ∈ , and −KX · ≤ dim X + 1. Theorem 2 ([Mor82]). Let X be a complex-projective manifold. If KX is not nef, then X contains rational curves. More precisely, if C ⊂ X is a curve with KX · C < 0, and x ∈ C any point, then there exists a rational curve ⊂ X that contains x. We refer to [CKM88] or [Deb01] for an accessible introduction. These results, and the subsequently developed “minimal model program” allowed, in dimension three, to give a positive answer to a long-standing conjecture attributed to Mumford that characterizes manifolds covered by rational curves as those without pluricanonical forms. Conjecture 1. A projective manifold X is covered by rational curves if and only if κ(X) = −∞. In higher dimensions, the conjecture is still open, although the recent result of Boucksom, Demailly, Peternell and P˘ aun is considered a serious step forward. The first Chern class of TX used in Theorems 1 and 2 is, however, a rather coarse measure of positivity. For instance, if Y is a Fano manifold and Z

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a torus, then the tangent bundle of the product X = Y × Z splits into a direct sum TX = F ⊕ G where F = p∗1 (TY ) has positivity properties and identifies tangent directions to the rational curves contained in X. In general, rather than looking at KX , one would often like to deduce the existence of rational curves from positivity properties of subsheaves F ⊂ TX and relate the geometry of those curves to that of F . The most fundamental result in this direction is Miyaoka’s criterion of uniruledness. In order to state Miyaoka’s result, we recall the theorem of Mehta-Ramanathan for normal varieties. Theorem 3 (Mehta-Ramanathan, Flenner, [MR82, Fle84]). Let X be a normal variety of dimension n, and L1 , . . . Ln−1 ∈ Pic(X) be ample line bundles. If m1 , . . . , mn−1 ∈ N are large enough and Hi are general elements of the linear systems |mi · Li |, then the curve C = H1 ∩ · · · ∩ Hn−1 is smooth, reduced and irreducible, and the restriction of the HarderNarasimhan filtration of TX to C is the Harder-Narasimhan filtration of TX |C . We refer to [Lan04b, Lan04a] for a discussion and an explicit bound for the mi . A detailed account of slope, semistability and of the Harder-Narasimhan filtration of vector bundles on curves is found in [Ses82]. Definition 1. We call a curve C ⊂ X as in Theorem 3 a general complete intersection curve in the sense of Mehta-Ramanathan. Miyaoka’s result then goes as follows. Theorem 4 ([Miy87, thm. 8.5]). Let X be a normal projective variety, and C ⊂ X a general complete intersection curve in the sense of Mehta1 Ramanathan. Then ΩX |C is a semi-positive vector bundle unless X is uniruled. We refer the reader to [MP97] for a detailed overview of Miyaoka’s theory of foliations in positive characteristic. The relation between negative directions 1 of ΩX |C and tangents to rational curves has been studied in [Kol92, sect. 9]. We give a full account in Sect. 2.2. Outline of the Section In Sect. 2.1, we study the case where X is a complex manifold and F ⊂ TX is a (possibly singular) foliation. The main result —which appeared first in the preprint [BM01] of Bogomolov and McQuillan— gives a criterion to guarantee that the leaves of F are compact and rationally connected. Miyaoka’s characterization of uniruledness, Theorem 4, and the statements of [Kol92, sect. 9] follow as immediate corollaries. Apart from a simple vanishing theorem for


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Fig. 1. The points x and y are joined by a chain of rational curves of length k

vector bundles in positive characteristic, the proof employs only standard techniques of Mori theory that are well discussed in the literature. In particular, it will not be necessary to make any reference to the more involved properties of foliations in characteristic p. We also mention a sufficient condition to ensure that all leaves of a given foliation are algebraic. In Sects. 2.2 and 2.3 we discuss the relation with the rationally connected quotient. The results of Sect. 2.1 are applied to show that Q-Fano varieties with unstable tangent bundles always admit a sequence of partial rational quotients naturally associated to the Harder-Narasimhan filtration of the tangent bundle. We will later need to discuss an analog of Mumford’s Conjecture 1 for quasi-projective varieties. Here the logarithmic Kodaira dimension takes the role of the regular Kodaira dimension, and rational curves are replaced by C or C∗ . For surfaces, this setup has been studied by Keel and McKernan. We recall their results in Sect. 2.4. 2.1 Rationally Connected Foliations In the previous section we have mentioned that positivity properties of TX imply the existence of rational curves in X. Here, we will study how Mori’s ideas can be applied to foliations on complex varieties. We recall the notion of rational connectivity and fix notation first. Definition 2. A normal variety X is rationally chain connected if any two general points x, y ∈ X can be joined by a chain of rational curves, as shown in Fig. 1. The variety X is rationally connected if for any two general points x, y ∈ X there exists a single rational curve that contains both. Remark 1. For smooth varieties, the two notions rationally chain connected and rationally connected agree, see [Deb01, sect. 4.7]. Definition 3. In this survey, a foliation F on a normal variety X is a saturated, integrable subsheaf of TX . A leaf of F is a maximal F -invariant connected subset of the set X ◦ where both X and F are regular. A leaf is called algebraic if it is open in its Zariski closure. The main result of this section asserts that positivity properties of F imply algebraicity and rational connectivity of the leaves. In particular, it gives a criterion for a manifold to be covered by rational curves.

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Theorem 5. Let X be a normal complex projective variety, C ⊂ X a complete curve which is entirely contained in the smooth locus Xreg , and F ⊂ TX a (possibly singular) foliation which is regular along C. Assume that the restriction F |C is an ample vector bundle on C. If x ∈ C is any point, the leaf through x is algebraic. If x ∈ C is general, the closure of the leaf is rationally connected. The statement appeared first in the preprint [BM01] by Bogomolov and McQuillan. Below we sketch a simple proof which recently appeared in [KST06]. Remark 2. In Theorem 5, if x ∈ C is any point, it is not generally true that the closure of the leaf through x is rationally connected —this was wrongly claimed in [BM01] and in the first preprint versions of [KST06]. The classical Reeb stability theorem for foliations [CLN85, thm. IV.3], the fact that rationally connected manifolds are simply connected [Deb01, cor. 4.18], and the openness of rational connectivity [KMM92a, cor. 2.4] immediately yield the following4 . Theorem 6 ([KST06, thm. 2]). In the setup of Theorem 5, if F is regular, then all leaves are rationally connected submanifolds. In fact, a stronger statement holds that guarantees that most leaves are algebraic and rationally connected if there exists a single leaf through C whose closure does not intersect the singular locus of F , see [KST06, thm. 28]. The following characterization of rational connectivity is a straightforward corollary of Theorem 5. Corollary 1. Let X be a complex projective variety and let f : C → X be a curve whose image is contained in the smooth locus of X and such that TX |C is ample. Then X is rationally connected. Preparation for the Proof of Theorem 5: The Rational Quotient Map Before sketching a proof of Theorem 5, we recall a few facts and notations associated with the rationally connected quotient of a normal variety, introduced by Campana and Koll´ ar-Miyaoka-Mori, [Cam92, KMM92b]. See [Deb01], [Kol96] or [Ara05] for an introduction. Theorem 7 (Campana, and Koll´ ar-Miyaoka-Mori). Let X be a smooth projective variety. Then there exists a rational map q : X Q, with the following properties. 4

H¨ oring has independently obtained similar results, [Hör05].


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1. The map q is almost holomorphic5 , i.e., there exists an open set X ◦ ⊂ X such that q|X ◦ is a proper morphism. 2. If XF ⊂ X is a general fiber of q, then XF is a compact, rationally chain connected manifold, i.e., any two points in XF can be joined by a chain of rational curves. 3. If XF ⊂ X is a general fiber of q and x ∈ XF a general point, then any rational curve that contains x is automatically contained in XF . Properties (1)–(3) define q uniquely up to birational equivalence.

Definition 4. Any map q : X Q for which properties (1)–(3) of Theorem 7 hold is called a maximally rationally chain connected fibration of X, or MRCC fibration, for short. Let X be a normal projective variety, and let q : X Q be the rational map defined through the maximally rationally chain connected fibration of a desingularization of X. We call q the rationally connected quotient of X. Remark 3. If X is normal, the rationally connected quotient is again defined uniquely up to birational equivalence. Remark 4. Rational chain connectivity is not a birational invariant. For instance, a cone over an elliptic curve is rationally chain connected, while the ruled surface obtained by blowing up the vertex is not. It is therefore important in Definition 4 to pass to a desingularization. In the proof of Theorem 5, we will need two important properties of the rationally connected quotient. Theorem 8 (Universal property of the maximally rationally chain connected fibration, [Kol96, thm. IV.5.5]). Let X1 , X2 be projective manifolds, and fX : X1 X2 dominant. If qi : Xi Qi are the maximally rationally chain connected fibrations, then there exists a commutative diagram as follows. fX X 1 _ _ _/ X 2 q2 q1 Q1 _ _ _/ Q2 fQ

Theorem 9 (Graber-Harris-Starr, [GHS03]). If X is a normal projective variety and q : X Q the rationally connected quotient, then Q is not covered by rational curves.

5

[Deb01] and [Kol96] use the word “fibration” for an almost holomorphic map.

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Stefan Kebekus and Luis Sol´ a Conde ˜ Y =X ×C X C p1

-

C

KA A leaves

6 σ:=(ν,id)

p2 =:π

? ˜ C

Fig. 2. Reduction to the case of a normal foliation

Sketch of Proof of Theorem 5 Step 1: Reduction to the case where C is transversal to F Following an idea of Bogomolov and McQuillan, we consider a non-constant ˜ > 0. morphism ν : C˜ → C from a smooth curve C˜ of positive genus g(C) ˜ Let Y denote the product X × C with projections p1 and p2 and consider the following diagram, depicted in Fig. 2. p1

EY σ=(ν,id)

/X

p2

C˜

It is obviously enough to show Theorem 5 for the variety Y , the curve C := ˜ and the foliation FY := p∗ (F ) ⊂ T ˜ ⊂ TY , which is ample along C . σ(C) 1 Y |C The advantage lies in the smoothness of C , and in the transversality of FY to C . Both properties are required for the next step. Step 2: Algebraicity of the leaves Since C is everywhere transversal to FY , we can apply the classical Frobenius theorem: there exists an analytic submanifold W ⊂ Y which contains C and has the property that its fibers over C˜ are analytic open sets of the leaves of FY . Let W be the Zariski closure of W .


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Using the transversality of C and FY and the fact that FY |C is ample, a theorem of Hartshorne, [Har68, thm. 6.7], asserts that dim W = dim W . Accordingly, every leaf is algebraic. Step 2 again follows [BM01], but see also [Bos01, thm. 3.5]. Step 3: Setup of notation Replacing X by a desingularization of the normalization of W , we are reduced to prove Theorem 5 under the following extra assumptions: X is smooth, C is a smooth curve of genus g(C) > 0, and there exists a morphism π : X → C such that • π has connected fibers and is smooth along C, • the foliation F is the foliation associated to π, i.e. F = TX|C wherever π is smooth, and • π admits a section, σ : C → X. We will have to show that the general π-fiber is rationally connected. To this end, consider the rationally connected quotient q : X Z of X. The universal property of the maximally rationally chain connected fibration, Theorem 8, then yields a diagram as follows. _ _q _/ Z π t o ~β C

EX σ

(1)

We finally fix a very ample line bundle HZ on Z, and we denote by HX its pull-back to X. We can then consider q as the rational map associated to a certain linear subsystem of H 0 (X, HX ). Observe that to prove Theorem 5, it suffices to show that dim Z = 1. Namely, if dim Z = 1, then π is itself a rationally connected quotient, and its general fiber will therefore be rationally chain connected, hence rationally connected. We assume the contrary and suppose that dim Z > 1. Below we will then show the following. Proposition 1. Assume that dim(Z) ≥ 2. Then Z is uniruled with curves of HZ -degrees at most d := 2 deg σ ∗ (HX ) · dim X. This clearly contradicts Theorem 9, and concludes the proof of Theorem 5. Step 4: Strategy of proof for Proposition 1 and Theorem 5 Assume for the moment that the morphism σ admits a large number of deformations. More precisely, assume that there exists a component H ⊂ Hom(C, X) that contains σ, and an open set Ω ⊂ H such that the following holds:

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1. If T ⊂ X is any set of codimension ≤ 2, then the set of morphisms that avoid T , A := {τ ∈ Ω | τ −1 (T ) = ∅}, is a non-empty open set in Ω. 2. If τ ∈ Ω and x ⊂ C a point, then the evaluation morphism associated to the set Ωτ (x) = {τ ∈ Ω | τ (x) = τ (x)} still dominates X. If (1) and (2) hold true, choosing T to be the indeterminacy locus of q the natural morphism A → Hom(C, Z) provides a family C ⊂ Hom(C, Z) verifying: • The evaluation morphism associated to C dominates Z, and • if x ∈ C and τ ∈ C are general points, then the evaluation morphism associated to the set Cx = {τ ∈ C | τ (x) = τ (x)} still dominates Z. If dim Z ≥ 2, we can then use Mori’s Bend-and-Break argument. Definition 5. Let f : C → Z be any morphism and B ⊂ C a subscheme of finite length. The space of morphisms that agree with f on B is denoted as Hom(C, Z, f |B ). Theorem 10 (Mori’s Bend-and-Break, [MM86], [Kol91, prop. 3.3]). Let Z be a projective variety and let HZ be a nef R-divisor on Z. Let C be a smooth, projective and irreducible curve, B ⊂ Z a finite subscheme and f : C → Z a non-constant morphism. Assume that Z is smooth along f (C). If dim[f ] Hom(C, Z, f |B ) ≥ 1 then there exists a rational curve R on Z meeting f (B) and such that 2HZ · C . HZ · R ≤ #B If HZ is ample, the above inequality can be made strict.

Remark 5. Theorem 10 works for varieties defined over algebraically closed fields of arbitrary characteristic. In our situation, Theorem 10 with B = (∗) and f a general element of C would show that Z is uniruled and hence prove Proposition 1 by contradiction with Theorem 9 —for this, a simpler version of Mori’s Bend-and-Break would suffice, but we will need the full force of Theorem 10 later. Unfortunately, the ampleness of the normal bundle of C in X does not guarantee the existence of large family of deformations of σ that satisfy conditions (1) and (2). We can circumvent this problem using reduction modulo p. Step 5: Reduction modulo p In view of Mori’s standard argument using reduction modulo p, see [Mor79], [Deb01] or [CKM88], it is enough to prove Proposition 1 over an algebraically closed field k of characteristic p, for p large enough. We use a subindex k to denote the reductions modulo p of all the objects defined above, and let F denote the k-linear Frobenius morphism.


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Step 6: A vanishing result in characteristic p We briefly recall the language of Q-twisted vector bundles, as explained in [Laz04, II, 6.2]. This notion is a generalization of the concept of a Q-divisor to higher rank. It allows us to make a finer use of the positivity of a vector bundle. We identify rational numbers δ with numerical classes δ · [P ] ∈ NQ1 (C), where P is a point in C. For every δ ∈ Q, the Q-twist Eδ is defined as the ordered pair of E and δ. A Q-twisted vector bundle is said to be ample if the class c1 OPC (E) (1) + π ∗ (δ) is ample on the projectivized bundle PC (E), where π denotes the natural projection. One defines the degree deg(Eδ) := deg(E) + rank(E)δ. A quotient of Eδ is a Q-twisted vector bundle of the form E δ where E is a quotient of E. Pull-backs of Q-twisted vector bundles are defined in the obvious way. We can now formulate the following vanishing result in characteristic p that will be used later on. Proposition 2 ([KST06, prop. 9]). Let Ck be a curve defined over an algebraically closed field of characteristic p > 0. Let Ek be a vector bundle of rank r over Ck , and δ a positive rational number. Assume that Ek −δ is ample and that the “vanishing threshold” bp (δ) := pδ − 2g(C) + 1 is non-negative. Let F : Ck [1] → Ck be the k-linear Frobenius morphism. Then for every subscheme B ⊂ Ck [1] of length smaller than or equal to bp (δ) we have H 1 Ck [1], F ∗ (Ek ) ⊗ IB = {0}. Further, F ∗ (Ek ) ⊗ IB is globally generated. Proof. To prove both vanishing and global generation, it is enough to show that H 1 Ck [1], F ∗ (Ek ) ⊗ IB = {0} for #(B) ≤ bp (δ) + 1. Since F is finite, the pull-back F ∗ (Ek −δ) = F ∗ (Ek )−pδ is also ample, and so is every quotient of rank one. In particular, HomCk F ∗ (Ek ), OCk [1] (B) ⊗ ωCk [1] = {0} if deg(OC[1] (B) ⊗ ωC[1] ) ≤ pδ. The proof is concluded by applying Serre duality. Step 7: Proof in characteristic p Back to the proof of Theorem 5. We would like to apply the vanishing result of Proposition 2 to describe the space of relative deformations of τ := σk ◦ F : Ck [1] → Xk . over Ck . To this end, recall the following standard description of the Homscheme.

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Theorem 11. Let H := HomCk (Ck [1], Xk ) be the space of relative deformations of τ , and let ν ∈ H be any element. 1. If H 1 (Ck [1], ν ∗ (TXk |Ck )) = 0, then H is smooth at ν, and has dimension H 0 (Ck [1], ν ∗ (TXk |Ck )). 2. Let T ⊂ Xk be any set of codimension ≤ 2. If τ ∗ (TXk |Ck ) is globally generated and H 1 (Ck [1], τ ∗ (TXk |Ck )) = 0, then the set of morphisms whose images avoid T is a non-empty open subset of H. 3. If B ⊂ Ck is any subscheme of finite length, if ν ∗ (TXk |Ck ) ⊗ IB is globally generated and H 1 (Ck [1], ν ∗ (TXk |Ck ) ⊗ IB ) = 0, then the images of morphisms ν that agree with ν along B dominate X. By Hartshorne’s characterization of ampleness, [Laz04, Thm. 6.4.15], the ampleness of σ ∗ (TX|C ) is equivalent to the ampleness of the Qtwisted vector bundle σ ∗ (TX|C )−1/ dim X. Note also that the ampleness of σ ∗ (TX|C )−1/ dim X is preserved by the general reduction modulo p, for p sufficiently large. Apply Proposition 2 to the vector bundle σk∗ (TXk |Ck )−1/ dim X. Observe that by semicontinuity, the vanishing and global generation results obtained in Proposition 2 extend to general deformations of τ = σk ◦ F . Theorem 11 then immediately yields the following. Corollary 2. There is an open neighborhood Ω ⊂ HomCk (Ck [1], Xk ) of τ such that 1. If [ν] ∈ Ω is any morphism and B ⊂ Ck [1] any subscheme of length #(B) ≤ bp (1/ dim X), then the relative deformations of ν over Ck fixing B dominate Xk . 2. If T ⊂ X is the set of fundamental points of the birational map q then the subset Ω 0 = {[ν] ∈ Ω | (ν)−1 (Tk ) = ∅}

of morphisms whose images avoid Tk is again open in HomCk (Ck [1], Xk ).

In particular, Corollary 2 states the following: given a general element [ν] ∈ Ω 0 and a subscheme B ⊂ Ck [1] of length #(B) ≤ bp (1/ dim X), the deformations of q ◦ ν that fix B dominate Zk . Using Mori’s Bend-and-Break, Theorem 10, we obtain that Zk is uniruled in curves of HZ,k -degree at most 2 deg(q ◦ ν)∗ (HZ,k )/(bp (1/ dim X)) = 2 deg ν ∗ (HX,k )/(bp (1/ dim X)) = 2p · deg σ ∗ (HX )/(bp (1/ dim X)). The proof of Proposition 1 and Theorem 5 is finished if we note that this number is smaller than or equal to d, for p >> 0.


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2.2 An Effective Version of Miyaoka’s Criterion As an immediate corollary of Theorem 5, we deduce an effective version of Miyaoka’s characterization of uniruledness. It asserts that positive parts in the restriction of TX to a general complete intersection curve are tangent to rational curves. More precisely, we use the following definition. Definition 6. Let X be a normal projective variety, and C ⊂ X a subvariety which is not contained in the singular locus of X, and not contained in the indeterminacy locus of the rationally connected quotient q : X Q. If F ⊂ TX |C is any subsheaf, we say that F is vertical with respect to the rationally connected quotient, if F is contained in TX|Q at the general point of C. The effective version of Miyaoka’s criterion is then formulated as follows. Corollary 3. Let X be a normal complex projective variety and C ⊂ X a general complete intersection curve. Assume that the restriction TX |C contains an ample locally free subsheaf FC . Then FC is vertical with respect to the rationally connected quotient of X. This statement appeared first implicitly in [Kol92, chap. 9], but we believe there are issues with the proof, see [KST06, rem. 23]. To our best knowledge, the argument presented here gives the first complete proof of this important result. Vector Bundles over Complex Curves The proof of Corollary 3 relies on a number of facts about the HarderNarasimhan filtration of vector bundles on curves, which are known to the experts. For lack of an adequate reference we include full proofs here. To start, we show that any vector bundle on a smooth curve contains a maximally ample subbundle. Proposition 3. Let C be a smooth complex-projective curve and E a vector bundle on C, with Harder-Narasimhan filtration 0 = E0 ⊂ E1 ⊂ . . . ⊂ Er = E. Let µi := µ(Ei /Ei−1 ) be the slopes of the Harder-Narasimhan quotients. Suppose that µ1 > 0 and let k := max{ i | µi > 0}. Then Ei is ample for all 1 ≤ i ≤ k and every ample subsheaf of E is contained in Ek . Definition 7. In the setup of Proposition 3, the bundle Ek is called the maximal ample subbundle of E.

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Proof (of Proposition 3). Hartshorne’s characterization of ampleness, [Har71, thm. 2.4], says that Ei is ample iff all its quotients have positive degree. Dualizing, we have to prove that every subbundle of Ei∨ has negative degree, or, equivalently, that its maximal destabilizing subsheaf has negative slope, see [HL97, 1.3.4]. This, however, holds because the uniqueness of the HarderNarasimhan filtration of Ei∨ , [HL97, 1.3.5], implies that µmax (Ei∨ ) = −µi < 0. To show the second statement, let F ⊂ E be any ample subsheaf of E and set j := min{ i | F ⊂ Ei , 1 ≤ i ≤ r}. We need to check that j ≤ k. By the definition of j and the ampleness of F , the image of F in Ej /Ej−1 has positive slope. The semi-stability of Ej /Ej−1 therefore implies µj > 0 and j ≤ k. Proposition 3 says that the first few terms in the Harder-Narasimhan filtration are ample. The following, related statement will be used in the proof of Corollary 3 to construct foliations on X. Proposition 4. In the setup of Proposition 3, the vector bundles Ej ⊗ . ∨ E E are ample for all 0 < j ≤ i < r. In particular, if Ei is any ample i . term in the Harder-Narasimhan Filtration of E, then Hom Ei , E Ei and . Hom Ei ⊗ Ei , E Ei are both zero. Remark 6. If X is a polarized manifold whose tangent bundle contains a subsheaf of positive slope, Proposition 4 shows that the first terms in the Harder-Narasimhan filtration of TX are special foliations in the sense of Miyaoka, [Miy87, sect. 8]. By [Miy87, thm. 8.5], this already implies that X is dominated by rational curves that are tangent to these foliations. Proof (of Proposition 4). As a first step, we show that the vector bundle . ∨ . Fi,j := Ej Ej−1 ⊗ E Ei is ample. Assume not. Then, by Hartshorne’s ampleness criterion [Har71, prop. 2.1(ii)], there exists a quotient A of Fi,j of degree degC A ≤ 0. Equivalently, there exists a non-trivial subbundle ∨ . . ∨ = Ej Ej−1 ⊗ E Ei α : B → Fi,j with degC B ≥ 0. Replacing B by its maximally destabilizing subbundle, if necessary, we can assume without loss of generality that B is semistable. In particular, B has non-negative slope µ(B ≥ 0. On the other hand, we have . that Ej Ej−1 is semistable. The slope of the image of the induced morphism . . B ⊗ Ej Ej−1 → E Ei


373

. . will thus be larger than µmax E Ei = µ Ei+1 Ei . This shows that α must be zero, a contradiction which proves the amplitude of Fi,j . With this preparation we will now prove Proposition 4 inductively. If j = 1, then the above claim and the statement of Proposition 4 agree. Now let 1 < j ≤ i < r and assume that the statement was already shown for j − 1. Then consider the sequence . . . . 0 → Ej−1 ⊗ E Ei )∨ → Ej ⊗ E Ei )∨ → ( Ej Ej−1 ) ⊗ ( E Ei )∨ → 0 ample

ample

But then also the middle term is ample, which shows Proposition 4.

Proof of Corollary 3 We will show that the sheaf FC , which is defined only on the curve C is contained in a foliation F which is regular along C and whose restriction to C is likewise ample. Corollary 3 then follows immediately from Theorem 5. An application of Proposition 3 to E := TX |C yields the existence of a locally free term Ei ⊂ TX |C in the Harder-Narasimhan filtration of TX |C which contains FC and is ample. The choice of C then guarantees that Ei extends to a saturated subsheaf F ⊂ TX . The proof is thus finished if we show that F is a foliation, i.e. closed under the Lie bracket. Equivalently, we need to show that the associated O’Neill tensor6 . N : F ⊗ F → TX F vanishes. By Proposition 4, the restriction of the bundle . . Hom F ⊗ F, TX F ∼ = (F ⊗ F)∨ ⊗ TX F to C is anti-ample. In particular, . N |C ∈ H 0 C, Hom F ⊗ F, TX F |C = {0}. Ampleness is an open property, [Gro66, cor. 9.6.4], so that the restriction of N to deformations (Ct )t∈T of C stays zero for most t ∈ T . Since the Ct dominate X, the claim follows. This ends the proof of Corollary 3. 2.3 The Stability of the Tangent Bundle, and Partial Rationally Connected Quotients Recall that a complex variety X is called Q-Fano if a sufficiently high multiple of the anticanonical divisor −KX is Cartier and ample. The methods introduced above immediately yield that Q-Fano varieties whose tangent bundles are unstable allow sequences of rational maps with rationally connected fibers. 6

The Lie bracket is of course not OX -linear. However, an elementary computation show that N is well-defined and linear.

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Corollary 4. Let X be a normal complex Q-Fano L1 , . . . , Ldim X−1 ∈ Pic(X) be ample line bundles. Let

variety

and

{0} = E−1 = E0 ⊂ E1 ⊂ · · · ⊂ Em = TX be the Harder-Narasimhan filtration of the tangent sheaf with respect to L1 , . . . , Ldim X−1 and set k := max{0 ≤ i ≤ m | µ(Ei /Ei−1 ) > 0}. Then k > 0, and there exists a commutative diagram of dominant rational maps ··· (2) X X X q1 q2 qk _ _ _ / _ _ _ / _ _ _ / ··· Q2 Qk , Q1 with the following property: if x ∈ X is a general point, and Fi the closure of the qi -fiber through x, then Fi is rationally connected, and its tangent space at x is exactly Ei , i.e., TFi |x = Ei |x . Proof. Let C ⊂ X be a general complete intersection curve with respect to L1 , . . . , Ldim X−1 . Since c1 (TX ) · C > 0, Proposition 3 implies k > 0 and that the restrictions E1 |C , . . . , Ek |C are ample vector bundles. We have further seen in Theorem 5 that the (Ei )1≤i≤k give a sequence of foliations with algebraic and rationally connected leaves. To end the construction of Diagram (2), let qi : X Chow(X) be the map that sends a point x to the Ei -leaf through x, and let Qi := Image(qi ). Remark 7. Corollary 4 also holds in the more general setup where X is a normal variety whose anti-canonical class is represented by a Weil divisor with positive rational coefficients. Open Problems It is of course conjectured that the tangent bundle of a Fano manifold X with b2 (X) = 1 is stable. We are therefore interested in a converse to Corollary 4 and ask the following. Question 1. Given a Q-Fano variety and a sequence of rational maps with rationally connected fibers as in Diagram (2), when does the diagram come from the unstability of TX with respect to a certain polarization? Is Diagram (2) characterized by universal properties? Question 2. To what extent does Diagram (2) depend on the polarization chosen? Question 3. If X is a uniruled manifold or variety, is there a polarization such that the rational quotient map comes from the Harder-Narasimhan filtration of TX ?


375

2.4 Rational Curves on Quasi-projective Manifolds Quasi-projective varieties appear naturally in a number of settings, e.g., as moduli spaces or modular varieties. In this setup, it is often not reasonable to ask if a given variety S ◦ contains complete rational curves7 . Instead, one is interested in hyperbolicity properties of S ◦ , i.e., one asks if there are nonconstant morphisms C → U , or if S ◦ is dominated by images of such morphisms —we refer to [Siu04] for a general discussion, and to Sect. 6 for a very brief overview of the hyperbolicity question for moduli of canonically polarized manifolds and for applications. In this respect, a famous conjecture attributed to Miyanishi8 suggests that the following logarithmic analog of Conjecture 1 holds true. Conjecture 2 (Miyanishi). Let S ◦ be a smooth quasi-projective variety. Then κ(S ◦ ) = −∞ if and only if S ◦ is dominated by images of C. We briefly recall the definition of the logarithmic Kodaira dimension. Definition 8. Let S ◦ be a smooth quasi-projective variety and S a smooth projective compactification of S ◦ such that D := S \ S ◦ is a divisor with simple normal crossings. The logarithmic Kodaira dimension of S ◦ , denoted by κ(S ◦ ), is defined as the Kodaira-Iitaka dimension κ(KS + D) of the line bundle OS (KS + D) ∈ Pic(S). The variety S ◦ is called of log general type if κ(S ◦ ) = dim S ◦ , i.e., if the divisor KS + D is big. It is a standard fact in logarithmic geometry that a compactification S with the described properties exists, and that the logarithmic Kodaira dimension does not depend on the choice of the compactification, [Iit82, chap. 11]. Conjecture 2 was studied by Miyanishi and Tsunoda in [MT83, MT84] and a number of further papers, and by Zhang [Zha88]. Complete results for surfaces were obtained by Keel-McKernan, as follows. Theorem 12 ([KMcK99, thm. 1.1]). Let S ◦ be a smooth quasi-projective variety of dimension at most two. Then κ(S ◦ ) = −∞ if and only if S ◦ is dominated by images of C. Keel-McKernan also give conditions that guarantee that U is dominated by images of C∗ . Theorem 13 ([KMcK99, prop. 1.4]). Let S be a normal projective surface, let D ⊂ S be a reduced curve, and set U := S \ (D ∪ Sing(S)). Consider the following conditions: 7 8

But see [KMcK99, cor. 5.9] for criteria that can sometimes be used if S ◦ can be compactified by a finite number of points. See [GZ94] for a partial case.

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1. KS + D is numerically trivial, but not log-canonical. 2. KS + D is numerically trivial, and D = ∅. 3. KS is numerically trivial, D = ∅, and S has a singularity which is not a quotient singularity. If any of the above hold, then U is dominated by images of C∗ .

Open Problems Theorems 12 and 13 were shown in [KMcK99] using deformation theory on non-separated algebraic spaces and a rather involved case-by-case analysis of possible curve configurations on S. The analysis alone covers more than a hundred pages, and a generalization to higher dimensions seems out of the question. We would therefore like to pose the following problem. Problem 1. Find a more conceptual proof of Theorem 12, perhaps using methods introduced in Sect. 2.1.

3 Geometry of Rational Curves on Projective Manifolds In Sect. 2 we have reviewed criteria to guarantee that a given variety is covered by rational curves. In this section, we assume that we are given a variety X that is covered by rational curves, and study the geometry of curves on X in more detail. Outline of the Section In Sect. 3.1 we review a number of known concepts concerning rational curves on X. In particular, we give the definition of a family of rational curves and fix the notation that will be used later on. Section 3.2 deals with the locus of singular curves of a dominating family of minimal rational curves. We apply the results of [Keb02b] in Sect. 3.3 to ensure the existence and finiteness of the tangent morphism, which maps every minimal rational curve passing by a point x to its tangent direction at x. The image, i.e., the set of tangent directions to which there exists a minimal degree rational curve, is called variety of minimal rational tangents, or VMRT. Recently, Hwang and Mok have shown that the general minimal degree rational curve on X is uniquely determined by its tangent direction at a point. This result gives more information about the VMRT. We sketch their argument in Sect. 3.4. We have claimed in the introduction that the VMRT determines the geometry of X to a large degree. In order to substantiate this claim, we mention, in Sect. 3.5, a number of results in that direction. One particular result, an estimate for the minimal number of rational curves required to connect two points on a Fano manifold, is explained in Sect. 3.6 in more detail.


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3.1 The Space of Rational Curves, Setup of Notation We begin by defining some well known concepts about parameter spaces of rational curves in projective varieties. We refer the reader to [Kol96, II.2] for a detailed account. Definition 9. Let X be a normal complex projective variety, and RatCurves(X) ⊂ Chow(X) be the quasi-projective subvariety whose points correspond to irreducible and generically reduced rational curves in X. Let RatCurvesn (X) be its normalization and U the normalization of the universal family over RatCurves(X). We obtain a diagram as follows. U

ι evaluation morphism

/X

(3)

π

RatCurvesn (X) Remark 8. The normalization morphism RatCurvesn (X) → Chow(X) is finite and generically injective, but not necessarily injective. It is therefore possible that points in RatCurvesn (X) are not in 1:1 correspondence with actual curves in X. We use the notation [] to denote points in RatCurvesn (X), and for the associated curves. A standard cohomological argument for families of irreducible and generically reduced rational curves shows that the morphism π is in fact a P1 -bundle. The space of rational curves is often described in terms of the Hom-scheme. The following theorem establishes the link. Theorem 14 ([Kol96, II thm. 2.15]). There exists a diagram as follows univ. morphism

Homnbir (P1 , X) × P1

quotient by natl. action of Aut(P1 )

ι evaluation morphism

/* X

π P1 −bundle

projection

Homnbir (P1 , X)

/U

quotient by natl. action of Aut(P1 )

/ RatCurvesn (X)

where Hombir (P1 , X) ⊂ Hom(P1 , X) is the scheme parametrizing birational morphisms from P1 to X and Homnbir (P1 , X) its normalization. Definition 10. A maximal family of rational curves is an irreducible component H ⊂ RatCurvesn (X). A maximal family H is called dominating, if ι|π−1 (H) dominates X. A dominating family H is a dominating family of rational curves of minimal degrees if the degrees of the associated rational curves on X are minimal among all dominating families.

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Example 1. Let X ⊂ P3 be a general cubic surface. It is classically known that X contains 27 lines, and that there exists a dominating family of conics, i.e., rational curves of degree 2, on X. In this case, the family of conics would be a dominating family of rational curves of minimal degree. Example 1 shows that a dominating family of rational curves of minimal degrees needs not be proper —there are sequences of conics on X whose limit cycle is a union of two lines. It is, however, true that given a sequence of rational curves on any X, the limit cycle is composed of rational curves. This observation immediately gives the following properness statement for families of curves of minimal degrees. Theorem 15 ([Kol96, IV cor. 2.9]). Let H ⊂ RatCurvesn (X) be a dominating family of rational curves of minimal degrees. If x ∈ X is a general point, then the subspace of curves through x, Hx := π ι−1 (x) ∩ H = { ∈ H | x ∈ } is proper.

Definition 11. Let H ⊂ RatCurvesn (X) be a dominating family of rational curves. If H is proper, it is often called unsplit. If for a a general point x ∈ X the associated subspace Hx is proper, the family is called generically unsplit. One of the key tools in the description of rational curves on a manifold X is an analysis of the restriction of TX to the rational curves in question —recall that any vector bundle on P1 can be written as a sum of line bundles. The following proposition summarizes the most important facts. Proposition 5 ([Deb01, Prop. 4.14], [Kol96, IV Cor. 2.9]). Let X be a smooth complex projective variety and H ⊂ RatCurvesn (X) be a dominating family of rational curves of minimal degrees. If x ∈ X is a general point and [] ∈ Hx any element, with normalization η : P1 → , then the following holds. • • • •

H is smooth at []. ˜ x of Hx is smooth. The normalization H is free, i.e. the vector bundle η ∗ (TX ) is nef. If [] is a general point of Hx , then is standard, i.e., there exists a number p such that η ∗ (TX ) ∼ = OP1 (2) ⊕ OP1 (1)⊕p ⊕ OP1

⊕ dim(X)−1−p

.


379

Open Problems Problem 2. To what extent are dominating families of rational curves of minimal degrees characterized by their generic unsplitness? Problem 3. Let X be smooth and H be a dominating family of rational curves of minimal degrees. Assume that for general x ∈ X, the space Hx is positive dimensional. Is it true that Hx is irreducible? See [KK04, sect. 5.1] for a partial case.

3.2 Singular Rational Curves Let X ⊂ Pn be a projective manifold that is covered by lines. If x ∈ X is any point, we can consider the space Cx ⊂ P(TX |∨ x ) of tangent directions that are tangent to lines through x. The so-defined variety of minimal rational tangents is a very classical and important object that has been studied in the past by Cartan and Fubini, and a number of other projective differential geometers. We will, in this section, give a similar construction for families of rational curves of minimal degrees. The main obstacle is that these curves may be singular, which makes it difficult to properly define tangents to them. It is, however, well understood that minimal degree curves have only mild singularities at the general point of X. Definition 12. We say that a curve C is immersed, if its normalization morphism C˜ → C has rank 1 at every point. Theorem 16 ([Keb02b, thm. 3.3]). Let X be normal and H ⊂ RatCurvesn (X) a dominating family of rational curves of minimal degrees. Further, let x ∈ X be a general point, and consider the closed subvarieties HxSing := {[] ∈ Hx | is singular} and HxSing,x := {[] ∈ Hx | is singular at x}. Then the following holds. 1. The space HxSing has dimension at most one, and the subspace HxSing,x is at most finite. Moreover, if HxSing,x is not empty, the associated curves are immersed. 2. If there exists a line bundle L ∈ Pic(X) that intersects the curves with multiplicity 2 then HxSing is at most finite and HxSing,x is empty. The main idea in the proof of Theorem 16 is the observation that an arbitrary family of singular rational curves, like any family of higher genus curves, is hardly ever projective —see [Keb02c] for worked examples. An analysis of the projectivity condition yields the statement.

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Sketch of Proof of Theorem 16 As usual, we subdivide the proof into several steps. We will only give an idea how to show that the subspace HxSing,x is at most finite. Step 1: Dimension count We assume that HxSing,x is not empty because otherwise there is nothing to prove. A technical dimension count —which we are not going to detail in this sketch— shows that dim HxSing ≥ dim HxSing,x + 1. Thus, the assumption implies that dim HxSing ≥ 1. We fix a proper 1dimensional subfamily H ⊂ HxSing . Step 2: A partial resolution of singularities Recall that H ⊂ RatCurvesn (X) has a natural morphism into the Chow variety of X. Let π : U → H be the pull-back of the universal family. We aim to replace U by a family where all fibers are singular plane cubics. For that, consider the normalization diagram. / U

η

˜ U

normalization

π ˜

'

H

w

(4)

π

After performing a series of finite base changes, if necessary, we can assume that the following holds: H is smooth. ˜ is a P1 -bundle over H —see [Kol96, thm. II.2.8]. U contained in the singular locus of U such There exists a curve s ⊂ USing that π|s is an isomorphism. For this, let s be the normalization of a suitable . component of USing ˜ -fibers is of 4. There exists a subscheme s˜ ⊂ η −1 (s) whose restriction to all π length 2. For this, let s˜ be the normalization of a curve in Hilb2 (η −1 (s)/H ) and note that the relative Hilb-functor commutes with base change.

1. 2. 3.

We would like to extend the diagram (4) to η / ˜ M U normalization p8 U MM p MM pp p ∃α ∃β MM & pp ˆ U π π ˜ ∃ˆ π # { H


ˆ U

˜ U α

U

/

β

P1 -bundle

π ˆ

)

/ family of very singular curves

family of plane cubics π ˜

381

π

w

H

Fig. 3. Replacing singular curves by plane cubics

where all fibers of π ˆ are rational curves with a single cusp or node, i.e., isomorphic to a plane cubic. Figure 3 depicts this setup. Here we explain only how to do this locally. ˜ is a P1 -bundle over H , we find an (analytic) open set Knowing that U ˜ −1 (V ) with V × C, V ⊂ H with coordinate v, identify an open subset of π choose a bundle coordinate u and write s˜ = {u2 = f (v)} where f is a function on V . We would then define α to be α : V × C −→ V × C2 (v, u) → (v, u2 − f (v), u(u2 − f (v))) A direct calculation shows that these locally defined morphisms glue together ˜ →U ˆ , that a morphism β : U ˆ → U exists to give a global morphism α : U and that the induced map π ˆ := π ◦ β has the desired properties. Step 3: Ruling out several cases In order to conclude in the next step, we have to rule out several possibilities ˆ . We do this in every case by a reduction to the absurd. for the geometry of U Step 3a: The case where all curves are immersed If all curves associated with H are immersed, then the construction outlined ˆ where all fibers are isomorphic to above will automatically give a family U ˜ be the section which is connodal plane cubics —see Fig. 4. Let σ∞ ⊂ U tracted to the point x ∈ X (drawn as a solid line) and consider the preimage ˆSing ). After another finite base change, if necesof the singular locus α−1 (U sary, we may assume that this set decomposes into two disjoint components

382


ˆ U

˜ U α normalization

/

P1 -bundle

family of immersed curves π ˜

% w

π ˆ

H

Fig. 4. Family of immersed curves

ˆSing ) = σ0 ∪ σ1 , drawn as dashed lines. That way we obtain three secα−1 (U ˜ , where σ∞ can be contracted to a point and σ0 , tions σ0 , σ1 and σ∞ in U σ1 are disjoint. But then it follows from an elementary calculation with intersection numbers that either σ0 = σ∞ or that σ1 = σ∞ . This however, is impossible, because then the Stein factorization of ιx ◦ η would contract both σ0 and σ1 , but ruled surfaces allow at most a single contractible section. Remark 9. This setup has already been considered by several authors. See e.g. [CS95]. Step 3b: The case where no curve is immersed If none of the curves associated with H is immersed, then the curves s and s˜ ˆ → H is a family of cuspidal plane in the construction can be chosen so that U ˜ be the section which can be contracted cubics, see Fig. 5. Again, let σ∞ ⊂ U and let σ0 be the preimage of the singularities. That way we obtain two sections. In order to obtain a third one, remark that if C ⊂ P2 is a cuspidal plane cubic and H ∈ Pic(C) a line bundle of positive degree k > 0, then there exists a unique point9 y ∈ C, contained in the smooth locus of C, such that OC (ky) ∼ = H. Thus, using the pull-back of an ample line bundle L ∈ β ∗ Pic(X), we obtain a third section σ1 which is disjoint from σ0 . Now conclude as above. This time, however, it is not obvious that neither σ0 nor σ1 coincides with σ∞ . Actually, this is true because x ∈ X was chosen to be a general point. The proof of this is rather technical therefore omitted.

9

If H = OP2 (1)|C , this is the classical inflection point. In general, this will be the hyperosculating point associated with the embedding given by H.


383

ˆ U

˜ U α normalization

/

P1 -bundle

family of non-immersed curves π ˜

% w

π ˆ

H

Fig. 5. Family of non-immersed curves

Step 4: End of proof To end the proof, we argue by contradiction and assume that dim HxSing,x ≥ 1. We have seen in Step 1 that this implies dim HxSing ≥ 2. The argumentation of Step 3 implies that points of dim HxSing ≥ 2 correspond to both immersed and non-immersed singular rational curves in X. Elementary deformation theory, however, shows that the closed subfamily of non-immersed curves is always of codimension at least one. Thus, the subfamily HxSing,ni ⊂ HxSing of non-immersed curves is proper and positivedimensional. This again has been ruled out in Step 3b. Open Problems Problem 4. Examples show that Theorem 16 is optimal for normal varieties X. It is not clear to us if Theorem 16 can be improved if X is smooth. 3.3 The Tangent Morphism and the Variety of Minimal Rational Tangents We apply Theorem 16 to show the existence and finiteness of the tangent morphism, an important tool in the study of uniruled manifolds that encodes the infinitesimal behavior of H near a general point x ∈ X. Definition 13. Given a dominating family H of rational curves of minimal degrees on X, and a general point x ∈ X we consider a rational tangent map tx : Hx P(TX |∨ x) → P(T |∨ x) associating to a curve through x its tangent direction at x.

384


Theorem 17 ([Keb02b, thm. 3.4]). With the notation as above, let x ∈ X ˜ x be the normalization of Hx , and let be a general point, let H ˜ x P(TX |∨ τx : H x) be the composition of tx with the normalization morphism. Then τx is a finite morphism, called tangent morphism. Proof. Consider the pull-back of Diagram (3) from page 377, Ux

ιx evaluation

/X

πx

˜x H

∼ Theorem 16 asserts that the preimage ι−1 x (x) contains a reduced section σ∞ = ˜ Hx , and at most finitely many points. Since all curves are immersed at x, the tangent morphism of ιx gives a nowhere vanishing morphism of vector bundles, T ιx : TUx |H˜ x |σ∞ → ι∗x (TX |x ).

(5)

The tangent morphism is then given by the projectivization of (5). Assuming that τx is not finite, Equation (5) asserts that we can find a curve ˜ x such that Nσ ,U is trivial along C. But σ∞ can be contracted, and C⊂H ∞ x the normal bundle must thus be negative. With these preparations, we can now introduce one of the central objects of this survey, the variety of minimal rational tangents, or VMRT. ˜ x) ⊂ Definition 14. If x ∈ X is a general point, we call the image Cx := τx (H ) the variety of minimal tangents of H at x. The subvariety P(TX |∨ x ∨ Cx ⊂ P(TX ) C := closure of x general in X is called the total variety of minimal rational tangents of H. Remark 10. The projectivized tangent map of the evaluation morphism yields a diagram ∨ i e 1 C ⊂ P(TX ) τ m q v ρ, projection { ι /X U evaluation

that we will later also use to describe the tangent map τx at a general point. We call τ the global tangent map.


385

The variety of minimal rational tangents has been extensively studied by several authors, including Hwang and Mok. It can be computed in a number of examples of uniruled varieties, such as Fano hypersurfaces, rational homogeneous spaces and moduli spaces of vector bundles. We refer the reader to [Hwa01] for examples, but see also Sect. 4 below. For varieties covered by lines, the situation is particularly easy. Proposition 6 ([Hwa01, prop. 1.5]). Suppose that there exists an embedding X ⊂ PN such that the curves associated with points of H are lines. If x ∈ X is a general point, then τx is an embedding and Cx is smooth. If [] ∈ Hx is any line x, then the+ projective tangent space to Cx through ∨ , where TX | ⊂ TX | is the maximal ample at P(T |x ) is exactly P (TX |+ ) subbundle. In general, there is a direct and well understood relation between the splitting type of TX |+ and the Zariski tangent space of Cx . See [Hwa01, sect. 1] for details. Open Problems Problem 5. In all smooth examples that we are aware of, the variety Cx is smooth, and has good projective-geometrical properties. What can be said in general? See [Hwa01] for more detailed lists of problems. 3.4 Birationality of the Tangent Morphism We will later see that the projective geometry of the VMRT Cx ⊂ P(TX |∨ x) encodes a lot of the geometrical properties of X. One of the main difficulties in the applications is that the variety Cx of minimal rational tangents might be singular or reducible. To overcome this difficulty one often studies the tangent ˜ x → Cx . Since H ˜ x is smooth, one asks if τx is injective, and morphism τx : H if it has maximal rank —this question can sometimes be answered in the examples. In general, it has been shown by Hwang and Mok that the normalization of Cx is smooth. This is a direct consequence of Proposition 5 and the main result of [HM04]. Theorem 18 ([HM04, thm. 1]). With the same notation as above, the global tangent map τ is generically injective. In particular, the normalization of Cx is smooth for general x ∈ X. See [KK04] for related criteria to guarantee that τx is in fact injective. Remark 11. A line in Pn is specified by giving a point x ∈ Pn and a tangent direction at x. Theorem 18 says that a similar statement holds for minimal degree curves. See [Keb02b, sect. 3.3] for the related question if a minimal degree curve can be specified by two points.

386


∨ Theorem 18 was known to be true in the case where Cx = P(TX |x ), where n it follows as a by-product of a characterization of P .

Theorem 19 ([CMSB02], [Keb02a]). Let X be an irreducible normal projective variety of dimension n. Let H ⊂ RatCurvesn (X) be a dominating fam∨ ). Then ily of rational curves of minimal degrees and assume that C = P(TX n there exists a finite morphism P → X, étale over X \Sing(X) that maps lines in Pn to curves parametrized by H. In particular, X ∼ = Pn if X is smooth. Hwang and Mok have applied the theory of differential systems to C in order to reduce the general case to that of Theorem 19. We give a very rough sketch of the proof and refer to [HM04] for details. Sketch of Proof of Theorem 18 Step 1: Differential systems in uniruled varieties Definition 15. A distribution on X is a saturated subsheaf of TX . Given a distribution D, its Cauchy characteristic distribution is the integrable subdistribution Ch(D) ⊂ D whose fiber at the general point x ∈ X is Ch(D)x := {v ∈ Dx ; N (v, Dx ) = 0} , where N denotes the O’Neill tensor , i.e., the OX -linear map N : D ⊗ D → TX /D induced by the Lie bracket. ∨ On C ⊂ P(TX ) we can consider a natural distribution P defined in the general point α ∈ C by: Pα := (T ρ)−1 (TCˆx,α ),

where TCˆx,α ⊂ TX |x is the tangent space to the affine cone of Cx along the ray C · α ⊂ TX |x determined by α. The following proposition is the technical core of Theorem 18. It is based in a detailed study of the distribution P and its relation with the family H, which is beyond the purpose of this survey. We refer to [HM04] for a detailed account. Proposition 7. With notation as above, the following holds. 1. The distribution Ch(P) ⊂ TC contains the tangent directions of the images in C of the curves parametrized by H. More precisely, if [] ∈ H is a general point, then the morphism τ |π−1 ([]) : P1 → C is immersive and its image is tangent to Ch(P). 2. Let y ∈ C be a general point and S the associated leaf of Ch(P). Then S is algebraic, and there exists a dense open set S0 such that W := π(S0 ) ⊂ X is quasi-projective, and such that π|S0 : S0 → W ∨ is a bundle of projective spaces, isomorphic to P(TW ).


387

3. Let W ⊂ X be the Zariski closure of W . The subvariety / 0 HW := [] ∈ H ⊂ W is a dominating family of rational curves of minimal degrees10 in W .

The subschemes of the form W are called Cauchy subvarieties of X with respect to H. Step 2: End of sketch of proof Let x ∈ X be a general point and c ∈ Cx ⊂ P(TX |∨ x ) be a general minimal rational tangent. We assume to the contrary and take two general curves [1 ] and [2 ] ∈ Hx that have c as tangent direction. Let S be the leaf of Ch(P) that contains c and let W ⊂ X be the corresponding Cauchy subvariety. By Proposition 7.(1), W contains both 1 and 2 . By Proposition 7.(2) and (3), the tangent map associated with HW , ∨ τW,x : HW,x P(TW |∨ x ) ⊂ P(TX |x ),

is surjective. Theorem 19 then applies to the normalization of W and asserts that τW must be birational. By general choice, 1 and 2 must then be equal, a contradiction. 3.5 The Importance of the VMRT Given a uniruled variety X and a dominating family H ⊂ RatCurvesn (X) of rational curves of minimal degrees, we have claimed that geometry of X is determined to a large degree by the projective geometry of the associated ∨ ). In this section we would like to name a few results that VMRT C ⊂ P(TX support the claim. Sect. 3.6 discusses one particular example in more detail. In view of the Minimal Model Program, we restrict ourselves to Fano manifolds of Picard number 1. In this case, Hwang and Mok have shown under some technical assumptions, that X is completely determined by the family of minimal rational tangents over an analytic open set. This “Cartan-Fubini” type result is stated as follows. Theorem 20 ([HM01]). Let X and X be Fano manifolds of Picard number 1 defined over the field of the complex numbers, and let H and H be dominating families of minimal rational curves in X and X , respectively. Let Cx and Cx denote the associated VMRT at x ∈ X and x ∈ X . Assume that Cx is positive-dimensional and that the Gauss map of the embedding Cx ⊂ P(ΩX |x ) 10

Strictly speaking, we have defined the notion of a family of rational curves only for normal varieties because the notion of the Chow-variety and its universal property is a little delicate for non-normal spaces. Although W need not be normal, we ignore this (slight) complication in this sketch for simplicity.

388


is finite for general x ∈ X. Then any biholomorphic map φ : V → V between analytic open neighborhoods V ⊂ X and V ⊂ X inducing an isomorphism for all x ∈ V can be extended to a biholomorphic map between Cx and Cφ(x) Φ:X→X . A detailed proof of Theorem 20 is given in the survey article [Hwa01]. There it is also discussed under what conditions even stronger results can be expected. The VMRT has also been used to attack the following problems. Stability of the tangent bundle: For Fano manifolds of Picard number 1, this property can be easily restated in terms of projective properties of Cx ⊂ P(TX |∨ x ) for general x, which can be checked in some cases, see e.g. [Hwa98, Hwa02] for low-dimensional Fano manifolds and moduli spaces of vector bundles, or [Keb05] for contact manifolds. Deformation rigidity: The VMRT have been used by Hwang and Mok to prove deformation rigidity of various types of varieties and morphisms. See, e.g., [Hwa97, Hwa01] or [HM98]. Uniqueness of contact structures: It has been shown in [Keb01] that contact structures on Fano manifolds of Picard number 1 are unique since their projectivization at a point coincides with the linear span of the VMRT. The Remmert–Van de Ven / Lazarsfeld problem: Theorem 20 can be used to classify smooth images of surjective morphisms from rational homogeneous spaces of Picard number 1, see [HM99]. 3.6 Higher Secants and the Length of a Uniruled Manifold As before, let X be a complex projective manifold and H ⊂ RatCurvesn (X) be a dominating family of rational curves of minimal degrees. If b2 (X) = 1, then X is rationally connected —see Definition 2 on page 363. In particular, if x, y ∈ X are any two general points, there exists a connected chain of rational curves 1 , 2 , . . . , k ∈ H such that x ∈ 1 , y ∈ k —see Fig. 1 on page 363. The number k, i.e., the minimal length of chains of H-curves needed to connect two general points is called the length of X with respect to H. The length is an important invariant; among other applications, it was used by Nadel in the proof of the boundedness of the degrees of Fano manifolds of Picard number one, [Nad91]. The main aim of this section is to introduce an effective method that allows to compute the length for a number of interesting varieties. We relate the length of X to the projective geometry of the variety of minimal rational tangents. The length of X can then be computed in situations where the secant defect of the VMRT is known. We will employ these results in Sect. 4 below to give a bound on the multiplicities of divisors at a general point of the moduli of stable bundles of rank two on a curve.


smooth

-

nodal

389

x

-

smooth

-

Fig. 6. Nodal curves causing reducibility of Cx1

Statement of Result To formulate the result precisely, let us fix our assumptions first. For the remainder of the present section, let X be a complex projective manifold of arbitrary Picard number and H ⊂ RatCurvesn (X) a dominating family of rational curves of minimal degrees, as in Sect. 3.1. If x ∈ X is a general point, assume additionally that the space Hx is irreducible11 . We also need to consider a few following auxiliary spaces. Definition 16. Consider the irreducible subvarieties loc1 (x) := closure of C,

and

general C ∈ Hx

lock+1 (x) := closure of

C.

general C ∈ Hy general y ∈ lock (x)

Set dk := dim(lock (x)); we call it the k-th spanning dimension of the family k H. Finally, let Cxk ⊂ P(TX |∨ x ) be the tangent cone to loc (x) at x. Remark 12. Even though Cx is assumed to be irreducible, Cx1 can have several components. This might be the case if some of this curves associated with points in Hx are nodal, see Fig. 6. It is clear that the spanning dimensions of H do not depend on the choice of the general point x. The tangent cones Cxk ⊂ P(TX |∨ x ) have pure dimension dim Cxk = dk − 1 —we refer to [Har95, lect. 20] for an elementary introduction to tangent cones. Remark 13. The spanning dimensions are natural invariants of the pair (X, H). In many situations, however, there is a canonical choice of H, so that we can actually view the dk as invariants of X. 11

The irreducibility assumption is posed for simplicity of exposition. It holds for all examples we will encounter. See [KK04, thm. 5.1] for a general irreducibility criterion.

390


Definition 17. If there exists a number l > 0 such that dl = dim(X) and dl−1 < dim(X), we call X rationally connected by H-curves, write lengthH (X) = l and say that X has length l with respect to H. Remark 14 ( [Nad91]). If there exists a number k such that dk = dk+1 , it is clear from the definition that lock (x) = lock+1 (x) = lock+2 (x) = · · · . In particular, if X is rationally connected by H-curves, then lengthH (X) ≤ dim X. Definition 18 ([CC02, Zak93]). Let S k Cx be the k-th secant variety of Cx , i.e., the closure of the union of k-dimensional linear subspaces of P(TX |∨ x) determined by general k + 1 points on Y . With this notation, the main result is formulated as follows. Theorem 21 ([HK05, thms. 3.11–14]). Let X be as above. Then the following holds. 1. We have Cx ⊂ Cx1 ⊂ S 1 Cx . If none of the curves of H has nodal singularities at x then Cx = Cx1 . 2. For each k ≥ 1, we have S k Cx ⊂ Cxk+1 . In particular, dk+1 ≥ dim(S k Cx ) + 1. 3. If X admits an embedding X → PN such that the curves parametrized by H are mapped to lines, then S 1 Cx = Cx2 , and so d2 = dim(S 1 Cx ) + 1. We will give a rough sketch of a proof below. In view of Theorem 21, one might be tempted to conjecture that Cxk+1 = S k Cx . Remark 12 and the following remark show that this is not the case. Remark 15. The inequality in Theorem 21.(2) can be strict. For an example, let X ∼ Pn be a Fano manifold with Picard number one which carries a = complex contact structure12 , and H a dominating family of rational curves of minimal degrees. In this setup, Cx is linearly degenerate. In particular, we have that dim(S k Cx ) + 1 < dim X = lengthH (X) for all k. Generalizing Nadel’s product theorem, [Nad91], the knowledge of the length can be used to bound multiplicities of sections of vector bundles at general points of X, as follows. Proposition 8 ([HK05, prop. 2.6]). Assume that X is rationally connected by curves of the family H and let V be a vector bundle on X. Consider a general curve C ∈ H, let ν : P1 → C be its normalization, and write 12

See e.g. [Bea99, KPSW00] and the references therein for an introduction to complex contact manifolds.


ν ∗ (V ) ∼ = O(a1 ) ⊕ · · · ⊕ O(ar ),

391

with a1 ≥ · · · ≥ ar .

If x ∈ X is a general point and σ ∈ H (X, V ) any non-zero section, then the order of vanishing of σ at x satisfies multx (σ) ≤ lengthH (X) · a1 . 0

Sketch of Proof of Theorem 21.(1) Let x ∈ X be a general point. The first inclusion, Cx ⊂ Cx1 , is immediate from the definition. ˜ x be the normalization of Hx and For the other inclusion, let again H consider the pull-back of Diagram (3) from page 377, Ux

ιx evaluation

/X

πx

˜x H

If none of the curves of Hx has nodal singularities at x, Theorem 16 asserts ∼ ˜ that the preimage ι−1 x (x) is exactly a reduced section σ∞ = Hx . By the universal property of blowing up, the evaluation morphism ιx factors via the blow-up of X at x and the equality Cx1 = Cx follows. If some of the curves in Hx are nodal, ι−1 x (x) contains the section σ∞ and finitely many points. A somewhat technical analysis of the tangent morphism T ιx , and a comparison between positive directions in the restriction of TX to a nodal curve and the Zariski tangent space to the VMRT then yields the inclusion Cx1 ⊂ S 1 Cx . Sketch of Proof of Theorem 21.(2) Here we only give an idea of why the first secant variety of the VMRT, S 1 Cx is contained in the tangent cone Cx2 to the locus of length-2 chains of rational curves. If v and w ∈ Cx are any two general elements, we need to show that the 2 line in P(TX |∨ x ) through v and w is contained in the tangent cone to loc (x). To this end, let [v ] and [w ] ∈ Hx be two rational curves that have v and w as tangent directions, respectively. By general choice of v and w, the curves v and w are smooth at x. Consider a small unit disk ∆ ⊂ v , centered about x. By general choice of x, we can find a smooth holomorphic arc γ : ∆ → H such that • γ(0) = [w ], • for any point y ∈ ∆ ⊂ v ⊂ X, the curve γ(y) contains y, i.e., γ(y) ∈ Hy . The situation is described in Fig. 7 above. The unions of the curves γ(∆) then forms a surface S ⊂ loc2 (x). By general choice, it can be seen that S is smooth at x and contains both v and w are tangent directions —see Fig. 7. As a consequence, we obtain that the line in P(TX |∨ x ) through v and w is also ∨ 2 ). Since P(T | ) ⊂ C , this shows the claim. contained in P(TS |∨ S x x x

392

Stefan Kebekus and Luis Sol´ a Conde lv

P(TS |∨ x) S⊂loc2 (x)

∆

v

1 Cx

A U A x w lw

P(TX |∨ x)

X

Fig. 7. Secants to Cx1

Sketch of Proof of Theorem 21.(3) Let v ∈ Cx2 be any element in the tangent cone to loc2 (x). We need to show that v is contained in the first secant variety to Cx , i.e., that v ∈ S 1 Cx . Recall from [Har95, lect. 20] that the tangent cone Cx2 to loc2 (x) is settheoretically exactly the union of the tangent lines to holomorphic arcs γ : ∆ → loc2 (x) that are centered about x. Recall also that if the arc γ is not smooth at 0, then the tangent line at 0 is the limit of the tangent lines to γ at points where γ is smooth. We can thus find an arc γ : ∆ → loc2 (x) with γ(0) = x and tangent v. Recall that loc2 (x) is the locus of chains of rational curves of length 2 that contain x. Replacing ∆ by a finite covering, if necessary, we can then find arcs G : ∆ → Hx

and

F : ∆→H

such that for all t ∈ ∆ the curves G(t) ∪ F (t) form a chain of length two that contains both x and γ(t) —this construction is depicted in Fig. 8. Further, we consider arcs, defined for general t as follows. g : ∆ → loc1 (x) t → G(t) ∩ F (t)

P : ∆ → Grass(2, PN ) t → plane spanned by G(t) and F (t)

Observe that this makes g and P well-defined because the target varieties are proper. We distinguish two cases. Case (i), G(0) = F (0): In this case, x = g(0) = G(0) ∩ F (0), and [v] ∈ ∨ P(TP (0) |∨ x ). Since for general t ∈ ∆, the line P(TP (t) |x ) is contained in 1 ∨ 1 the secant variety S Cg(t) , we have P(TP (0) |x ) ⊂ S Cx . This shows the claim. Case (ii), G(0) = F (0): Since curves of H are lines by assumption, Proposition 6 applies to the point y = g(0) and the line = G(0).


393

g(t) G(t) γ(t)

F (t) G(t)

g(t) g(0)=y

G(0)

F (t) G(0) =F (0)

γ(t)

γ(0)=x F (0)

γ(0)=x

Case (i)

Case (ii)

2

Fig. 8. Arcs in loc (x) ∨ Since P(TP (t) |∨ g(t) ) is secant to Cg(t) for general t, we obtain that P(TP (0) |y ) ∨ is secant to Cy . Moreover, since = G(0) = F (0), the line P(TP (0) |y ) lies in projective tangent space to Cy at the point corresponding to []. To end, observe that the plane P (0) is tangent to X all along . Its tan∨ gent directions determine a subbundle of TX |+ . This implies P(TP (0) |x ) ⊂ ∨ P(TX |+ |x ) and concludes the proof.

Open Problems Problem 6. In view of Remarks 12 and 15, is it possible to improve on Theorem 21 and find a formula that computes the spanning dimensions in all cases? Perhaps one needs to take into account how the VMRT (or its linear span) deforms as the base point changes.

4 Examples of Uniruled Varieties: Moduli Spaces of Vector Bundles In this section we apply the results of Sect. 3 to one particular example of a uniruled variety, namely the moduli space of stable vector bundles on a curve. We first recall a classic construction of rational curves on the moduli space, and then show by example how the understanding of the VMRT can be used to study geometric properties of the moduli space. 4.1 Setup of Notation. Definition of Hecke Curve Let C be a smooth projective curve of genus g ≥ 3 and L be a line bundle of C of degree d. We will denote by M r (L), or simply by M r if there is no

394


possible confusion, the moduli scheme of semistable vector bundles of rank r and determinant L. We can assume without loss of generality that 0 ≤ d ≤ r − 1, otherwise twist with a line bundle. For simplicity, let us assume that (r, d) = 1, in which case M r is a smooth Fano manifold of Picard number one. We remark that most of the arguments and constructions presented here work the same for the general case. Definition 19. Given two integers k and l, we say that a vector bundle E of rank r is (k, l)-stable if every locally free subsheaf F ⊂ E verifies: deg(F ) + k deg E + k − l < . rank(F ) rank(E) Let M r (k, l) ⊂ M r the subset parameterizing (k, l)-stable vector bundles of rank r. Remark 16. The subsets M r (k, l) are open in M r . For instance M r (0, 0) ⊂ M r is the open set parametrizing stable bundles. Narasimhan and Ramanan showed that the subset M r (1, 1) is non-empty for g ≥ 3 —see [NR78, prop. 5.4] for a precise statement. For every (1, 1)stable bundle E they constructed a 1-dimensional family deformations of E that corresponds to a rational curve in M r . The so-constructed curves are called Hecke curves. We briefly review their definition below. Construction of the Hecke curves Let E be a vector bundle of rank r and let x be any point of C. Every element p = [α] ∈ P(Ex ) provides an exact sequence of OC -modules, φ

0 → E p −→ E −→ Ox → 0,

(6)

which is induced by the map α : Ex → C. The kernel E depends only on the class p = [α] and it is called the elementary transform of E at p. Let Hp (E) := (E p )∨ be its dual. Considering the associated projective bundles, elementary transforms can be described in terms of blow-ups and blow-downs. We have depicted the case r = 2 in Fig. 9. Dualizing the exact sequence (6) we obtain the following. p

φ∨

0 → E ∨ −→ Hp (E) −→ Ox → 0 ∨

(7)

Sequence (7) expresses E as the elementary transformation of Hp (E) at the point p = coker φ∨ x ∈ P(Hp (E)x ), that is Hp (Hp (E)) = E. We can then consider the Hq (Hp (E)) as deformations of E parametrized by q ∈ P(Hp (E)x ). It is easy to see that if E is (k, l)-stable, then Hp (E) is (l − 1, k)-stable —see [NR78, lem. 5.5]. In particular, when E is (1, 1)-stable, then every element of the form Hq (Hp (E)) is stable. A line L ⊂ P(Hp (E)x ) through p will therefore determine a rational curve through [E] in M r . We denote this curve by C(E, p, L).


395

P(E x)

+

Exc.

blowing-up p

p

Q Q blowing-down P(E x) Q Q Q s Q

p

P(Ex )

P(E)

P(Ep )

Fig. 9. Elementary transform of a ruled surface. Moving the point p along P(Ex ) gives a rational curve in M r .

Definition 20. A curve of the form C(E, p, L) constructed as above is called a Hecke curve through [E]. 4.2 Minimality of the Hecke Curves Notice that lines in P(Hp (E)x ) through p are in one-to-one correspondence with the set of hyperplanes in P(Ex ) that contain p, i.e., with P(TP(Ex )|p ). The next proposition states basic properties of Hecke curves that have been originally proved by Narasimhan and Ramanan—we refer the interested reader to [NR78, sect. 5] for details. Proposition 9. With the same notation as above, the following holds. •

[NR78, 5.13]: The map that sends (p, L) to the Hecke curve C(E, p, L) is injective for every [E] ∈ M r (1, 1). In particular, Hecke curves through [E] are naturally parametrized by the points of the projectivization of the relative tangent bundle P(TP(E),C ). • [NR78, 5.9, 5.15, 5.16]: Hecke curves are free smooth rational curves of anticanonical degree 2r —see Proposition 5 on page 378 for the notion of a free rational curve. Definition 21. Set HE := P(TP(E),C ), and let O(1) be the associated tautological line bundle. Further, let ρ : HE → C be the natural projection. If [E] ∈ M r (1, 1) is any point, we can view HE as a subset of RatCurvesn (M r ). Let H ⊂ RatCurvesn (M r ) be the closure of the union of the HE , for all [E] ∈ M r (1, 1). Minimality of the Hecke curves was shown by Hwang in [Hwa00, prop. 9] for the case r = 2. Recently Sun has proven that Hecke curves are minimal curves in M r for any r, and that the converse is also true:

396


Theorem 22 ( [Sun05, thm. 1]). Any rational curve C ⊂ M r passing through a general point of M r has anticanonical degree at least 2r. If g ≥ 3, then the anticanonical degree of C is 2r if and only if it is a Hecke curve. In particular, Sun shows that the variety H is a dominating family of rational curves of minimal degrees in M r . If [E] ∈ M r (1, 1) is a general point, the associated subspace of curves through [E] is exactly HE . 4.3 VMRT Associated to M r The variety of minimal rational tangents to M r at [E] ∈ M r (1, 1) is the image of the tangent map: ∨ τE : HE −→ P(TM r ,[E] ) (p, L) → [TC(E,p,L),[E] ]

We would like to understand τE in two ways, namely • in terms of the Kodaira-Spencer map associated to Hecke curves, as studied by Narasimhan and Ramanan in [NR78], and • in terms of linear systems on HE . For the first item, recall the standard description of the tangent space to the moduli scheme, TM r ,[E] ∼ = H 0 C, KC ⊗ ρ∗ (O(1)) , = H 1 C, ad(E) ∼ where ad(E) is the sheaf of traceless endomorphisms of E. The morphism τE is then described as follows. Proposition Lemma 5.10]). The Kodaira-Spencer map 10 ([NR78, TL,p → H 1 C, ad(E) coincides, up to sign, with the composition of the following two morphisms. • The natural map α : TL,p → Ex , and • the connecting morphism β : Ex → H 1 (E ⊗ E ∨ ) of the exact sequence (7) tensored with E. For the other description of τE , we mention another result of Hwang. Theorem 23 ([Hwa02, thms. 3-4], [Hwa00, prop. 11]). With the same notation as above, the morphism τE is given by the complete linear system associated to the line bundle ρ∗ (KC ) ⊗ O(1). If moreover g > 2r + 1 then τE is an embedding. By the minimality of Hecke curves, Theorem 17 implies that τE is a finite map, that is, that the line bundle ρ∗ (KC ) ⊗ O(1) is ample.


397

4.4 Applications The description of the VMRT given in Sect. 4.3 has interesting consequences. For instance, Hwang has used the projective properties of the tangent morphism τE to deduce the following properties of the tangent bundle to the moduli space. Theorem 24 ([Hwa00, thm. 1], [Hwa02, cor. 3]). Let M 2 be the moduli space of stable bundles of rank 2 with a fixed determinant of odd degree over an algebraic curve of genus g ≥ 2. Then the tangent bundle of M 2 is stable. Let M r be the moduli space of semistable bundles of rank r with fixed determinant and (M r )0 its smooth locus. Then T(M r )0 is simple for g ≥ 4. The above description of the variety of minimal rational tangents also allows to deduce some of its projective properties, for instance its secant defect. This has been done in the case r = 2. Proposition 11 ([HK05, prop. 6.10]). Let M 2 be the moduli space of stable bundles of rank 2 with a fixed determinant of odd degree over an algebraic curve of genus g ≥ 4. The variety of minimal rational tangents at a general point of M 2 has no secant defect. This result, combined with Proposition 8 from page 390, provides a bound on the multiplicity of divisors in the moduli space, perhaps similar in spirit to the classical Riemann singularity theorem. Corollary 5 ([HK05, cor. 6.12]). With the same notation as above, let x ∈ M 2 be a general point, and L be the ample generator of Pic(M 2 ), and D ∈ |mL|, m ≥ 1 be any divisor. Then multx (D) ≥ 2m(g − 1). The fact that the only minimal rational curves at the general point are Hecke curves has also very important corollaries, as pointed out by Sun. To begin with, it allows us to state a Torelli-type theorem for moduli spaces: Theorem 25 ([Sun05, cor. 1.3]). Let C and C be two smooth projective curves of genus g ≥ 4. Let M r and (M )r be two irreducible components of the moduli schemes of vector bundles of rank r over C and C , respectively. If Mr ∼ = C . = (M )r then C ∼ Second, it can be used to describe the automorphism group of M r . Theorem 26 ( [Sun05, cor. 1.4]). Let C be a smooth projective curve of genus g ≥ 4, L a line bundle on C. If r > 2, then the group of automorphisms of M r (L) is generated by:

398


• automorphisms induced by automorphisms of C, and • automorphisms of the form E → E ⊗ L , where L is an r-torsion element of Pic0 (C). For r = 2, we need additional generators of the form E → E ∨ ⊗ L where L is a line bundle verifying (L )⊗2 ∼ = L⊗2 .

5 Consequences of Non-uniruledness I: Deformations of Surjective Morphisms and a Refinement of Stein Factorization Let f : X → Y be a surjective morphism between normal complex projective varieties. A classical problem of complex geometry asks for a criterion to guarantee the (non-)existence of deformations of the morphism f , with X and Y fixed. More generally, one is interested in a description of the connected component Homf (X, Y ) ⊂ Hom(X, Y ) of the space of morphisms. For instance, if X is of general type, it is well-known that the automorphism group is finite. It is more generally true that surjective morphisms between projective manifolds X, Y of general type are always infinitesimally rigid so that the associated connected components of Hom(X, Y ) are reduced points. Similar questions were discussed in the complex-analytic setup by Borel and Narasimhan, [BN67]. We will show here how Miyaoka’s characterization of uniruledness, or the existence result for rational curves contained in Theorem 5/Corollary 3 can be used to give a rather satisfactory answer in the projective case. Before giving an idea of the methods employed, we will first, in Sects. 5.1 and 5.2, state and explain the result. 5.1 Description of Homf (X, Y ) if Y is not Uniruled If f : X → Y is as above, it is obvious that f can always be deformed if the target variety Y has positive-dimensional automorphism group; this is because the composition morphism f ◦ : Aut0 (Y ) → Homf (X, Y ) g → g◦f

(8)

is clearly injective. One could na¨ıvely hope that all deformations of f come from automorphisms of the target. While this hope does not hold true in general, we will show, however, that it is almost true: if Y is not uniruled, f always factors via an intermediate variety Z whose automorphism group is positive-dimensional and induces all deformations of f . More precisely, the following holds.


399

Theorem 27 ([HKP03, thm. 1.2]). Let f : X → Y be a surjective morphism between normal complex-projective varieties, and assume that Y is not uniruled. Then there exists a factorization of f , f

X

α

/Z

β

%/

Y ,

such that: 1. the morphism β is unbranched away from the singularities of X and Y , and 2. the natural morphism ; Aut0 (Z) Deck(Z/Y ) → Homf (X, Y ) g → β ◦ g ◦ α is an isomorphism of schemes, where Deck(Z/Y ) is the group of Deck transformations, i.e., relative automorphisms. In particular, f deforms unobstructedly, and the associated component Homf (X, Y ) is a smooth abelian variety. The following is an immediate corollary whose proof we omit for brevity. Corollary 6 ([HKP03, cor. 1.3]). In the setup of Theorem 27, if the target variety Y is smooth, then Y admits a finite, étale covering of the form T × W , where T is a torus of dimension dim T = h0 (X, f ∗ (TY )). Additionally, we have dim Homf (X, Y ) ≤ dim Y − κ(Y ), where κ(Y ) is the Kodaira dimension.

We will later, in Sect. 5.3 give an idea of the proof of Theorem 27. 5.2 Description of Homf (X, Y ) in the General Case If Y is rationally connected, partial descriptions of the Hom-scheme are known —the results of [HM03, thm. 1] and [HM04, thm. 3] assert that whenever Y is a Fano manifold of Picard number 1 whose variety of minimal rational tangents is finite, or not linear, then all deformations of f come from automorphisms of Y . This covers all examples of Fano manifolds of Picard number one that we have encountered in practice. If Y is covered by rational curves, but not rationally connected, we consider the rationally connected quotient qY : Y QY which is explained in Definition 4. It is shown in [KP05] that f can be factored via an intermediate variety Z, in a manner similar to that of Theorem 27, such that a covering of Homf (X, Y ) decomposes into

400


• an abelian variety, which comes from the automorphism group of Z, and • the space of deformations that are relative over the rationally connected f := {f ∈ Homf (X, Y )red | qY ◦ f = qY ◦ f }. quotient, i.e., Hvert To formulate the result precisely, we recall a result that yields a factorization of f and may be of independent interest. Theorem 28 ([KP05, thm. 1.4]). Let f : X → Y be a surjective morphism between normal projective varieties. Then there exists a factorization f

X

α

/Z

β

)/

(9)

Y

where β is finite and étale in codimension one13 , and where the following universal property holds: for any factorization f = β ◦ α , where β : Z → Y is finite and étale in codimension 1, there exists a morphism γ : Z → Z such that β = β ◦ γ. It follows immediately from the universal property that the factorization (9) is unique up to isomorphism. We call (9) the maximally étale factorization of f . The maximally étale factorization can be seen as a natural refinement of the Stein factorization. More precisely, we can say that a surjection f : X → Y of normal projective varieties decomposes as follows. f

X

conn. fibers

/W

finite

/Z

max. ´ etale

/, Y

The paper [KP05] discusses the maximally étale factorization in more detail. Its stability under deformations of f is shown [KP05, sect. 1.B], and a characterization in terms of the positivity of the push-forward sheaf f∗ (OX ) is given, [KP05, sect. 4]. The main result is then formulated as follows. Theorem 29 ([KP05, thm. 1.10]). In the setup of Theorem 28, let T ⊂ Aut0 (Z) be a maximal compact abelian subgroup. Then there exists a normal ˜ and an étale morphism variety H ˜ → Homnf (X, Y ) T ×H f ˜ to the preimage of Hvert . If Y is smooth or if f is itself that maps {e} × H f ˜ surjects onto the preimage of Hvert maximally étale, then {e} × H .

In Theorem 29, we discuss the maximal compact abelian subgroup of an algebraic group. Recall the following basic fact of group theory. 13

I.e., where β is finite and étale away from a set of codimension two.


401

Remark 17. Let G be an algebraic group. Then there exists a maximal compact abelian subgroup, i.e., an abelian variety T ⊂ G which is a subgroup and such that no intermediate subgroup T ⊂ S ⊂ G, T = S, is an abelian variety. A maximal compact abelian subgroup is unique up to conjugation. Remark 18. In the setup of Theorem 29, it need not be true that Aut0 (Z) is itself an abelian variety. Unlike Theorem 27, Theorem 29 does not make any statement about the scheme structure of Homf (X, Y ). While the methods used to show Theorems 27 and 29 are obviously related, the fact that the rational quotient is generally not a morphism, and the lack of a good parameter space for rational maps makes the proof of Theorem 29 technically more involved. We have thus decided to restrict to a sketch of a proof of Theorem 27 only. 5.3 Sketch of Proof of Theorem 27 We give only a rough idea of a proof for Theorem 27 —our main intention is to show how the existence result for rational curves comes into the picture. The interested reader is referred to the rather short original article [HKP03], and perhaps to the more detailed survey [Keb04]. Step 1: Simplifying Assumption As we are only interested in a presentation of the core of the argumentation, we will show Theorem 27 under the simplifying assumption that the surjective morphism f : X → Y is a finite morphism between complex-projective manifolds. Step 2: The Tangent Map to f ◦ Consider again the natural morphism f ◦ : Aut0 (Y ) → Homf (X, Y ), as introduced in Equation (8) on page 398. The universal properties of Hom(X, Y ) and of the automorphism group Aut0 (Y ) yield the following description of the tangent map T f ◦ at the point e ∈ Aut0 (Y ), T f ◦ |e : TAut(Y ) |e → THom |f .

(10)

Namely, the natural identifications TAut(Y ) |e ∼ = H 0 (Y, TY ) and

THom |f ∼ = H 0 (X, f ∗ (TY ))

associate the tangent morphism (10) with the pull-back map T f ◦ |e = f ∗ : H 0 (Y, TY ) → H 0 (X, f ∗ TY ).

(11)

402


The following observation is now immediate: Lemma 1. The morphism (10) is injective. If the pull-back morphism (11) is surjective, i.e., if any infinitesimal deformation of f comes from a vector field on Y , then (10) yields an isomorphism of Zariski tangent spaces. Similar considerations also yield a description of the tangent morphism at an arbitrary point g ∈ Aut0 (Y ), T f ◦ |g = (g ◦ f )∗ : H 0 (Y, TY ) → H 0 X, (g ◦ f )∗ (TY ) . Since g : Y → Y has maximal rank everywhere, it can be shown that the rank of T f ◦ is constant on all of Aut0 (Y ). Using that Aut0 (Y ) is smooth, an elementary argumentation then yields the following. Corollary 7. If the pull-back morphism (11) is surjective, then the composition map f ◦ : Aut0 (Y ) → Homf (X, Y ) is isomorphic. In this case, the proof of Theorem 27 is finished by setting Z := Y . Step 3, Central Step in the Proof: Construction of a Covering Corollary 7 allows us to assume without loss of generality that there exists a section σ ∈ H 0 (X, f ∗ (TY )) that does not come from a vector field on Y . Under this assumption, we will then construct an unbranched covering of Y , which factors the surjection f . The main tool is the following negativity theorem of Lazarsfeld for the push-forward sheaf f∗ (OX ) which can, in our setup, be seen as a competing statement to Corollary 3 of page 371, the characterization of uniruledness. Theorem 30 ([Laz80], [PS00, thm. A]). The trace map tr : f∗ (OX ) → OY yields a natural splitting f∗ (OX ) ∼ = OY ⊕ E ∨ , where E is a vector bundle with the following positivity property: if C ⊂ Y is any curve not contained in the branch locus of f , then E|C is nef. The curve C intersects the branch locus of f if and only if deg(E|C ) > 0. Corollary 8. Choose an ample line bundle H ∈ Pic(Y ) and let C ⊂ Y be an associated general complete intersection curve in the sense of MehtaRamanathan, as in Definition 1 of page 362. Then the following holds: β

• If X → Z − → Y is a factorization of f with β étale, then β∗ (OZ ) ⊂ f∗ (OY ) is a subbundle which is closed under the multiplication map µ : f∗ (OX ) ⊗ f∗ (OX ) → f∗ (OX ), and satisfies deg β∗ (OZ )|C = 0.


403

• Conversely, if F ⊂ f∗ (OX ) is a subbundle that is closed under multiplication, and deg(F |C ) = 0, then f factors via Z := Spec(F ), and Z is étale over Y . Observe that the projection formula gives an identification H 0 (X, f ∗ (TY )) = H 0 (Y, f∗ (f ∗ (TY ))) = H 0 (Y, TY ) ⊕ H 0 (Y, E ∨ ⊗ TY ) =HomY (E,TY )

Since σ does not come from a vector field on Y , the section σ is not contained in the component H 0 (Y, TY ). Thus, we obtain a non-trivial morphism σ : E → TY . Now choose H and C as in Corollary 8. Lazarsfeld’s Theorem 30 then implies that Image(σ)|C is nef. On the other hand, Corollary 3 asserts that Image(σ)|C cannot be ample. In summary we have the following. Lemma 2. The restricted vector bundle E|C is nef, but not ample.

Definition 22. Let VC ⊂ E|C be the maximal ample subbundle, as discussed in Proposition 3 and Definition 7. Further, let FC ⊂ E ∨ |C be the kernel of the associated morphism E ∨ |C → VC∨ . ; Remark 19. The bundle FC is dual to the quotient E|C VC . In particular, FC is nef and has degree 0. With these preparations we will now construct the factorization of f . We construct the factorization first over C, and then extend it to all of Y . Lemma 3. The sub-bundle OC ⊕ FC ⊂ f∗ (Ox )|C is closed under the multiplication map µ : f∗ (OX ) ⊗ f∗ (OX ) |C → f∗ (OX )|C , because the associated morphism ; ∨ µ : (OC ⊕ FC ) ⊕ (OC ⊕ FC ) −→ OC ⊕ E |C OC ⊕ FC deg 0,nef

is necessarily trivial.

∨ , anti-ample ∼ =VC

By Corollary 8, the subbundle OC ⊕FC ⊂ f∗ (Ox )|C induces a factorization of the restricted morphism f |C : f −1 (C) → C via an étale covering of C. In order to extend this covering from C to all of Y , if suffices to extend the maximal ample subbundle VC ⊂ E|C to a subbundle V ⊂ E that has the property that the restriction V|C ⊂ E|C to any complete intersection curve C ⊂ Y is exactly the maximal ample subbundle of E|C . But since the maximal ample subbundle is by definition a term of the Harder-Narasimhan filtration of E|C , Theorem 3 of Flenner and Mehta-Ramanathan, exactly asserts that this is possible.

404


Corollary 9. If the pull-back morphism H 0 (Y, TY ) → H 0 (X, f ∗ (TY )) is not surjective, then there exists a factorization of f , f

X where β is étale.

α

/ Y (1)

β

&/

Y ,

Step 4: End of Proof Assuming that the pull-back map H 0 (Y, TY ) → H 0 (X, f ∗ (TY )) was not surjective, we have in Step 3 constructed a factorization of f via an étale covering β : Y (1) → Y . The étalité obviously implies f ∗ (TY ) = α∗ (TY (1) ). We ask again if the pull-back morphism α∗ : H 0 Y (1) , TY (1) → H 0 (X, f ∗ (TY )) is surjective. Yes → Following the considerations of Step 2, the proof is finished if we set Z := Y (1) . No → repeat Step 3, using the morphism α : X → Y (1) rather than f : X → Y. Repeating this procedure, we construct a sequence of étale coverings f

X

/ Y (d)

/ Y (d−1)

/ ...

/ Y (1)

/+ Y.

The sequence, however, must terminate after finitely many steps, simply because the number of leaves is finite. The same line of argumentation that led to Corollary 7 shows that we can end the proof is we set Z = Y (d) . This finishes the proof of Theorem 27 —under the simplifying assumption that f is a finite morphism between complex manifolds. 5.4 Open Problems It is not quite clear to us if projectivity or if complex number field is really used in an essential manner. We would therefore like to ask the following. Question 4. Does Theorem 27 hold in positive characteristic? Question 5. Does it hold for K¨ ahler manifolds or complex spaces? Theorem 27 can also be interpreted as follows: all obstructions to deformations of surjective morphisms come from rational curves in the target. Is it possible to make this statement precise?


405

6 Consequences of Non-uniruledness II: Families of Canonically Polarized Varieties Let B ◦ be a smooth quasi-projective curve, defined over an algebraically closed field of characteristic 0 and q > 1 a positive integer. In his famous paper [Sha63], Shafarevich considered the set of families of curves of genus q over B ◦ . More precisely, he considered isomorphism classes of smooth proper morphisms f : S → B ◦ whose fibers are connected curves of genus q. He conjectured the following. Finiteness conjecture: There are only finitely many isomorphism classes of non-isotrivial families of smooth projective curves of genus q over B ◦ — recall that a family is called isotrivial if any two fibers are isomorphic. Hyperbolicity conjecture: If κ(B ◦ ) ≤ 0, then no such families exist. These conjectures, which later played an important role in Faltings’ proof of the Mordell conjecture, were confirmed by Parshin [Par68] for projective bases B ◦ and by Arakelov [Ara71] in general. We refer the reader to the survey articles [Vie01] and [Kov03] for a historical overview and references to related results. It is a natural and important question whether similar statements hold for families of higher dimensional varieties over higher dimensional bases. Families over a curve have been studied by several authors in recent years and they are now fairly well understood—the strongest results known were obtained in [VZ01, VZ02], and [Kov02]. For higher dimensional bases, however, a complete picture is still missing and no good understanding of subvarieties of the corresponding moduli stacks is available. As a first step toward a better understanding, Viehweg conjectured the following: Conjecture 3 ([Vie01, 6.3]). Let f ◦ : X ◦ → S ◦ be a smooth family of canonically polarized varieties. If f ◦ is of maximal variation, then S ◦ is of log general type —see Definition 8 on page 375 for the logarithmic Kodaira dimension and log general type. For the reader’s convenience, we briefly recall the definition of variation that was introduced by Koll´ ar and Viehweg. Definition 23. Let f ◦ : X ◦ → S ◦ be a family of canonically polarized varieties and µf ◦ : S ◦ M the induced map to the corresponding moduli scheme. The variation of f ◦ is defined as Var(f ◦ ) := dim µf ◦ (S ◦ ). The family f ◦ is called isotrivial if Var(f ◦ ) = 0. It is called of maximal variation if Var(f ◦ ) = dim S ◦ . 6.1 Statement of Result Using a result of Viehweg-Zuo and Keel-McKernan’s proof of the Miyanishi conjecture in dimension two, we describe families of canonically polarized

406


varieties over quasi-projective surfaces. We relate the variation of the family to the logarithmic Kodaira dimension of the base and give an affirmative answer to Viehweg’s Conjecture 3 for families over surfaces. Theorem 31 ([KK05, thm. 1.4]). Let S ◦ be a smooth quasi-projective surface and f ◦ : X ◦ → S ◦ a smooth non-isotrivial family of canonically polarized varieties, all defined over C. Then the following holds. 1. If κ(S ◦ ) = −∞, then Var(f ◦ ) ≤ 1. 2. If κ(S ◦ ) ≥ 0, then Var(f ◦ ) ≤ κ(S ◦ ). In particular, Viehweg’s Conjecture holds true for families over surfaces. Remark 20. Notice that in the case of κ(S ◦ ) = −∞ one cannot expect a stronger statement. For an easy example take any non-isotrivial smooth family of canonically polarized varieties over a curve g : Z → C, set X := Z × P1 , S ◦ := C ×P1 , and let f ◦ := g ×idP1 be the obvious morphism. Then we clearly have κ(S ◦ ) = −∞ and Var(f ) = 1. 6.2 Open Problems Theorem 31 and its proof seem to suggest that the logarithmic Kodaira dimension of a variety S ◦ gives an upper bound for the variation of any family of canonically polarized varieties over S ◦ , unless κ(S ◦ ) = −∞. We would thus like to propose the following generalization of Viehweg’s conjecture. Conjecture 4 ( [KK05, conj. 1.6]). Let f ◦ : X ◦ → S ◦ be a smooth family of canonically polarized varieties. Then either κ(S ◦ ) = −∞ and Var(f ◦ ) < dim S ◦ , or Var(f ◦ ) ≤ κ(S ◦ ). 6.3 Sketch of Proof of Theorem 31 Again we give only an incomplete proof of Theorem 31. We restrict ourselves to the case where κ(S ◦ ) = 0 and show only that Var(f ◦ ) ≤ 1. We will then, at the end of the present section, give a rough idea of how isotriviality can be concluded. Setup of Notation Let S be a compactification of S ◦ as in Definition 8, and let D := S \ S ◦ be the boundary divisor, which has at worst simple normal crossings. We assume κ(S ◦ ) = κ(KS + D) = 0. A part of the argumentation involves the log minimal model of (S, D) —we refer to [KM98] and [Mat02] for details on the log minimal model program for surfaces. If κ(S ◦ ) = −∞, we denote the birational morphism from S to its logarithmic minimal model by


407

φ : (S, D) → (Sλ , Dλ ), where Dλ is the cycle-theoretic image of D in Sλ . We briefly recall a few important facts of minimal model theory in dimension two that can be found in the standard literature, e.g. [KM98] or [Mat02]. Proposition 12. The minimal model has the following properties: 1. The pair (Sλ , Dλ ) has only log-canonical singularities and Sλ itself has only log-terminal singularities. In particular, Sλ has only quotient singularities and is therefore Q-factorial. 2. The log-canonical divisor KSλ + Dλ is nef. 3. The log Kodaira dimension remains unchanged, i.e., κ(KSλ + Dλ ) = κ(KS + D). 4. Logarithmic Abundance in Dimension 2: The linear system |n(KSλ + Dλ )| is basepoint-free for sufficiently large and divisible n ∈ N. A Result of Viehweg and Zuo Before starting the proof we recall an important result of Viehweg and Zuo that describes the sheaf of logarithmic differentials on the base of a family of canonically polarized varieties in our setup. Theorem 32 ( [VZ02, thm. 1.4(i)]). In the setup introduced above, there exists an integer n > 0 and an invertible subsheaf A ⊂ Symn ΩS1 (log D) of Kodaira dimension κ(A) ≥ Var(f ◦ ).

Reduction to the Uniruled Case A surface S with logarithmic Kodaira dimension zero need of course not be uniruled. Using the result of Viehweg-Zuo we can show, however, that any family of canonically polarized varieties over a non-uniruled surface with Kodaira dimension zero is isotrivial. Proposition 13. If S is not uniruled, then Var(f ◦ ) = 0. We prove Proposition 13 using three lemmas that describe the sheaves of log-differentials on the minimal model Sλ . Lemma 4. If n ∈ N is sufficiently large and divisible, then OSλ (n(KSλ + Dλ )) = OSλ .

(12)

In particular, the log-canonical Q-divisor KSλ + Dλ is numerically trivial. Proof. Equation (12) is an immediate consequence of the assumption κ(S ◦ ) = 0 and the logarithmic abundance theorem in dimension 2, which asserts that the linear system |n(KSλ + Dλ )| is basepoint-free.

408


Lemma 5. If κ(S) ≥ 0, then Sλ is Q-Gorenstein, KSλ is numerically trivial and Dλ = ∅. Proof. Lemma 4 together with the assumption that |nKS | = ∅ for large n imply that φ contracts all irreducible components of D, and all divisors in any linear system |nKS |, for all n ∈ N. The claim follows. Lemma 6. Assume that κ(S) ≥ 0 and Var(f ◦ ) ≥ 1. If H ∈ Pic(Sλ ) is any ample line bundle, then the (reflexive) sheaf of differentials (ΩS1 λ )∨∨ has slope µH (ΩS1 λ )∨∨ = 0, but it is not semistable with respect to H. Proof. Fix a sufficiently large number m > 0 and a general curve Cλ ∈ |mH|. Theorem 3 ensures that if (ΩS1 λ )∨∨ is semistable, then so is its restriction ΩS1 λ |Cλ . By general choice, Cλ is contained in the smooth locus of Sλ and stays off the fundamental points of φ−1 . The birational morphism φ will thus be well-defined and isomorphic along C := φ−1 (Cλ ). Lemma 5 then asserts that KSλ · Cλ = 0, µH (ΩS1 λ )∨∨ = 2 which shows the first claim. Similarly, Lemma 5 implies that codimSλ φ(D) ≥ 2, and so C is disjoint from D. The unstability of (ΩS1 λ )∨∨ can therefore be checked using the identifications (13) (ΩS1 λ )∨∨ |Cλ ∼ = ΩS1 λ Cλ ∼ = ΩS1 C ∼ = ΩS1 (log D)|C . Since symmetric powers of semistable vector bundles over curves are again semistable [HL97, cor. 3.2.10], in order to prove Lemma 6, it suffices to show that there exists a number n ∈ N such that Symn ΩS1 (log D)|C is not semistable. For that, use the identifications (13) to compute degC Symn ΩS1 (log D)|C = const+ · degC ΩS1 C = const+ · degCλ (ΩS1 λ )∨∨ |Cλ

Isomorphisms (13)

= const · (KSλ · Cλ ) = 0.

Lemma 5

+

Hence, to prove unstability it suffices to show that Symn ΩS1 (log D)|C contains a subsheaf of positive degree. Theorem 32 implies that there exists an integer n > 0 such that Symn ΩS1 (log D) contains an invertible subsheaf A of Kodaira dimension κ(A) ≥ 1. But by general choice of Cλ , this in turn implies that degC (A|C ) > 0, which shows the required unstability. This ends the proof of Lemma 6. With these preparations, the proof of Proposition 13 is now quite short. Proof (of Proposition 13). We argue by contradiction and assume to the contrary that both S is not uniruled, i.e., κ(S) ≥ 0, and Var(f ◦ ) ≥ 1. Again, let H ∈ Pic(Sλ ) be any ample line bundle.


409

Lemma 6 implies that ΩS1 λ |Cλ has a subsheaf of positive degree or, equivalently, that it has a quotient of negative degree. On the other hand, Corollary 3 then asserts that S is uniruled, leading to a contradiction. This ends the proof of Proposition 13. Images of C∗ on S ◦ , End of Proof In view of Proposition 13, to prove that Var(f ◦ ) ≤ 1, we can assume without loss of generality that Var(f ◦ ) > 0 and therefore S is uniruled. Since families of canonically polarized varieties over C∗ are always isotrivial, [Kov00, thm. 0.2], the result then follows from the following proposition. Proposition 14. If Var(f ◦ ) > 0, then S ◦ is dominated by images of C∗ . In particular Var(f ◦ ) ≤ 1. As a first step in the proof of Proposition, recall the following fact. If X is a projective manifold, and H ⊂ RatCurvesn (X) is an irreducible component of the space of rational curves such that the associated curves dominate X, it is well understood that a general point of H corresponds to a free curve, i.e., a rational curve whose deformations are not obstructed (cf. [KMM92a, 1.1], [Kol96, II Thm. 3.11], see also Prop. 5). In particular, if E ⊂ X is an algebraic set of codimension codimX E ≥ 2, then the subset of curves that avoid E, H := { ∈ H | E ∩ = ∅}, is Zariski-open, not empty, and curves associated with H still dominate X. A similar statement, which we quote as a fact without giving a proof, also holds if X is a surface with mild singularities, and for quasi-projective varieties that are dominated by C∗ rather than complete rational curves. Proposition 15 (Small Set Avoidance, [KK05, prop. 2.7]). Let X be a smooth projective surface, and E ⊂ X a divisor with simple normal crossings. Assume that X \ E is dominated by images of C∗ , and let F ⊂ X \ E be any finite set. Then X \ (E ∪ F ) is also dominated by images of C∗ . Lemma 7. In the setup of Proposition 14, the quasi-projective surface Sλ \ (Sing(Sλ ) ∪ Dλ ) is dominated by images of C∗ . Proof. We aim to apply Theorem 13.(2), and so we need to show that • the log-canonical divisor KSλ + Dλ is numerically trivial, and that • the boundary divisor Dλ is not empty. The numerical triviality of KSλ + Dλ has been shown in Lemma 4 above. To show that Dλ = ∅, we argue by contradiction, and assume that Dλ = ∅. Set Sλ◦ := Sλ \ φ(exceptional set of φ) . finite, contains φ(D)

410


Then Sλ◦ is the complement of a finite set and φ−1 |Sλ◦ is a well-defined open immersion. Let fλ := φ ◦ f . Then X := X ◦ |f −1 (S ◦ ) → Sλ◦ is a smooth family λ λ of canonically polarized varieties. Consider the following diagram: Xo

˜ := X ×S S˜ X λ f˜

fλ

Sλ o

α index-one-cover

β S˜λ o log resolution

˜ S

where α is the index-one-cover described in [KM98, 5.19] or [Rei87, sect. 3.5], and β is the minimal desingularization of S˜λ composed with blow-ups of smooth points so that β −1 (S˜λ \ α−1 (Sλ◦ )) is a divisor with at most simple normal crossings. By Lemma 4, KSλ is torsion. Since α is étale in codimension one this implies that KS˜λ is trivial. Furthermore, S˜λ has only canonical singularities: we have already noted in Proposition 12 that the singularities of Sλ are log-terminal, i.e., they have minimal discrepancy > −1. Then by [KM98, Prop. 5.20] the minimal discrepancy of the singularities of S˜λ is also > −1, and as KS˜λ is Cartier, the discrepancies actually must be integral and hence ≥ 0, cf. [KM98, proof of Cor. 5.21]. Consequently, KS˜ = β ∗ (KS˜λ ) +(effective and β-exceptional).

(14)

∼ =OS˜

This in turn has two further consequences: i) κ(KS˜ ) = 0. In particular, S˜ is not uniruled. ˜ ◦ := f˜−1 (S˜◦ ) then X ˜ ◦ → S˜◦ is again ii) If we set S˜◦ := (α ◦ β)−1 (Sλ◦ ) and X ˜ := S˜ \ S˜◦ a smooth family of canonically polarized varieties. Letting D ˜ then D is exactly the β-exceptional set, and (14) implies that ˜ ) = 0. κ(S˜◦ ) = κ(KS˜ + D effective, β-exceptional

˜ → S˜ and shows that In particular, Proposition 13 applies to f˜ : X V ar(f˜|X˜ ◦ ) = 0. This is a contradiction and thus ends the proof of Lemma 7. Observe that Lemma 7 does not immediately imply Proposition 14. The problem is that the boundary divisor D ⊂ S can contain connected components that are contracted by φ to points. These points do not appear in the cycle-theoretic image divisor Dλ , and it is a priori possible that all morphisms C∗ → Sλ \ Sing(Sλ ) contain these points in the image. Taking the strict transforms would then give morphisms C∗ → S \ φ−1 (Sing(Sλ ) ∪ Dλ ), but φ−1 (Sing(Sλ ) ∪ Dλ ) = D. An application of Proposition 15 will solve this problem.


411

Proof (of Proposition 14). If φ(D) ⊂ Dλ ∪ Sing(Sλ ), i.e., if all connected components of D are either mapped to singular points, or to divisors, Lemma 7 immediately implies Proposition 14. Likewise, if Sλ was smooth, Proposition 15 on small set avoidance would imply that almost all curves in the family stay off the isolated zero-dimensional components of φ(D), and Proposition 14 would again hold. In the general case, where Sλ is singular, and d1 , . . . , dn are smooth points of Sλ that appear as connected components of φ(D), a little more argumentation is required. If D is the union of connected components of D which are contracted to the set of points {d1 , . . . , dn } ⊂ Sλ , it is clear that the birational morphism φ : S → Sλ factors via the contraction of D , i.e., there exists a diagram φ

S

α

/ S

β

*/

Sλ

where S is smooth, and α maps the connected components of D to points d1 , . . . d2 ∈ S and is isomorphic outside of D . Now, if D := D \ D , the above argument shows that S is dominated by rational curves that intersect α(D ) in two points. Since S is smooth, Proposition 15 applies and shows that almost all of these curves do not intersect any of the di . In summary, we have seen that most of the curves in question intersect α(D) in two points. This completes the proof of Proposition 14. Sketch of Further Argument We have seen above that S ◦ is dominated by images of C∗ , which implies Var(f ◦ ) ≤ 1. If S ◦ is connected by C∗ , i.e., if there exists an open subset Ω ⊂ S ◦ such that any two points x, y ∈ Ω can be joined by a chain of C∗ , then it is clear that Var(f ◦ ) = 0, and Theorem 31 is shown in case κ(S ◦ ) = 0. Since dim S ◦ = 2, we can thus assume without loss of generality that a general point of S ◦ is contained in exactly one curve which is an image of C∗ . Blowing up points on the boundary D if necessary, we can thus assume that S is a birationally ruled surface, with a map π : S → C to a curve C, and that the boundary D is a divisor that intersects the general π-fiber F in exactly two points, which implies that the restriction of the vector bundle ΩS1 (log D) to F is trivial. If Var(f ◦ ) > 0, the result of Viehweg-Zuo, Theorem 32, then implies that the restriction of ΩS1 (log D) to non-fiber components of D cannot be stable of degree zero. However, a detailed analysis of the self-intersection graph of D, and the standard description of the restriction of ΩS1 (log D) to components of D shows both stability and zero-degree.

412


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Special Families of Curves, of Abelian Varieties, and of Certain Minimal Manifolds over Curves Martin M¨ oller1 , Eckart Viehweg1 and Kang Zuo2 1

2

Universit¨ at Duisburg-Essen, Mathematik, 45117 Essen, Germany [email protected] [email protected] Universit¨ at Mainz, Fachbereich 17, Mathematik, 55099 Mainz, Germany [email protected]

Introduction Let f : X → Y be a surjective morphism from an n + 1 dimensional complex projective manifold X to a curve Y . In this article we want to present some recent results about the structure of f , in particular about its discriminant locus S, i.e. the set of points s in Y with f −1 (s) singular. So f : V = f −1 (U ) −−→ U = Y \ S will always be smooth. We will usually require ∆ = f ∗ S to be a normal crossing divisor. If Y = P1 and if X is a curve of genus g(X) > 0 the Hurwitz formula implies that #S ≥ 3. In the extremal case #S = 3, X is obviously defined over a number field. G.V. Bely˘ı has shown in [Be79] that the existence of f with #S = 3 characterizes curves defined over a number field. As conjectured by F. Catanese and M. Schneider, the first part can be generalized for n > 0. Theorem 0.1 ( [VZ01]). Let X be a complex projective manifold of Kodaira dimension κ(X) ≥ 0 and let f : X → P1 be a surjective morphism. Then f has at least 3 singular fibres. It is more reasonable to assume that the general fibre F of f is connected, or as we will say, that f : X → Y is a family of n-dimensional manifolds, and to use the Hurwitz formula for the Stein factorization to get hold of the general

This work has been supported by the “DFG-Schwerpunktprogramm Globale Methoden in der Komplexen Geometrie”, and by the DFG-Leibniz program.

418

Martin M¨ oller, Eckart Viehweg and Kang Zuo

case. Y will be again a curve of arbitrary genus. Putting together results due to Parshin-Arakelov, Migliorini, Kov´ acs, Bedulev-Viehweg, Oguiso-Viehweg and Viehweg-Zuo (see [VZ01] and the references given there) one has: Theorem 0.2. Let f : X → Y be a non-isotrivial family of n-folds, with general fibre F . Assume either a. κ(F ) = dim(F ), or b. ωF semiample. Then deg(ΩY1 (log S)) = 2g(Y ) − 2 + #S > 0. Here f is isotrivial, if over some finite covering Y → Y the pullback X ×Y Y is birational to F × Y . As a byproduct, the proof of 0.2 in [VZ01] (see also [Kov02]) gives some explicit lower bounds for the degree of deg(ΩY1 (log S)). Writing δ for the number of singular fibres of f , which are not semistable, i.e. not reduced normal crossing divisors, one finds constants ν and e, depending only on the Hilbert polynomial h of some polarization of the family, with ν deg(f∗ ωX/Y ) ν rank(f∗ ωX/Y )

≤ (n · (2g(Y ) − 2 + #S) + δ) · ν · e.

In Section 2 we will use [VZ05, 3.4] to get a similar bound for arbitrary semistable families. Theorem 0.3. Assume that f : X → Y is a semistable family of n-folds. If ν Y = P1 assume in addition that #S ≥ 2. Then for all ν ≥ 1 with f∗ ωX/Y = 0 ν ) deg(f∗ ωX/Y ν rank(f∗ ωX/Y )

≤

n·ν n·ν · (2g(Y ) − 2 + #S) = · deg(ΩY1 (log S)). 2 2

The assumption #S ≥ 2 is just added to guaranty that deg(ΩY1 (log S)) is non-negative, and that the fundamental group of Y \ S sufficiently large. Of course it will always hold true if one declares one or two smooth fibres to be singular. There are families of Abelian varieties and of Calabi-Yau manifolds, with ν deg(f∗ ωX/Y

)=

ν ) deg(f∗ ωX/Y ν rank(f∗ ωX/Y )

=

n·ν · deg(ΩY1 (log S)). 2

We do not know, whether there are families of manifolds of higher Kodaira dimension with such an equality. Theorem 0.3 implies Theorem 0.2. In fact, if deg(ΩY1 (log S)) ≤ 0 the open set U is either an elliptic curve or it contains C∗ . In the first case the family is smooth, in the second one one finds a new family over an étale covering C∗ → C∗ , which is semistable. For a semistable family f : X → Y Theorem

Families of Manifolds over Curves

419

ν 0.3 implies that deg(f∗ ωX/Y ) ≤ 0. On the other hand, the assumptions a) or b) in Theorem 0.2 imply that for ν ≥ 2 ν deg(f∗ ωX/Y ) ≥ 0, ν ) = 0 if and only if f is isotrivial. The same argument and that deg(f∗ ωX/Y is used to show Theorem 0.1: If there is a morphism with 2 singular fibres, one uses the Hurwitz formula to show that the Stein factorization of X → P1 is P1 , and that the induced morphism has again at most 2 singular fibres. Here the positivity of ν ) follows from κ(X) ≥ 0, and as above one obtains a contradiction deg(f∗ ωX/Y to the inequality stated in Theorem 0.3. In addition, the bound in Theorem 0.3 implies some weak analogue of the Shafarevich conjecture on the finiteness of smooth families of curves over a fixed curve U .

Corollary 0.4. Fix (Y, S) and a polynomial h. Then up to deformation there are only finitely many non-isotrivial families f : X → Y smooth over U = Y \ S, such that: 1. ωF is ample with Hilbert polynomial h. 2. ωFρ = OF for some ρ > 0, and the smooth part V → U allows a polarization with Hilbert polynomial h. In fact, consider the moduli scheme Mh parameterizing either canonically polarized manifolds, or polarized manifolds with a torsion canonical divisor, and with Hilbert polynomial h. In both cases one can show (see [Vi05]) that ¯ h and for a given ν ≥ 2 with h(ν) = 0 there is a projective compactification M (p) ¯ for some p an invertible sheaf λν on Mh with: (p)

1. λν is numerically effective, and ample with respect to Mh . 2. If f : X → Y is a semistable family over a curve Y , whose general fibre ¯ h is the extension of the belongs to the moduli functor, and if ϕ : Y → M induced morphism to Mh , then ν p ϕ∗ λ(p) ν = det(f∗ ωX/Y ) .

Here an invertible sheaf λ is “ample with respect to Mh ” if there exists some ¯ h , some µ ' 1 and an effective divisor E on M ¯, ¯ → M modification τ : M h h −1 ¯ ∗ supported in τ (Mh \ Mh ), such that τ λ ⊗ OM¯ h (−E) is ample. So Theorem 0.3 says that for given Y and S the degree of ϕ∗ λν is bounded, which implies Corollary 0.4. Using a different argument (see [VZ02, 6.2]) one can extend 0.4 to families of minimal models of arbitrary (nonnegative) Kodaira dimension. As we will see in Section 2 the Theorems 0.1, 0.2 and 0.3 follow from the study of variations of Hodge structures of certain families obtained as cyclic coverings of X → Y . (p)

420


It remains the question, what is special about families with few singular fibres. Here the only results are for families controlled by their variation of Hodge structures Rn f∗ CV (or R1 f∗ CV for families of Abelian varieties). Moreover, we will assume that the family has semistable reduction, or slightly weaker, that the local monodromies in all s ∈ S are unipotent. Before stating the results, and before giving “few singular fibres” a precise meaning, we will need some notations. Let V ⊂ Rk CV be a C sub variation of Hodge structures. V⊗C OU extends to a locally free sheaf H on Y in such a way that the Gauß-Manin connection acquires logarithmic singularities. We choose for H the Deligne extension, i.e. an extension such that the real part of the local residues are zero. The Hodge filtration extends to a holomorphic filtration on H, and the extended GaußManin connection defines on the associated graded bundle the structure of a logarithmic Higgs bundle (F, τ ), i.e. a locally free sheaf F together with a collection of maps τp,q : F p,q −−→ F p−1,q+1 ⊗ ΩY1 (log S),

p + q = k.

The bundle maps τp,q can be iterated to obtain τ () : F k,0 −−→ F k−1,1 ⊗ ΩY1 (log S) −−→ · · · −−→ F k−, ⊗ S (ΩY1 (log S)). Definition 0.5. i. We call τ (k) : F k,0 → F 0,k ⊗ S k (ΩY1 (log S)) the Griffiths-Yukawa coupling of V (or of f in case V = Rk f∗ CV ). ii. The Higgs field is strictly maximal, if F k,0 = 0 and if the τp,q are all isomorphisms. As it will turn out, the property ii) is of numerical nature. Lemma 0.6 ( [VZ05]). Assume that V is a variation of polarized complex Hodge structures of weight n with unipotent local monodromy in all s ∈ S, and with logarithmic Higgs bundle ( F p,q , τp,q ). If F k,0 = 0 one has the Arakelov inequality k deg(F k,0 ) ≤ · rank(F k,0 ) · deg(ΩY1 (log S)), (1) 2 and (1) is an equality if and only if one has a decomposition V = V1 ⊕ V2 where the Higgs field of V1 is strictly maximal and where V2 is a variation of polarized complex Hodge structures, zero in bidegree (k, 0). For families of curves, the inequality (1) is due to Arakelov, and for families of Abelian varieties it has been shown by Faltings in [Fa83]. If #S is even, [VZ03, 3.4] gives a more precise description of V1 . Choose a logarithmic theta characteristic, i.e. an invertible sheaf L with L2 = ΩY1 (log S). The existence of V in 0.6 implies that deg(ΩY1 (log S)) > 0. Then the Higgs bundle L ⊕ L−1 with Higgs field


421

L −−→ L−1 ⊗ ΩY1 (log S) τ

is stable. As recalled in Section 1 it must be the Higgs bundle of a polarized variation of Hodge structures L of weight one. L is unique up to the tensor product with a unitary rank one local system, induced by a two-division point of Pic0 (Y ). Lemma 0.7. Assume that #S is even, and that V1 is a variation of Hodge structures with a strictly maximal Higgs field and with unipotent local monodromies. Then there exists a unitary local system T on Y , regarded as a variation of Hodge structures in bidegree (0, 0), with V1 = S k (L) ⊗C T|U . So “special families” or “families with few singular fibres” will be those where the variation of Hodge structures Rk CV contains an irreducible sub variation V with a strictly maximal Higgs field, or equivalently a sub variation V for whose Higgs bundle the equality deg(F k,0 ) =

k · rank(F k,0 ) · deg(ΩY1 (log S)) 2

(2)

holds. Here k = n, except for families of Abelian varieties, where we usually choose k = 1. Let us first consider a semistable family f : X → Y of Abelian varieties, smooth over U = Y \ S, and V = f −1 (U ). Theorem 0.8 ( [VZ04]). Assume that each irreducible and non-unitary sub variation V of Hodge structures in R1 f∗ CV has a strictly maximal Higgs field. Then (replacing U by an étale covering and V by the pullback family) U is a Shimura curve of Hodge type, and f : V → U the corresponding universal family. The construction of Shimura curves will be recalled in Section 3. Let us just mention here, that a Shimura curve U is an étale covering of a certain moduli space of Abelian varieties with prescribed Mumford-Tate group and a suitable level structure. So whenever we talk about Shimura curves, this is a property up to étale coverings of U . This allows in particular to assume that #S is even. As we will recall in Section 4 the converse of Theorem 0.8 was shown in [Moe3]: Theorem 0.9. If f : V → U is the universal family over a Shimura curve its Higgs field is strictly maximal. Because of Kollár’s decomposition [Kol87] (see Lemma 1.9 below) one can 1 (log S) as a direct sum of an ample sheaf A and a subsheaf B, write f∗ ΩX/Y flat for the Gauß-Manin connection. Correspondingly on the local system side one has a decomposition R1 f∗ CV = W ⊕ U, where U is unitary and invariant under complex conjugation, and where the (1, 0) component of W is ample.

422


Hence the Higgs bundle corresponding to W is of the form (A ⊕ A∨ , θ) and using 0.6 one can restate the Theorem 0.8 as: If deg(A) = 12 · rank(A) · deg(ΩY1 (log S)), then U is a Shimura curve of Hodge type, and f : V → U the corresponding universal family. The Lemma 0.7 implies that the Arakelov equality forces W = L ⊗C T for a unitary system T. In particular, if S = ∅, replacing Y by an étale covering, ⊕2·n−2·r . As in [VZ04] one deduces one can assume that T = C⊕r U and U = CU the next Corollary. Corollary 0.10. If S = ∅ consists of an even number of points, and if deg(A) =

1 · rank(A) · deg(ΩY1 (log S)), 2

then (replacing Y by an étale covering) f : X → Y is isogenous to E ×Y · · · ×Y E × B, where B is an Abelian variety of dimension n − r and where E → Y is a modular family of elliptic curves. The proof of Corollary 0.10 is easy if one assumes that the maximal unitary subsystem of R1 f∗ CV is defined over Q, and it will be sketched in Section 3. This condition holds true if S = ∅. For S = ∅ the proof of Theorem 0.8 in [VZ04] also gives back the known classification of Shimura curves. As a supplement to 0.8 we will show in Section 3: Corollary 0.11. If the maximal unitary local subsystem U of R1 f∗ CV is triv¯ ial, then the family f : V → U is rigid. In particular it is defined over Q. Let us consider next families of curves of genus g ≥ 2, and let us return to the local systems L, defined by logarithmic theta characteristics L. Obviously L has a strictly maximal Higgs field. Theorem 0.12 ( [Moe2]). Let f : X → Y be a semistable family of curves of genus g ≥ 2, smooth over U = Y \ S, and V = f −1 (U ). 1. R1 f∗ CV contains a rank two sub variation of Hodge structures L with a strictly maximal Higgs field if and only if U is a Teichm¨ uller curve, and V → U the corresponding universal family. 2. Teichm¨ uller curves are defined over number fields. The definition of a Teichm¨ uller curve will be given in Section 4. Roughly speaking, one considers geodesics in the Teichm¨ uller space, constructed by an Sl(2, R)-action on the real and imaginary part of a given holomorphic differential form. If the quotient by a suitable lattice in Sl(2, R) is an algebraic curve, it is called a Teichm¨ uller curve in Mg . As for Shimura curves, we always allow ourselves to replace the lattice by a smaller one, hence the Teichm¨ uller


423

curve by an étale cover. We say that f : V → U is the universal family if the morphism U → Mg is induced by f . It is a striking fact, that in spite of the differential geometric definition of Teichm¨ uller curves, they can be characterized analytically, or even by a numerical property of the Higgs bundle. This has a number of applications in the theory of Teichm¨ uller curves (see [Moe2], [Moe3] and [BM05]), and it allows to construct new examples of such special curves in Mg . At the same time, Theorem 0.12 allows to use differential geometric properties of Teichm¨ uller curves to study variations of Hodge structures for families of curves. For example, at the present moment the only known proof of the next Corollary relies on Theorem 0.12, and on differential geometric methods. Corollary 0.13. In Theorem 0.12 the variation of Hodge structures R1 f∗ CV can not contain two different sub variation of Hodge structures L1 and L2 , both with a strictly maximal Higgs field. There remains the question whether there are higher rank irreducible sub variations of Hodge structures V with a strictly maximal Higgs field. By Lemma 0.7 those are of the form L ⊗C T, where T is an irreducible unitary system, at least after replacing U by an étale covering of degree two. As we will see in Section 1 rank(T) ≥ 2 implies that S = ∅, and squeezing the methods used to prove 0.2 a bit more, one can show that this is not possible (see [VZ05]). So putting everything together one obtains: Corollary 0.14. Let f : X → Y be a semistable family of curves of genus g ≥ 2, smooth over U = Y \ S. Then R1 f∗ CV does not contain a sub variation of Hodge structures V with a strictly maximal Higgs field and with rank(V) > 2. If S = ∅ then R1 f∗ CV does not contain any sub variation of Hodge structures V with a strictly maximal Higgs field. As we will see in Sect. 4, the rank two local subsystem L ⊂ R1 f∗ CV can be defined over a number field K, i.e. it is of the form LK ⊗K C for a local subsystem LK ∈ R1 f∗ KV , but in general it will not be defined over Q. Replacing K by its Galois hull, for σ ∈ Gal(K/Q) the conjugate local system Lσ can only have a strictly maximal Higgs field, if L = Lσ . The defect in the inequality (1) for Lσ has been studied in [BM05]. In general things are changing, if one replaces f : X → Y by its family of Jacobians. Passing from a family of curves to its Jacobian, one might replace the discriminant locus S by a subset S . Singular fibres with a compact Jacobian, for example those with two components meeting in one point, are not contributing to the discriminant locus of the family of Jacobians. If f : V → U is the universal family over a Teichm¨ uller curve, then the existence of the local subsystem L together with the inequality (1) applied to the family of Jacobians, shows that S = S, hence that all singular fibres of f have a non-compact Jacobian. One can ask, whether there are any Teichm¨ uller curves

424


U ⊂ Mg in the moduli space of curves of genus g which are Shimura curves in the moduli space Ag of Abelian varieties (with a suitable level structure). By [Moe3] the answer is “no” for g ≥ 4 and for g = 2, whereas for g = 3 there exists one example, essentially unique. On the other hand, by the Corollaries 0.13 and 0.14 we know, that the only curves U ⊂ Mg , which are Shimura curves in Ag have to be induced by a family f : X → Y with R1 f∗ CV = L⊕U with rank(L) = 2 and with U unitary. This implies that U is a Teichm¨ uller curve, and that U is not projective. Theorem 0.15. a. For g ≥ 2 the moduli space Mg does not contain any compact Shimura curve. b. Mg contains a non-compact Shimura curve U if and only if g = 3. The uller curve, essentially unique. Shimura curve U ∈ M3 is a Teichm¨ In this article, we will explain the main ingredients used in the introduction, and we will sketch the proofs of some of the results mentioned above. In Section 1 we explain the Simpson correspondence for variations of Hodge structures over curves, and we will prove Lemma 0.6 and Lemma 0.7 for k = 1. For k > 1, Lemma 0.6 will be shown under the additional assumption that rank(F k,0 ) = 1. In Section 2 we prove Theorem 0.3, and we give some hints, how one obtains Corollary 0.14 from Corollary 0.13. The characterization of Shimura curves will be discussed in Section 3. We prove Theorem 0.8 for families where R1 f∗ CV itself has a strictly maximal Higgs field, hence we exclude unitary direct factors. We sketch the proofs of Corollaries 0.10 and 0.11 and we also reproduce the proof of Theorem 0.9 from [Moe3]. Section 4 gives an introduction to the theory of Teichm¨ uller curves. In particular we sketch the proof of Theorem 0.12 and of the Corollary 0.13. Finally the non-existence of Teichm¨ uller curves in Mg , for g = 2 and g ≥ 4 and the proof of Theorem 0.15, b), will be discussed in Section 5.

1 Higgs Bundles over Curves and Arakelov Inequalities We will frequently use C. Simpson’s correspondence between polystable logarithmic Higgs bundles of degree zero and representations of the fundamental group π1 (U, ∗). Recall that a logarithmic Higgs bundle is a locally free sheaf E on Y together with an OY linear morphism θ : E → E ⊗ ΩY1 (log S) with θ ∧ θ = 0. The usual definitions of stability (and semistability) for locally free sheaves extend to Higgs bundles, by requiring that µ(F ) =

deg(E) deg(F ) < µ(E) = rank(F ) rank(E)

(or µ(F ) ≤ µ(E)) for all subsheaves F with θ(F ) ⊂ F ⊗ ΩY1 (log S).


425

Theorem 1.1 (C. Simpson [Si90]). There exists a natural equivalence between the category of direct sums of stable filtered regular Higgs bundles of degree zero, and of direct sums of stable filtered local systems of degree zero. We will not recall the definition of a “filtered regular” Higgs bundle [Si90, page 717], and just remark that for a Higgs bundle corresponding to a local system V with unipotent monodromy around the points in S the filtration is trivial, and automatically deg(V) = 0. In general it is impossible to describe the Higgs bundle (E, θ) explicitly in terms of the corresponding local system V, with two exceptions: Example 1.2. If T is a unitary local system on U with unipotent monodromy operators in s ∈ S, then the corresponding Higgs bundle is the Deligne extension N of N0 = T ⊗ OU and the Higgs field θ is the zero map. The second example already occurred in the Introduction. Example 1.3. Let V be a polarized C variation of Hodge structures of weight k and with unipotent local monodromy operators. The F -filtration on F0 = V ⊗C OU extends to a locally splitting filtration on the Deligne extension F F k+1 ⊂ F k ⊂ · · · ⊂ F 0 . We will usually assume that F k+1 = 0 and F 0 = F . The Griffiths transversality condition for the Gauß-Manin connection ∇ says that ∇(F p ) ⊂ F p−1 ⊗ ΩY1 (log S), and hence ∇ induces a OY linear map θp,k−p : F p,k−p = F p /F p+1 −−→ F p−1,k−p+1 = F p−1 /F p ⊗ ΩY1 (log S). So

F = F p,k−p , θ = θp,k−p p

is a Higgs bundle, and it is the image of V under the Simpson correspondence in Theorem 1.1. Properties 1.4. Let V be a direct sum of irreducible C local system and let (F, θ) be the corresponding Higgs bundle. 1. F = F p,q with θ(F p,q ) ⊂ F p−1,q+1 ⊗ ΩY1 (log S) if and only if V is a p+q=k

local system underlying a complex polarized variation of Hodge structures. 2. Under the equivalent conditions in 1) let T be a local subsystem of V with Higgs bundle N= N p,k−p , 0 ⊂ F = F p,k−p , θ = θp,k−p . p

Then T is unitary.

p

426


Proof. The condition on (F, θ) in 1) is just saying, that it is a system of Hodge bundles, as defined in [Si88]. So the first part follows from [Si88] and [Si90]. Let Θ(N, h) denote the curvature of the Hodge metric h on F restricted to N, then by [Gr84], chapter II we have Θ(N, h|N ) = −θN ∧ θ¯N − θ¯N ∧ θN = 0. This means that h|N is a flat metric. Hence, T is a unitary local system.

The Simpson correspondence does not say anything about the field of definition for V. Here we say that V is defined over a subfield K of C if there is a K-local system VK with V = VK ⊗K C. In different terms, for µ = rank(V) the representation γV : π0 (U, ∗) −−→ Gl(µ, C) is conjugate to one factoring like γV : π0 (U, ∗) −−→ Gl(µ, K) −−→ Gl(µ, C). If V is defined over K, and if σ : K → K is an isomorphism, we will write VσK for the local system defined by γV : π0 (U, ∗) −−→ Gl(µ, K) −−→ Gl(µ, K ), σ

and Vσ = VσK ⊗K C. By [De71] (for K = Q) and by [De87] the category of polarized Kvariations of Hodge structures is semisimple. By [De87, Proposition 1.13] one has: Lemma 1.5. A local system V, underlying a polarized variation of Hodge structures, decomposes as V=

r

(Vi ⊗ Wi ),

i=1

where Vi are pairwise non-isomorphic irreducible C-local systems and Wi are non-zero C-vector spaces. Moreover the Vi and the Wi carry polarized variations of Hodge structures, whose tensor product and sum gives back the Hodge structure on V. The Hodge structure on the Vi (and Wi ) is unique up to a shift of the bigrading. Suppose that W is a local system defined over a number field L. The local system WL is given by a representation ρ : π1 (U, ∗) → Gl(WL ) for the fibre WL of WL over the base point ∗. Fixing a positive integer r < µ let G(r, W) denote the set of all rankr local subsystems of W and let Grass(r, WL ) be the Grassmann variety of r-dimensional subspaces. Then G(r, W) is the subvariety of Grass(r, WL ) ×Spec(L) Spec(C)


427

consisting of the π1 (U, ∗)-invariant points. In particular, it is a projective variety defined over L. A K-valued point of G(r, W) corresponds to a local subsystem of WK = WL ⊗L K. One obtains the following well known property. ¯ Lemma 1.6. If [V] ∈ G(r, W) is an isolated point, then V is defined over Q. Lemma 1.7. Let W be a polarized variation of Hodge structures defined over L, and let V ⊂ W be an irreducible local subsystem of rank r defined over C,. Then V can be deformed to a local subsystem Vt ⊂ W, which is isomorphic to V and which is defined over a finite extension of L. Proof. By Lemma 1.5 W is completely reducible over C. Hence we have a decomposition W = V ⊕ V . The space G(r, W) of rank r local subsystems of W is defined over L and the subset pr1

{Vt ∈ G(r, W); the composite Vt ⊂ V ⊕ V −−→ V is non zero } forms a Zariski open subset. So there exists some Vt in this subset, which is defined over some finite extension of L. Since p : Vt → V is non zero, since rank(Vt ) = rank(V), and since V is irreducible, p is an isomorphism. The Lemmata 1.6 and 1.7 are just the starting points to show, that certain local subsystems of Rk f∗ CV are defined over number fields (see [VZ04]). Let us give a typical example: Lemma 1.8. Let W be defined over a (real) number field, and let W = V ⊕ U be a decomposition such that the Higgs field of V is maximal, and such that U is unitary. 1. Then V and U are defined over a (real) number field, as well as the decomposition. 2. If S = ∅, then U is defined over Q and it trivializes over an étale covering of Y . Proof. Consider a family Vt of local subsystems of W for t in a small disk ∆, and with V0 = V. Since the Higgs field of V0 is maximal, one may assume that the one of Vt is maximal for all t ∈ ∆. Then the projection Vt → U must be zero. Otherwise, the complete reducibility of local systems coming from variations of Hodge structures in 1.5 implies that Vt and U have a common factor, obviously a contradiction. ¯ The same arguSo V is rigid as a local subsystem, hence defined over Q. ment works for U instead of V. ¯ → U again has to be the zero map. So V If W is defined over R, then V and U have to be defined over R. Assume now that S = ∅. By (1) we know that the decomposition is defined over a number field. For the local monodromy operators “unitary and

428


unipotent” implies that the nilpotent part of the monodromy is zero. This is ¯ invariant under conjugation by σ ∈ Gal(Q/Q), and Uσ = U. Finally, since U is a local subsystem of a variation of Hodge structures, it carries a Z-structure. Then the image of the corresponding monodromy representation is finite. Lemma 1.5 together with Theorem 1.1 implies that the Higgs bundle (F, θ) of a variation of Hodge structures V is polystable (as a Higgs bundle) and of degree zero. As a first application one obtains Kollár’s decomposition of the sheaf F k,0 , mentioned in the introduction. Lemma 1.9. Let (F = F k,0 ⊕ · · · ⊕ F 0,k , θ) be the logarithmic Higgs bundle of a variation of Hodge structures V with unipotent local monodromies. Then F k,0 = A ⊕ B with A ample and with B flat. Proof. Replacing V by one of its irreducible direct factors, one has to show that F k,0 is ample, provided that θ = 0. A quotient sheaf N of F k,0 gives rise to a quotient Higgs bundle (N , 0) of (F, θ), hence 1.1 implies that deg(N ) ≥ 0. If deg(N ) = 0, then it corresponds to a local subsystem, which is excluded by the irreducibility of V. Hence all quotients of F k,0 have strictly positive degree, which implies that F k,0 is ample. Let us consider for a moment the case k = 1 and V = R1 f∗ CV . Then (B ⊕ B∨ , 0) is a sub Higgs bundle, hence by Lemma 1.4, 2), it corresponds to a unitary subbundle of V. The latter is defined over Q and by Lemma 1.8, (1), ¯ ∩ R. this decomposition is defined over Q The Arakelov inequality stated in Lemma 0.6 follows from the polystability of the Higgs bundles (see [VZ03] and [VZ05]). We will indicate the proof just in two simple cases, the one of weight one variations of Hodge structures, and the one where the (k, 0) part is one dimensional. Proof of Lemma 0.6 for weight one. Let A ⊂ F 1,0 be a subsheaf, and let C ⊗ ΩY1 (log S) be its image under θ1,0 . Then A ⊕ C is a Higgs subbundle of F 1,0 ⊕ F 0,1 , and 1.1 implies that deg(A) + deg(C) ≤ 0. Since (ker(θ|A ), 0) is a sub Higgs bundle, one has deg(A) ≤ deg(C) + rank(C) · deg(ΩY1 (log S)) ≤ deg(C) + rank(A) · deg(ΩY1 (log S)) ≤ − deg(A) + rank(A) · deg(ΩY1 (log S)), (3) and

deg(A) 1 ≤ deg(ΩY1 (log S)). rank(A) 2

For A = F 1,0 one obtains the inequality (1) in 0.6.


429

If (1) is an equality, all the inequalities in (3) are equalities. This implies that rank(A) = rank(C), hence that θ|A is injective. Furthermore deg(A) + deg(C) = 0, hence A⊕C corresponds to a sub-local system of V with a maximal Higgs field. In fact we have shown a bit more. Assume that (1) in Lemma 0.6 is an equality. If V is irreducible 1 deg(F 1,0 ) deg(A) < deg(ΩY1 (log S)) = , rank(A) 2 g0 except for A = F 1,0 . Hence F 1,0 is stable. By duality one obtains the same for F 0,1 . If V is not irreducible, applying this argument to all the direct factors of V one finds: Lemma 1.10. If (F 1,0 ⊕F 0,1 , θ) is the logarithmic Higgs bundle of a variation of Hodge structures V with unipotent local monodromies and with a strictly maximal Higgs field, then the sheaves F 1,0 and F 0,1 are polystable and the Higgs field θ : F 1,0 → F 0,1 ⊗ ΩY1 (log S) is a morphism between polystable sheaves of the same slope. Although in this survey we only consider families over curves, let us make one remark about families over a higher dimensional basis. One can hope that the polystability of the Hodge bundles F 1,0 and F 0,1 is again enforced by a numerical condition on the degrees with respect to ωY (S), and that it forces U = Y \ S to be a Shimura variety. Some first results in this direction have been obtained in [VZ05b]. Proof of Lemma 0.7 for k = 1. By assumption #S is even, so we can choose a logarithmic theta characteristic L and an isomorphism τ : L → L−1 ⊗ ΩY1 (log S), giving us the C-variation of Hodge structures L. By Lemma 1.10, the sheaf T = F 1,0 ⊗ L−1 is polystable of degree zero, hence isomorphic to T ⊗ OY for a unitary local system T. The isomorphism θ

T ⊗ L = F 1,0 −−→ F 0,1 ⊗ ΩY1 (log S) −−→ F 0,1 ⊗ L2

induces an isomorphism φ : T ⊗ L−1 −−→ F 0,1 , such that θ = φ ◦ (idT ⊗ τ ). Hence the Higgs bundles (F 1,0 ⊕ F 0,1 , θ) and (T ⊗ (L ⊕ L−1 ), idT ⊗ τ ) are isomorphic, and V T ⊗C L. Proof of Lemma 0.6 under the additional assumption rank(F k,0 ) = 1. For later use, let us consider a slightly more general situation: V is a variation of polarized complex Hodge structures and with unipotent

430


local monodromy in s ∈ S, and with logarithmic Higgs bundle ( F k,0 , θ). Let H ⊂ F k,0 be an invertible subsheaf, and let E ⊂ F be the sub Higgs bundle generated by H. Writing TY (− log S) for the dual of ΩY1 (log S), for some q0 H ⊗ TY (− log S)q for q ≤ q0 k−q,q k−q,q E =F ∩E = . 0 for q > q0 . Theorem 1.1 implies that deg(E) = (q0 + 1) · deg(H) −

q0 · (q0 + 1) deg(ΩY1 (log S)) ≤ 0, 2

hence

k q0 deg(ΩY1 (log S)) ≤ deg(ΩY1 (log S)). (4) 2 2 In Lemma 0.6 we choose H = F k,0 , and we obtain the Arakelov inequality. If this is an equality, then q0 = k and E corresponds to a local subsystem, by construction with a strictly maximal Higgs field. deg(H) ≤

2 Coverings and Bounds for Subsheaves of the Direct Images Let us recall from [VZ05] the proof of the following proposition, as a first application of the theory of Higgs bundles. Proposition 2.1. Let f : X → Y be a semistable family of n-folds over a curve Y ,and smooth over U = Y \ S. If Y = P1 assume that #S ≥ 2, and ν . Then consider for ν ≥ 1 an invertible subsheaf H of f∗ ωX/Y deg H ≤


Before sketching the proof of the proposition, let us show that it implies Theorem 0.3. Proof of Theorem 0.3. Since either g(Y ) ≥ 1, or Y = P1 and #S ≥ 2, one can always find a covering Y → Y , which is unramified over U and which has a prescribed ramification order in s ∈ S. Since f is semistable, the inequality in Theorem 0.3 is compatible with replacing Y by Y and f by a desingularization of the pullback family. ν ) consider the family f (r) : X (r) → Y , obtained as a For r = rank(f∗ ωX/Y desingularization of the total space of the r fold product X r = X ×Y · · · ×Y X −−→ Y. Remark that X r is normal, Gorenstein with at most rational singularities. Hence flat base change implies that

Families of Manifolds over Curves (r)

ν f∗ ω X (r) /Y =

r =

431

ν f∗ ωX/Y .

The family is perhaps not semistable, but replacing Y by a covering sufficiently ramified in s ∈ S the pullback will have this property (This step is not needed if one replaces in 2.1 “semistable” by “unipotent local monodromy”). (r) ν ν ) ⊂ f∗ ω X So Proposition 2.1 implies for det(f∗ ωX/Y (r) /Y that ν )≤ deg(f∗ ωX/Y

ν )·n·ν rank(f∗ ωX/Y

2

· deg(ΩY1 (log S)).

Proof of 2.1. For ν = 1 this is the inequality (4), obtained at the end of the last section. Hence we will only consider the case ν > 1 in the sequel. 1 Assume that deg(H) > n·ν 2 · deg(ΩY (log S)). Replacing Y by a finite covering, étale over U , one may assume that deg(H) = ν · ρ >


Let P be an effective divisor on Y of degree ρ. Then H ⊗ OY (−ν · P ) is in Pic0 (Y ), hence divisible. So for some invertible sheaf N of degree zero, for F = f ∗ P and for L = ωX/Y ⊗ f ∗ N ⊗ OX (−F ) the sheaf Lν has a non-zero section σ. It gives rise to a cyclic covering of X whose desingularization will ˆ (see [EV92], for example). Then for some divisor Tˆ the be denoted by W ˆ ˆ morphism h : W → Y will be smooth over Y \ Tˆ, but not semistable. Choose Y to be a covering, sufficiently ramified, such that the pullback family has a semistable model over Y . ˆ ×Y Y , and Next choose W to be a Z/ν equivariant desingularization of W Z to be a desingularization of the quotient. Finally let W be the normalization ˆ ×Y Y . So we have a diagram of Z in the function field of W τ

ϕ

δ

W −−−−→ Z −−−−→ X −−−−→ X ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ g f f

h

=

=

(5)

ϕ

Y −−−−→ Y −−−−→ Y −−−−→ Y. The ν-th power of the sheaf M = δ ∗ ϕ∗ L has the section σ = δ ∗ ϕ∗ (σ). The sum of its zero locus and the singular fibres will become a normal crossing divisor after a further blowing up. Replacing Y by a larger covering, one may assume that Z → Y is semistable, and that Z and D satisfy the assumption iii) stated below. For a suitable choice of T one has the following conditions: i. X = X ×Y Y , and τ : W → Z is the finite covering obtained by taking the ν-th root out of σ ∈ H 0 (Z, Mν ).

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ii. g and h are both smooth over Y \ T for a divisor T on Y containing ϕ−1 (S). Moreover g is semistable and the local monodromy of Rn h∗ CW \h−1 (T ) in t ∈ T are unipotent. iii. δ is a modification, and Z → Y is semistable. Writing ∆ = g ∗ T and D for the zero divisor of σ on Z, the divisor ∆ + D has normal crossing and Dred → Y is étale over Y \ T . iv. g∗ (ωZ/Y ⊗ M−1 ) = ϕ∗ (N −1 ⊗ OY (P )), In fact, since f : X → Y is semistable, X has at most rational double points. Then −1 −1 δ∗ (ωZ/Y ⊗ δ ∗ ϕ∗ ωX/Y ) = δ∗ (ωZ/Y ⊗ δ ∗ ωX /Y ) = δ∗ ωZ/X = OX ,

which implies iv). The properties i), ii) and iii) hold by construction. p ∗ ∗ 1 W might be singular, but the sheaf ΩW/Y (log τ ∆ ) = τ ΩZ/Y (log ∆ ) is locally free and compatible with desingularizations. The Galois group Z/ν p ∗ acts on the direct image sheaves τ∗ ΩW/Y (log τ ∆ ). As in [EV92] or [VZ05, Section 3] one has the following description of the sheaf of eigenspaces. Claim. Let Γ be the sum over all components of D, whose multiplicity is not divisible by ν. Then the sheaf p −1 ΩZ/Y ⊗ OZ (log(Γ + ∆ )) ⊗ M

> D ? , ν

p ∗ is a direct factor of τ∗ ΩW/Y (log τ ∆ ). Moreover the Z/ν action on W induces a Z/ν action on W = Rn h∗ CW \τ −1 ∆

and on its Higgs bundle. One has a decomposition of W in a direct sum of sub variations of Hodge structures, given by the eigenspaces n for this action, and the Higgs bundle of one of them is of the form G = q=0 Gn−q,q for p > D ? −1 . Gp,q = Rq g∗ ΩZ/Y ⊗ OZ (log(Γ + ∆ )) ⊗ M ν The Higgs field θp,q : Gp,q → Gp−1,q+1 ⊗ ΩY1 (log T ) is induced by the edge morphisms of the exact sequence p−1 ∗ 1 0 −−→ ΩZ/Y (log(Γ + ∆ )) ⊗ g ΩY (log T ) p −→ 0, (6) −−→ ΩZp (log(Γ + ∆ )) −−→ ΩZ/Y (log(Γ + ∆ )) −

tensorized with M−1 ⊗ OZ

> D ? . ν

The sheaf n > D ? −1 Gn,0 = g∗ ΩZ/Y ⊗ OZ (log(Γ + ∆ )) ⊗ M ν


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contains the invertible sheaf n −1 = g∗ (ωZ/Y ⊗ M−1 ) = ϕ∗ (N −1 ⊗ OY (P )). H = g∗ ΩZ/Y (log ∆ ) ⊗ M of degree deg(ϕ) · ρ. Claim. H generates a sub Higgs bundle (H =

n

H n−q,q , θ|H )

q=0

of (G, θ) where H n−q−1,q+1 = Im θ|H n−q,q : H n−q,q → Gn−q+1,q+1 ⊗ ωY (T ) ⊗ ϕ∗ ωY (S)−1 . For some q0 the sheaf H n−q,q is invertible of degree deg(ϕ) · ρ − q · deg(ΩY1 (log S)) for q ≤ q0 and zero for q > q0 . Proof. Writing ∆ = f ∗ (S) consider the tautological exact sequences p−1 p p 0 → ΩX/Y (log ∆) ⊗ f ∗ ΩY1 (log S) −−→ ΩX (log ∆) −−→ ΩX/Y (log ∆) → 0, (7)

tensorized with

−1 n ωX/Y = (ΩX/Y (log ∆))−1 .

Taking the edge morphisms one obtains a Higgs bundle starting with the (n, 0) part OY . The sub Higgs bundle generated by OY has ΩY1 (log S)−q in degree (n − q, q). Tensorizing with N −1 ⊗ OY (P ) one obtains a Higgs bundle H0 with H0n−q,q = N −1 ⊗ OY (P ) ⊗ ωY (S)−q . On the other hand, the pullback of the exact sequence (7) to Z is a subsequence of p−1 p p ∗ 1 0 → ΩZ/Y (log ∆ ) ⊗ g ΩY (log T ) → ΩZ (log ∆ ) → ΩZ/Y (log ∆ ) → 0,

hence of the sequence (6), as well. So the Higgs field of ϕ∗ H0 is induced by the edge morphism of the exact sequence (6), tensorized with −1 ⊗ f ∗ (N −1 ⊗ OY (P ))), ϕ∗ (ωX/Y

> ? . or with the larger sheaf M−1 ⊗ OZ D ν One obtains a morphism of Higgs bundles ϕ∗ H0 → G. By definition ⊂

ϕ∗ H0n,0 = ϕ∗ (N −1 ⊗ OY (P )) = H n,0 −−→ Gn,0 , and H is the image of ϕ∗ H0 in G.

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The last claim implies that the degree of H is q0 deg(ϕ) · (q0 + 1) · ρ − q · deg(ΩY1 (log S)) = q=0

q0 · (q0 + 1) · deg(ΩY1 (log S)) > deg(ϕ) · (q0 + 1) · ρ − 2 (q0 + 1) · n q0 · (q0 + 1) − · deg(ΩY1 (log S)) ≥ 0, deg(ϕ) · 2 2

contradicting the polystability of (G, θ).

Corollary 2.2. [VZ05, Theorem 6] Consider a semistable family f : X → Y of n-folds over a curve Y , smooth over U = Y \ S. Let V be a C-sub variation of Hodge structures in Rn f∗ CV , with (n, 0) part F n,0 and with a strictly maximal Higgs field. Then the rational map π : X → P(F n,0 ), defined by f ∗ F n,0 → ωX/Y , can not have a non-isotrivial image. In particular π can not be birational. Sketch of the proof. By Lemma 0.7 one may assume that F n,0 = Ln ⊗ T for a logarithmic theta characteristic L and for a unitary bundle T on Y . Let Bν denote the image of the multiplication map ν . m : S ν (F n,0 ) −−→ f∗ ωX/Y

If the image of π is non-isotrivial, the method used to prove [Vi95, Theorem 4.33] shows that deg(Bν ⊗ L−ν·n ) has to be larger than zero, for some ν ' 1. For r = rank(Bν ) one obtains an inclusion det(Bν ) −−→

r =

ν f∗ ωX/Y .

Since

r·n·ν ν · deg(ωX/Y ) 2 this contradicts Proposition 2.1, applied to a desingularization of the r fold fibre product X ×Y · · · ×Y X → Y . deg(Bν ) > r · ν · n · deg(L) =

Sketch of the proof of: “0.13 =⇒ 0.14”. We assume again that V has a maximal Higgs field, that rank(V) > 2 and we write F 1,0 for the first Hodge bundle. By Corollary 2.2 it only remains to consider the case where the image W of π is isotrivial. For n = 1 one shows (replacing Y by an étale covering) that W → Y is either of the form C × Y → Y , for some curve C of positive genus, or it is a P1 bundle. In the first case, it is easy to see, that P(F 1,0 ) is trivial, which implies by Lemma 0.7 that V is the direct sum of copies of L contradicting Corollary 0.13. A similar argument works in the second case, and we refer to [VZ05] for the details.


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3 Maximal Higgs Fields and Shimura Curves In this section f : X → Y is a family of g-dimensional Abelian varieties. We assume as usual that the family is semistable, or slightly weaker, that the local monodromies of R1 f∗ CV in s ∈ S are all unipotent. We will sketch the proof of Theorem 0.8 under the quite restrictive assumption Assumption 3.1. V = R1 f∗ CV has no unitary part. Of course we will also assume that the Higgs field of R1 f∗ CV is strictly maximal. So the Assumption 3.1 will exclude all non rigid families. The proof of Theorem 0.8 without it is unfortunately much more difficult. For S = ∅ the Assumption 3.1 is not really a restriction. If R1 f∗ CV = V ⊕ U where V has a strictly maximal Higgs field, and where U is unitary, the assumption on S implies by Lemma 1.8, (2), that (replacing U by an étale covering) this decomposition is defined over Q and that U is trivial. So f : X → Y is isogenous to the product of a family f : X → Y , with V as variation of Hodge structures, and a constant Abelian variety. In this case, the Assumption 3.1 holds if one replaces f : X → Y by its “moving part” f : X → Y . By Lemma 0.7 there is a decomposition R1 f∗ CV = L ⊗C T, and by the choice of L one has det(L) = C, hence det(T) = C, as well. Again one can show, that such a decomposition exists with L and T defined over real number fields. Here one applies the arguments used in the proof of Lemma 1.8 to the local system End(L ⊗ T). One has a description of g-fold wedge products of tensor products (see [FH91], for example) in terms of partitions λ = {λ1 , . . . , λν } of g. By definition λ1 , . . . , λν are natural numbers with g = λ1 + · · · + λν . The partition λ defines a Young diagram and a Schur functor Sλ . Some standard elementary calculations show that: Lemma 3.2. a. If k is odd, then for some partitions λc , k @

k−1

(L ⊗ T) =

2

S 2c+1 (L) ⊗ Sλ2c (T).

c=0

b. If k is even, then for some partitions λc , k @

2 −1 k

(L ⊗ T) = S (L) ⊕ S{2,...,2} (T) ⊕ k

c=1

Lemma 3.3. a. If k is odd, H 0 (Y,

k @

(L ⊗ T)) = 0

S 2c (L) ⊗ Sλ2c (T).

436


b. If k is even, say k = 2c, H 0 (Y,

k @

(L ⊗ T)) = H 0 (Y,

k @ (L ⊗ T))c,c = H 0 (Y, S{2,...,2} (T)).

Proof. S (L) has a strictly maximal Higgs field for > 0, whereas for all partitions λ the variation of Hodge structures Sλ (T) is again pure of bidegree 0, 0. The local system S (L) ⊗ Sλ (T) has again a strictly maximal Higgs field. A global section gives a trivial local subsystem, hence a Higgs subbundle of the form (OY , 0), contradicting the strict maximality of the Higgs field. So S (L) ⊗ Sλ (T) has no global sections. @

?p,p (H 1 (F, Q) ⊕ · · · ⊕ H 1 (F, Q)) .

For a smooth family of Abelian varieties f : V → U there exist the union Σ of countably many proper closed subvarieties of Y such that MT(f −1 (y)) is independent of y for y ∈ U \ Σ (see [De72], [Moo98] or [Sch96]). Let us fix such a very general point y ∈ U \Σ in the sequel and F = f −1 (y). One defines MT(R1 f∗ QV ) to be the Mumford-Tate group of F .


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Consider Hodge cycles η on F which remain Hodge cycles under parallel transform. Then MT(R1 f∗ QV ) is the largest Q-subgroup of Gl(H 1 (F, Q)) which leaves all those Hodge cycles invariant ([De82, §7] or [Sch96, 2.2]). For an algebraic group G the derived group Gder is the subgroup generated by all commutators, or equivalently the kernel of the homomorphism G → Gab for G to the maximal Abelian quotient. Let Mon0 be the algebraic monodromy group, i.e. the connected component of the Zariski closure of the image of the monodromy representation. Let us recall two results from [De82] and [An92] (see [Moo98, 1.4]). Proposition 3.4. a. Mon0 is a normal subgroup of the derived subgroup MT(F )der of MT(F ). b. If for some y ∈ Y the fibre f −1 (y ) has complex multiplication, then Mon0 = MT(R1 f∗ QV )der . Remark however, that up to now we do not know anything about the existence of points with complex multiplication for the families considered in Theorem 0.8. So instead of 3.4, b), we will use: Proposition 3.5. Let f : X → Y be a family of g-dimensional Abelian varieties. Assume that the local monodromies of R1 f∗ CV are all unipotent and that its Higgs field is strictly maximal. Then Mon0 = MT(R1 f∗ QV )der . Proof. Let us write MT = MT(R1 f∗ QV ) and let us fix a very general fibre F of f . By [Si92, 4.4] Mon0 is reductive. By [De82, 3.1 (c)] it is sufficient to show that each tensor η∈

k @ 1 H (F, Q) ⊕ · · · ⊕ H 1 (F, Q) = H k (F × · · · × F, Q)

which is invariant under Mon0 is also invariant under MTder . By abuse of notations, let us replace F × · · · × F by F . A section η ∈ H k (F, Q) \ {0} gives rise to a global section

η ∈ H U, 0

k @

L ⊗C T .

By Lemma 3.3, a) k must be even, and by part b) the section η is pure of bidegree ( k2 , k2 ). So η is a Hodge cycle, and by definition it is invariant under MT. Mumford defines in [Mu66] a Shimura variety X (Hg, u) of Hodge type as a moduli scheme of Abelian varieties (with a suitable level structure) whose Hodge group is contained in Hg. In [Mu69] he gives an explicit construction. X (Hg, u) is the image of centralizer of the Φ : HgR −−→ Sp(H 1 (F, Q), Q)R / = Hg complex structure u

438


divided by an arithmetic subgroup Γ ⊂ Hg. The kernel of Φ is a maximal compact subgroup. The monodromy group for X (Hg, u) is contained in Hg, hence equal to Hgder = MTder ∩ Hg. Since (MTder × MT)/MTder is isogenous to MT, we may replace in Mumford’s construction Hg by Hgder , and the dimension of X (Hg, u) is the dimension of Φ(Hgder ). Proof of Theorem 0.8 under the Assumption 3.1. The decomposition R1 f∗ CV = L ⊗C T implies that the representation defining the local system has values in the product of Sl(2, R) with a compact group. In particular, the dimension of X (Hg, u) is one, and the induced morphism ϕ : U → X (Hg, u) is dominant. Consider the composite ϕ ϕ

U −−→ X (Hg, u) −−→ Ag , where Ag denotes a fine moduli scheme of Abelian varieties, with a suitable level structure. If (F 1,0 ⊕ F 0,1 , θ) denotes the Higgs bundle of R1 f∗ CV the pullback of tangent sheaf of Ag is given by ϕ∗ TAg = S 2 (F 0,1 )|U , and TU → ϕ∗ TAg is the map ∨

−1 ∨ TU −−→ S 2 (F 0,1 |U ) −−→ (F 0,1 ⊗ F 1,0 )|U = L−1 0 ⊗ T0 ⊗ L0 ⊗ T0

induced by θ. Here L0 is the restriction to U of the logarithmic theta characteristic, and T0 = T ⊗ OU . So TU is just the subsheaf L−2 0 ⊗ OU , where OU ⊂ Hom(T0 , T0 ) is given by the homotheties. In particular TU is a direct factor of ϕ∗ TAg , and ϕ and ϕ are both étale. On the proof of Corollary 0.10. By Lemma 1.8, (2), the assumption S = ∅ implies that the unitary system T becomes trivial over some étale covering, as well as the maximal unitary sub-local system U in R1 f∗ CV . So we may assume, that this holds true on Y itself. So f : X → Y is isogenous to a constant Abelian variety, corresponding to the unitary subsystem U and the moving part f : X → Y . By abuse of notations, we will assume that f = f , hence that U = 0. Assume first that the general fibre F of f is simple, hence U = 0, and T = C⊕g . Sections in H 0 (U, End(R1 f∗ CV )) correspond to trivial Higgs subbundles of End((L ⊕ L−1 )⊕g ). The rang of this bundle is 4 · g 2 and a rank 3 · g 2 subsystem has a strictly maximal Higgs field. The rank of the largest sub Higgs bundle with a trivial Higgs field is g 2 , and it is concentrated in bidegree (0, 0). Since the corresponding local sub system of End(R1 f∗ CV ) is unitary, Lemma 1.8


439

implies that it is defined over Q and that it trivializes over an étale covering. So we may assume that 2

H 0 (U, End(R1 f∗ QV )) = H 0 (U, End(R1 f∗ QV ))0,0 = Qg , 2

hence that End(F )Q = Qg . Now take your favorite textbook on Abelian 2 varieties and look for simple ones, with End(F )Q = Qg , and with a positive dimensional non-compact moduli space. The only examples you will find are families of elliptic curves. So f : X → Y is a modular family of elliptic curves, completely determined by the local system L. If F is not simple, apply the argument indicated above to each simple factor of the family. Sketch of the proof of Corollary 0.11. The description R1 f∗ CV = L ⊗ T implies that H 0 (U, End(R1 f∗ CV )) = H 0 (U, End(R1 f∗ CV ))0,0 . Then the rigidity follows from [Fa83].

Let XHg be the Hg(R)+ -conjugacy class in Homalg.grp/R (ResC/R Gm , HgR ) containing u ◦ (ResC/R Gm → S 1 ). Here + denotes the topological connected component. For the reader’s convenience we note that (Hg, XHg ) is a Shimura datum in the sense of [De79, 2.1.1]: Hg is reductive and [De79] Proposition 1.1.14 derives the axioms (2.1.1.1) and (2.1.1.2) from the fact that u defines a complex structure compatible with the polarization. The axiom (2.1.1.3), i.e. the non-existence of a Q-factor in Hgad onto which h projects trivially, follows from Hg being the smallest Q-subgroup of Sp(H 1 (F, Q), Q) containing u. Sketch of the proof of Theorem 0.9. Since Hg is reductive we may split the representation Hg → Sp(H 1 (F, Q), Q) into a direct sum of irreducible representations. We may split off unitary representations over Q ([Kol87] Proposition 4.11). Claim. For each of the remaining irreducible representations there is an isogeny i : Sl2 (R) × K → HgR , where K is a compact group, such that the composition of i with HgR → Sp(H 1 (F, Q), Q)R is the tensor product of a representation of Sl2 (R) of weight one by a representation of K. Assuming the Claim, consider a maximal compact subgroup K1 of Sl2 (R) × K that maps to the centralizer of u under Sl2 (R) × K −−→ Sp(H 1 (F, Q), Q)R .

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The double quotient

X = Γ \(Sl2 (R) × K)/K1

is an unramified cover of X (Hg, U ). Since a strictly maximal Higgs field is characterized by the Arakelov equality we may as well prove that the pullback variation of Hodge structures has a strictly maximal Higgs field. Since the fundamental group of X acts via Γ this follows immediately from the claim and Lemma 2.1 in [VZ04]. Proof of the Claim. We first analyze Hgad and the Hgder (R)-conjugacy class ad ad of maps ResC/R Gm → Hgad containing (Hg → Hgad ) ◦ u. Note that XHg XHg is a connected component of XHg . Each Q-factor of Hgad onto which h projects non-trivially contributes to the dimension of H(Hg, u). Since we deal with Shimura curves Hgad is Q-simple by (2.1.1.3). Let Hgad Gi R = i∈I

ad be its decomposition into simple factors. Then XHg = Xi for Xi a Gi (R)conjugacy class of maps ResC/R Gm → Gi . For the same reason, only one of

the simple factors, say G1 , of Hgad R = i∈I Gi is non-compact. The possible complexifications (G1 )C are classified by Dynkin diagrams. The property ‘Shimura curve’, i.e. dimension one, implies that G1 ∼ = PSl(2, R). "1 → Now we determine the possible representations. The universal cover G G1 factors though G := Ker(Hg → Gi )0 . i∈I\{1}

We apply [De79] Section 1.3 to (G1 , XHg ) ← (G, XHg ) → (Sp, H). 1 factors through Sl(2, R), we Since a finite-dimensional representation of G conclude that G ∼ = Sl(2, R). Moreover, such a representation corresponds to a fundamental

weight, hence it is of weight one. Now we let K be the universal cover of i∈I\{1} Gi . Since Hg → Hgad is an isogeny, there is a lift of the → Hg. This lift factors though a quotient K of K such that universal cover K the natural map Sl(2, R) × K → HgR is an isogeny. Since we assumed the representation HgR → Sp(H 1 (F, Q), Q)R to be irreducible, also ρ : Sl(2, R) × K → HgR → Sp(H 1 (F, Q), Q)R is irreducible. Let W ⊂ H 1 (F, R) be an irreducible (necessarily weight one) representation of Sl(2, R) × {id}. Since K is reductive, hence its representations are semisimple, ρ is the tensor product of W and the representation HomSl(2,R)×{id} (W, H 1 (F, R)) of K. This completes the proof of the claim.


441

4 Teichm¨ uller Curves and their Variation of Hodge Structures We start with the assumptions of Theorem 0.12: Let f : X → Y be a semistable family of curves of genus g ≥ 2, smooth over U = Y \ S and let V = f −1 (U ). Suppose that R1 f∗ CV contains a sub variation of Hodge structures L of rank two with a strictly maximal Higgs field. Let us for the moment admit the following: Lemma 4.1. L is defined over R By Theorem 0.2 the universal covering of U is the upper half plane H. Let ϕ : H → H the period map for L. The strict maximality of the Higgs field is now equivalent to ϕ being an isomorphism (compare [VZ04] Lemma 2.1). In different terms ϕ is an isometry for the Kobayashi metric on H. The choice of a Teichm¨ uller marking (see below) on one of the smooth fibres of f defines a lift uller space. We denote this lift by of the moduli map m : U → Mg to Teichm¨ m : H → Tg and we let j : Tg → Hg the natural map to the Siegel half space, which associates to the curve its Jacobian and to the Teichm¨ uller marking the choice of a symplectic basis. Finally we let p11 : Hg → H the projection of the matrix in Hg to its (1, 1)-entry. If we have chosen the symplectic basis B suitably, i.e. such that the first pair in B spans the fibres of L, then ϕ = p11 ◦ j ◦ m. Since the composite map ϕ is a Kobayashi isometry, also m is an isometry, if we provide the image with the restriction of the Kobayashi metric on Tg . By a uller theorem of Royden the Kobayashi metric on Tg coincides with the Teichm¨ metric. To sum up: The image of U in Mg is an algebraic curve, whose lift in Tg is a complex geodesic for the Teichm¨ uller metric. These objects are called Teichm¨ uller curves. We have thus proved one implication of Theorem 0.12.1. Before showing the converse implication we need some background on geodesics for the Teichm¨ uller metric. Details may be found in [IT92]. One of the equivalent definitions of Teichm¨ uller space is the following. Let Tg be the space of pairs (R, h) of a Riemann surface plus a quasi-conformal mapuller marking) from a ‘reference ping h : R0 → R up to isotopy (the Teichm¨ uller’s theorem the isotopy class of h contains a surface’ R0 to R. By Teichm¨ unique representative with minimal maximal dilatation. Its Beltrami coefficient (a (−1, 1)-form) µ = hz /hz is of the form µ = t|q|/q for some quadratic 1 ⊗2 ) ) and |t| < 1. Conversely, one can solve the Beldifferential q ∈ Γ (R0 , (ΩR 0 trami equation for any µ of with |µ|∞ < 1. In particular for each µ = t|q|/q there is a quasi-conformal map h : R0 → R for some Riemann surface R whose Beltrami coefficient is µ. A quadratic differential q on a Riemann surface R0 determines an atlas of R minus the zero set of q0 such that the transition functions are translations

442


maybe composed by multiplication with ±1: The charts ψi : Ui → C are given by integrating by a square root of q that exists locally. The condition for h to have minimal maximal dilatation may be rephrased by saying that h is affine in the charts ψi for some q on R0 and some (’terminal’) quadratic differential on R. The Teichm¨ uller geodesic from id to h is given by the conformal maps with Beltrami coefficient ct|q|/q for c ∈ [0, 1]. Thus for each q, the set {µq (t) := t|q|/q, |t| < 1} defines totally geodesic subset of Tg . We briefly recall the construction of the complex structure on Tg that makes Tg → Mg holomorphic. Given (R, h) or rather its Beltrami coefficient µ on R0 , pull it back to universal covering H of R0 and extend it by zero to obtain a Beltrami coefficient µ ˜ on C. Solving the Beltrami equation for µ gives ˜ : C → C, conformal on the lower half plane H∗ . a quasi-conformal mapping h ˜ H∗ is a quadratic differential on H∗ . Moreover, The Schwarzian derivative of h| since µ was pulled back from R = H/ΓR , the Schwarzian derivative descends from H∗ to a quadratic differential 1 ⊗2 qµ∗ ∈ H 0 (H∗ /ΓR , (ΩH ). ∗ /Γ ) R

The map µ → qµ∗ factors through Teichm¨ uller space and the complex structure 1 ⊗2 ) is the one we want. on H 0 (H∗ /ΓR , (ΩH ∗ /Γ ) R The map µ → qµ∗ has a section locally around 0 (due to Bers) given by z ). This implies that a neighborhood of 0 of q ∗ (z) → −2Im(z)q ∗ (¯ {µq (t), |t| < 1} is a holomorphic submanifold of Tg . Since the complex structure on Tg is independent of the base point, all of {µq (t), |t| < 1} is a holomorphic submanifold. 1 ⊗2 ) ) fixed, To sum up: For a fixed Riemann surface R0 and q ∈ Γ (R0 , (ΩR 0 the set t|q|/q, |t| < 1 is a totally geodesic and holomorphic submanifold of Tg , a Teichm¨ uller disc. In order to characterize which (few!) Teichm¨ uller discs descend to Teichuller disc. m¨ uller curves in Mg we provide another description of a Teichm¨ There is a natural Sl(2, R)-action on the bundle of triples (R, ϕ, q) over Tg . Post-compose local charts ψi of R with A ∈ Sl(2, R) acting on C ∼ = R2 . Since overlapping charts differ only by translations and multiplication by ±1, the compositions /C= ∼C ∼ R2 A / R2 = ψi : Ui determine a new complex structure on R. We let this new Riemann surface be A · R. There is a unique quadratic differential A · q on A · R whose integration charts are ψi . The action of SO(2, R) does not change the complex structure of R. Hence the orbit Sl(2, R) · (R, ϕ, q) projects to a disc in Tg . If we lift


443

H → Sl(2, R) 1 Re τ τ → 0 Im τ one verifies that the Beltrami coefficient of the composition ψi is the image of the Sl(2, R)-orbit is a Teichm¨ uller disc.

i−τ |q| i+τ q .

Hence

Proposition 4.2. A Teichm¨ uller disc descends to a Teichm¨ uller curve, if and only if the setwise stabilizer of the disc in the mapping class group is a lattice in Aut(H) = Sl(2, R). This stabilizer coincides up to conjugation with a quotient (by a finite group) of the group of orientation-preserving diffeomorphisms that are affine in the charts provided by q. Proof. Proposition 3.2 and Proposition 3.3 in [McM03].

When referring to a Teichm¨ uller curve in the sequel we always assume that it is generated by q = ω 2 the square of a holomorphic one-form. If a Teichm¨ uller curve is generated by a quadratic differential on R, take the double covering of R where it admits a square root. The above criterion gives an easy way to check that this pair generates again a Teichm¨ uller curve. Proof of Corollary 0.13. This is [McM03] Theorem 4.2 in a different language. uller curve. Suppose the contrary was true. Then U → Mg defines a Teichm¨ Over the universal cover H of U we may choose sections ωj = ωj (τ ) for j = 1, 2 generating the (1, 0)-part of Lj . We may moreover choose a symplectic basis {ak , bk } for k = 1, . . . g of R1 (fH )∗ ZV such that bk ωj (τ ) = δj,k for τ ∈ H. We consider the maps Fj : H → H, τ → ωj (τ ) for j = 1, 2. aj

Since the local systems Lj are strictly maximal Higgs both F1 and F2 are isometries. uller disc Let R be the fibre of f over i ∈ H. Since H → Tg is the Teichm¨ generated by ω1 (say), Ahlfors’ variational formula implies that ⏐ dFj ⏐ ω1 (i) ⏐ . = ωj (i) dt ⏐t=i ω1 (i) R Hence the norm of dFj at τ = i in the hyperbolic metric equals ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ω1 (i) ⏐ ω1 (i) ⏐ ⏐ ⏐ ⏐ ⏐ ||dFj || = ⏐ ωj (i) |ω( i)|2 . ⏐ / ImFj (i) = ⏐ ωj (i) ⏐ / ⏐ R ⏐ R ω1 (i) ⏐ ω1 (i) ⏐ R Since Fj is an isometry, this implies by the Schwarz Lemma that ||dFj || = 1. Since ω2 is not proportional to ω1 this violates the Cauchy–Schwarz inequality.

444


Sketch of the proof of Lemma 4.1. Since L is also strictly maximal Higgs we have L ∼ = L by Corollary 0.13. It remains to check that L can indeed be defined over R, see [Moe2], proof of Theorem 5.3, or repeat the argument used in the proof of Lemma 1.8. 1 Proof of Theorem 0.12. We fix a fibre R of f and ω ∈ Γ (R, ΩR ) that generates the Teichm¨ uller curve. By its construction as Sl(2, R)-orbit the fundamental group of U maps to a lattice Γ in Sl(2, R) and the subspace

Re ω, Im ω ⊂ H 1 (R, R) is invariant under the action of the fundamental group. Let L be the corresponding irreducible rank two C-local system. As in Lemma 1.8 one finds that L is defined over a number field, whose Galois closure we denote by ˜ Moreover we let K the number field generated by the traces of Γ . If we K. apply Deligne’s semisimplicity (Lemma 1.5) to the polarized C-variation of Hodge structures R1 f∗ CV the local system L and all its Galois conjugates will appear. Moreover we may read the above argument of [VZ04] Lemma 2.1 backwards to conclude that L is strictly maximal Higgs. This proves 1. By Corollary 0.13 the local system R1 f∗ CV contains no other local subsystem isomorphic to L. We claim that R1 f∗ CV = W ⊕ M, WK˜ = Lσ , ˜ ˜ σ∈Gal(K/Q)/Gal( K/K)

where the summands W and M are defined over Q and Lσ are pairwise nonisomorphic rank two local systems. The fact that L and Lσ are isomorphic if and only if σ fixes K is not needed here, but see the remark at the end of this section. We now prove 2. The Q-sub variation W determines, after choosing a Zlattice and fixing a polarization of type δ, a family of Abelian varieties A → U of dimension r. By [Fa83] the tangent space to the space of deformations of the moduli map U → Ar,δ is a subspace of the global sections of the local system End(WC ) of bidegree (−1, 1). Using above decomposition of WC into irreducible summands, one checks ([Moe2] Lemma 3.3) that this tangent space is trivial. Hence U → Ar,δ is rigid and U is defined over a number field. The above way of characterizing Teichm¨ uller curves inverts the history of these objects: The first examples, starting with Veech ([Ve89]), of Teichm¨ uller curves were shown to have this property by checking that the stabilizer of uller curves were constructed as H → Tg is a lattice. Next, in [McM03], Teichm¨ the image in the moduli space M2 of the intersection of two higher-dimensional Sl(2, R)-invariant loci in the bundle of one-forms over M2 . Only recently ( [BM05]) Theorem 0.12 was used to construct an infinite series of Teichm¨ uller curves, including the examples of Veech. We address once again the decomposition of the variation of Hodge structures of a Teichm¨ uller curves. Obviously, if L ∼ = Lσ then σ fixes the trace field.


445

The converse is needed in 5.1. We reproduce the full proof, since it is, besides Corollary 0.13, the only argument in this context that relies essentially on Teichm¨ uller theory. By Proposition 4.2, elements in the fundamental group of U may be represented by diffeomorphisms that are affine in the ω-charts. We want to show that if σ does not fix the trace field, then L ∼ Lσ . Choose a hyperbolic ele= ment γ in the fundamental group whose trace t = trγ is not fixed by σ. Lift γ to a diffeomorphism φ of some smooth fibre R of f . Let φ∗ be the induced diffeomorphism of H 1 (R, R). The endomorphism φ∗ + (φ∗ )−1 acts on L|R by multiplication by t = trγ. We need to show that Ker(φ∗ + (φ∗ )−1 − t · id) has dimension two. It suffices to show that the eigenvalues λ+ (resp. λ− ) of φ∗ of maximal (resp. minimal) absolute values are unique and that these are the eigenvalues of φ∗ acting on L|R . This is shown in [McM03] Theorem 5.3: The diffomorphism φ is not of finite order nor does it fix a closed loop in R. Hence φ is pseudo-Ansov ([FLP79]). For such a diffeomorphism there are two transverse measured foliations µ+ , µ− on R, one is expanded by λ+ and one is contracted by λ− = (λ+ )−1 . These two foliations represent real cohomology classes in Re ω, Im ω = L|R . It remains to show that λ+ is simple and of largest absolute value. The corresponding statement for λ− follows by considering φ−1 . The cohomology H 1 (R, R) is spanned by R-linear combinations of the Poincaré duals C ∨ to simple closed curves. We know how ϕ acts on C ∨ : Since C is stretched in the direction of µ+ and contracted in the direction of µ− , we have for n → ∞ that (λ+ )−n (φ∗ )n C ∨ → αµ+ for some α ∈ R. This proves the nonexistence of an eigenspace for φ∗ in H 1 (R, R) with larger eigenvalue than λ+ and the simplicity of λ+ .

5 The only Teichm¨ uller-Shimura Curve This section is a comparison between Teichm¨ uller- and Shimura curves. Recall that in the proof of Theorem 0.12 we defined the family Abelian varieties A → U corresponding to the local system W. It is determined up to isogeny by the Teichm¨ uller curve and coincides with the Jacobian of the universal family over the Teichm¨ uller curve in case r = g. Theorem 5.1. The family A → U has real multiplication by K. The locus of real multiplication is the smallest Shimura subvariety of Ar that contains the image of the moduli map U → Ar . Proof. For each a ∈ K the cycle σ(a) · idLσ ∈ End(WK˜ ) = H 0 (U, End(WK˜ )) Ca := ˜ ˜ Gal(K/Q)/Gal( K/K)

is defined over Q and of bidegree (0, 0). It is hence an endomorphism of A/U ([De71] Remark 4.4.6). Since K is a trace field of a lattice in Sl(2, R) it is real.

446


By the classification of endomorphisms of Abelian varieties it is totally real. This proves the first claim. For the second statement we have to show that the local systems W⊗m ⊗ ∨ ⊗m do not contain Hodge cycles other than products and tensor powers (W ) of the cycles Ca . This is shown in [Moe2] Lemma 3.3. The spirit of this Lemma is similar to the decomposition into Schur functors of exterior powers of the variation of Hodge structures over a Shimura curve in Section 3. The difference consists of an essential use of the uniqueness of the strictly maximal Higgs local system at some point. We now address the question of classifying non-compact Shimura curves that lie entirely inside Mg for g ≥ 2. By Theorem 0.8 the variation of Hodge structures consists of unitary and strictly maximal Higgs local subsystems. By Corollary 0.14 the strictly maximal Higgs part splits off a rank two local subsystem. By Theorem 0.12 such a curve is automatically also a Teichm¨ uller curve. We may now look at the decomposition of the variation of Hodge structures in the proof of Theorem 0.12. Since the Galois conjugate of a noncocompact lattice in Sl(2, R) still contains non-trivial parabolic elements none of the local systems Lσ is unitary. By Theorem 0.8 again there is no Lσ for σ = id. To sum up: Lemma 5.2. Suppose U → Mg is a non-compact Shimura curve. Then there is an unramified cover U → U such that the pullback of the universal family over Mg to U has a family of Jacobians Jac(f) : J → U with fixed part of dimension g − 1, i.e. there is an Abelian variety A of dimension g − 1 such that A × U injects into J. Proof. The only thing left to remark is that the unitary parts in the variation of Hodge structures can be trivialized after an unramified cover U → U (see Lemma 1.8). The possible dimensions d of a fixed part in a family of Jacobians of dimension g over a one-dimensional base have been studied by Xiao. He proves in [Xi87] that 5g + 1 . d≤ 6 This implies that a fixed part of dimension g − 1 can only occur for g ≤ 7. Families of curves of genus g = 2, g = 3 and g − 4 with g − 1-dimensional fixed part are known to exist, but for g = 5, 6, 7 existence is still an open question. Theorem 0.15 may be considered as a non-existence statement for such families under the additional assumption ‘Shimura curve’. Sketch of the proof of Theorem 0.15, b. There are four main ingredients: First, the large fixed part limits the number of possible singular fibres the family f :X →Y. Second, the projection to J → J/A defines a covering π : V → E of a family of elliptic curves. By [Moe1] we may assume that U = X(d) is a modular curve and d = deg(π).


447

Third, all invariants, in particular some intersection numbers, of the complex surface E (or rather its completion) are known. We factor π = i ◦ πopt , where i is an isogeny and πopt does not factor through a non-trivial isogeny. The knowledge of intersection number on V will then be used to bound the degree dopt of πopt . Finally, the translation structure on a fibre R of f given by the one-form 1 ) that generates the Teichm¨ uller curve is shown to have very ω ∈ H 0 (R, ΩR special properties. In a case-by-case discussion depending on dopt and the number of zeros of ω we show that these properties are absurd, except for the case of the Legendre family and the family fW : y 4 = x(x − 1)(x − t) over X(2) ∼ = P1 \ {0, 1, ∞} with local coordinate t in genus three. We now give more details on the first step. Let P1 , . . . , Ps ∈ R be the zeros of ω and let ki (i = 1, . . . , s) be the multiplicity of the zero. Since U → Mg is a Teichm¨ uller curve, we may suppose by [GJ00] Theorem 5.5 that π|R is ramified over one point O ∈ E0 := |π(R) only. Equivalently, the translation structure determined by ω is such that π|R\{P1 ,...,Ps } is a square-tiled covering (see e.g. loc. cit.). One can deduce from this that singular fibres of f do not contain separating nodes ([Moe3] Proposition 2.3) and, more precisely, that the dual graph of a singular fibre is a ring, say of length (y) for y ∈ S = Y \U . (loc. cit. Lemma 2.1). Concerning the second step, we recall that we constantly consider Teichm¨ uller and Shimura curves up to unramified cover outside S, since this does not change the property ‘strictly maximal Higgs’. We may hence replace U by X(d) for d ≥ 3 sufficiently large and post-compose π by an isogeny to maintain the property d = deg(π). Concerning the third step, we remark that the genus of X(d), the number of cusps and the self-intersection of the zero-section of E are well-known as functions of d. From the construction of a Teichm¨ uller curve as Sl(2, R)-orbit one deduces that the zeros Pi extend to sections pi : U → V of f for i = 1, . . . , s. We denote the closure of the image of pi in X by Pi . Since we assume X to be a minimal semistable model, we have ( ' ωX/Y = OX Pi + d∆d · R + D , where R is a fibre of f , D is a divisor supported in the singular fibres of f and d2 1 ∆d = 1− 2 . 24 d p|d

Using the geometry of the singular fibres one checks ([Moe3] Lemma 3.13) that in fact D = 0. From the intersection numbers on E one deduces that on X we have

448


Pi = −d∆d /(ki + 1). Since L satisfies the Arakelov equality and since the remaining local subsystems of R1 f∗ C are trivial, we have deg f∗ ωX/Y = d∆d . We claim that for each of the singular fibres Ry for y ∈ S of f we have ∆χtop (Ry ) := χtop (Ry ) − χtop (R) = d/dopt . Putting all these data in the Noether formula 2 ∆χtop (Ry ) = ωX/Y 12 deg f∗ ωX/Y − y∈S

we obtain dopt = ' s

12 (

k12 i=1 k1 +1

+ 16 − 4g

.

s Since by definition i=1 ki = 2g − 2 there is only a finite number of integer solution to this equation. We thus obtain the desired bound on dopt . To prove the claim, notice first that ∆χtop (Ry ) coincides with the number of nodes of the dual graph of Ry or equivalently with the local component group of the Néron model of the family of Jacobians around at y. The size of the component group at y is known to be d for the universal family of elliptic curves over X(d). The details how to related the two groups are in [Moe3] Proposition 3.15. The reader is referred to [Moe3] for details on the last step. We remark that the exceptional family fW is generated by the degree 8 covering π|R of the once-punctured torus with Galois group the quaternion group. The differential ω is the pullback of a holomorphic differential on E0 . This covering π|R can also be obtained by a double covering of an elliptic curve ramified precisely over all four 2-torsion points post-composed by the multiplication by two.

References [An92] [Be79] [BM05] [De71]

André, Y.: Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part. Compositio Math. 82, 1–24 (1992) Bely˘ı, G.V.: Galois extensions of a maximal cyclotomic field. Izv. Akad. Nauk SSSR Ser. Mat. 43, 267–276 (1979) Bouw, I.I., M¨ oller, M.: Teichm¨ uller curves, triangle groups, and Lyapunov exponents. Preprint (2005) ´ Deligne, P.: Théorie de Hodge II. I.H.E.S. Publ. Math. 40, 5–57 (1971)

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Deligne, P.: La conjecture de Weil pour les surfaces K3. Invent. math. 15, 206–226 (1972) [De79] Deligne, P.: Variétés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques. Proc. Symp. Pure Math. 33, part II, 247–289, (1979) [De82] Deligne, P.: Hodge cycles on abelian varieties. (Notes by J. S. Milne). Springer Lecture Notes in Math. 900, 9–100 (1982) [De87] Deligne, P.: Un théorème de finitude pour la monodromie. Discrete Groups in Geometry and Analysis, Birkh¨ auser, Progress in Math. 67, 1–19 (1987) [EV92] Esnault, H., Viehweg, E.: Lectures on Vanishing Theorems. DMV-Seminar 20, Birkh¨ auser, Basel-Boston-Berlin (1992) [Fa83] Faltings, G.: Arakelov’s theorem for abelian varieties. Invent. math. 73, 337–348 (1983) [FH91] Fulton, W., Harris, J.: Representation Theory. A first course. Graduate Texts in Math. 129, Springer-Verlag, New-York (1991) [FLP79] Fahti, A., Laudenbach, F., Poénaru, V.: Traveaux de Thurston sur les surfaces. Astérique 66-67 (1979) [GJ00] Gutkin, E., Judge, C.: Affine mappings of translation surfaces. Duke Math. J. 103 (2000) [Gr84] Griffths, P.: Topics in transcendental algebraic geometry. Ann of Math. Stud. 106, Princeton Univ. Press. Princeton, N.J. (1984) [IT92] Imayhosi, Y., Taniguchi, M.: An introduction to Teichm¨ uller spaces. Springer Tokyo (1992) [Kol87] Koll´ ar, J.: Subadditivity of the Kodaira Dimension: Fibres of general type. In: Algebraic Geometry, Sendai, 1985. Advanced Studies in Pure Mathematics 10, 361–398 (1987) [Kov02] Kov´ acs, S.: Logarithmic vanishing theorems and Arakelov-Parshin boundedness for singular varieties. Compositio Math. 131, 291–317 (2002) [McM03] McMullen, C.: Billiards and Teichm¨ uller curves on Hilbert modular surfaces. Journal of the AMS 16, 857–885 (2003) [Moe1] M¨ oller, M.: Maximally irregularly fibred surfaces of general type. Manusc. Math. 116, 71–92 (2005) [Moe2] M¨ oller, M.: Variations of Hodge structures of Teichm¨ uller curves, JAMS 19, 327–344 (2006) [Moe3] M¨ oller, M.: Shimura and Teichm¨ uller curves, preprint (2005) [Moo98] Moonen, B.: Linearity properties of Shimura varieties. Part I. J. Algebraic Geom. 7, 539–567 (1998) [Mu66] Mumford, D.: Families of Abelian varieties. Proc. Sympos. Pure Math. 9, 347–351 (1966) [Mu69] Mumford, D.: A note of Shimura’s paper: Discontinuous groups and Abelian varietes. Math. Ann. 181, 345–351 (1969) [Sch96] Schoen, C.: Varieties dominated by product varieties. Int. J. Math. 7, 541–571 (1996) [Si88] Simpson, C.: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. Journal of the AMS 1, 867–918 (1988) [Si90] Simpson, C.: Harmonic bundles on noncompact curves. Journal of the AMS 3, 713–770 (1990) [Si92] Simpson, C.: Higgs bundles and local systems. Publ. Math. I.H.E.S 75 (1992), 5–95 (1992)

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[VZ05b] [Xi87]

Veech, W.: Teichm¨ uller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97, 533–583 (1989) Viehweg, E.: Quasi-projective Moduli for Polarized Manifolds. Ergebnisse der Mathematik, 3. Folge 30, Springer, Berlin-Heidelberg-New York (1995) Viehweg, E.: Compactifications of smooth families and of moduli spaces of polarized manifolds. Preprint 2005, final version 2006 Viehweg, E., Zuo, K.: On the isotriviality of families of projective manifolds over curves. J. Alg. Geom. 10, 781–799 (2001) Viehweg, E., Zuo K.: Base spaces of non-isotrivial families of smooth minimal models. In: Complex Geometry (Collection of Papers dedicated to Hans Grauert), 279–328 Springer, Berlin Heidelberg New York (2002) Viehweg, E., Zuo, K.: Families over curves with a strictly maximal Higgs field. Asian J. of Math. 7, 575 - 598 (2003) Viehweg, E., Zuo, K.: A characterization of certain Shimura curves in the moduli stack of abelian varieties. J. Diff. Geom. 66, (2004) 233–287 (2004) Viehweg, E., Zuo, K.: Numerical bounds for semistable families of curves or of certain higher dimensional manifolds. Preprint 2005, J. Alg. Geom. to appear Viehweg, E., Zuo, K.: Arakelov inequalities and the uniformization of certain rigid Shimura varieties. Preprint 2005 Xiao, G.: Fibred algebraic surfaces with low slope. Math. Ann. 276, 449– 466 (1987)

Hodge Theory and Algebraic Cycles Stefan J. M¨ uller-Stach Institut f¨ ur Mathematik, Fachbereich 08, Johannes Gutenberg–Universit¨ at Mainz, 55099 Mainz, [email protected]

1 Introduction Algebraic cycles and Hodge theory, in particular Chow groups, Deligne cohomology and the study of cycle class maps were some of the themes of the Schwerpunkt ”Globale Methoden in der Komplexen Geometrie”. In this survey we report about several projects around the structure of (higher) Chow groups CH p (X, n) [3] which the author has studied with his coauthors during this time by using different methods. In my opinion there are two interesting view points: first the internal structure of higher Chow groups, i.e., the existence of interesting elements and nontriviality of parts of their Bloch-Beilinson filtrations. This case has arithmetic and geometric features, and the groups in question show different properties depending whether X is the spectrum of a field or a local ring resp. a higher dimensional projective variety. The second point of view is the study of maps 2p−n (X, Z(p)). cp,n : CH p (X, n) → HD

from higher Chow groups to Deligne cohomology in the case X is defined over C as defined in [4]. These generalize Abel–Jacobi as well as regulator maps and are tractable via analytic methods and Hodge theory, even via differential equations, since cycles can often be deformed in families over moduli spaces. More precisely the contents are: structure results for higher Chow groups [18, 21], the discovery of explicit formulas for cp,n [30], higher Chow groups of complex surfaces [39, 38], differential equations associated to families of algebraic cycles [13, 15, 16] and Higgs cohomology of local systems on some locally symmetric varieties with applications to Chow–K¨ unneth decompositions [37].

452

Stefan J. M¨ uller-Stach

2 Higher Chow groups We give two different but equivalent definitions of higher Chow groups: 2.1 Simplicial Version: (Cf. [3], [5]) Let k be any field and k[t0 , . . . , tn ] ∆n = Spec ( ti − 1)

be the affine n-simplex over k with coordinates t0 , ..., tn , satisfying ti = 1. A face of ∆n of codimension m is a subsimplex ∆n−m ⊂ ∆n , obtained by setting m coordinates equal to zero. Let Z p (X, n) be the free abelian group generated by integral closed algebraic subvarieties of codimension p in X × ∆n which are admissible, i.e., meeting all faces of all codimensions again in codimension p (or in the empty set). These pieces form a simplicial abelian group: → → → → · · · Z (X, 3) Z r (X, 2)→Z r (X, 1) Z r (X, 0) → → → → r

and CH p (X, n) is defined to be the n-th homotopy group of this simplicial group. In other words, CH p (X, n) is the n-th homology group of the complex ∂

∂

∂

∂

... → Z p (X, n + 1)→Z p (X, n)→Z p (X, n − 1)→...→Z p (X, 0) where ∂ = (−1)i ∂i is given by the alternating sum of restriction maps to codimension one faces. 2.2 Cubical Version: (Cf. [32]) In the cubical version, the role of the algebraic simplex ∆n is played by the algebraic n–cube n = (P1k \ {1})n . The n–cube has 2n codimension one faces, defined by xi = 0 and xi = ∞, for 1 ≤ i ≤ n, while the boundary maps are given by ∂= (−1)i−1 (∂i0 − ∂i∞ ), where ∂i0 and ∂i∞ denote the restriction maps to the faces xi = 0 resp. xi = ∞. Zcp (X, n) is defined to be the quotient of the group of admissible cycles in X × n by the group of degenerate cycles as defined in [46], p.180 (where they are denoted by dp (X, n)). In [32], Theorem 4.7 (used for empty s), it is shown that the resulting complex is quasiisomorphic to the simplicial complex. Therefore we use the same notation CH p (X, n) for the n–th homology of the complex Zcp (X, ·) as for the simplicial case.

Hodge Theory and Algebraic Cycles

453

2.3 Formal Properties: 1. Functoriality [3]: The groups CH ∗ (X , ∗) are covariant for proper maps and contravariant for flat maps. They are contravariant for all maps between smooth affine schemes by [35](4.5.12.). 2. Products [3, 46]: If X is smooth, there is a product CH p (X , q) ⊗ CH r (X , s) −→ CH p+r (X , q + s) . 3. Homotopy Invariance [3]: For any equidimensional k-scheme X, we have CH ∗ (X , n) ∼ = CH ∗ (X × A1k , n). 4. Localization [3, 5]: If W ⊂ X is a closed subvariety of pure codimension r with X quasi–projective over a field, then one has localization: · · · → CH ∗−r (W , n) → CH ∗ (X , n) → CH ∗ (X − W , n) → · · · 5. In joint work with Philippe Elbaz-Vincent we extended the theorem of Nesterenko/Suslin and Totaro [46] to semi–local rings: if R is an essentially smooth, semi–local k-algebra over an infinite field k, then the graph morphism n n≥1 ϕn : KM n (R) −→ CH (R , n) , which associates to any tuple of functions its graph in the cubical higher Chow groups, is well–defined (all relations in Milnor K–theory also hold in CH n (R, n)) and surjective [18, 31]. 6. The most important property is given by the theorem of Bloch [3] (refined by Levine in [33]) which puts them into relation with (the weight-graded pieces of) Quillen K–theory in the case of a smooth, quasi–projective variety X of dimension d over a field k: Kn (X)(p) ⊗ Z[

1 1 ]∼ ]. = CH p (X, n) ⊗ Z[ (n + d − 1)! (n + d − 1)!

More generally, one has a spectral sequence [34] CH −q (X, −p − q) =⇒ G−p−q (X) (X any equidimensional scheme over a field). 7. Higher Chow groups are related to K-cohomology using the Gersten resolution for Milnor K-theory and the isomorphism of [18]. For example one gets for smooth varieties that CH p (X, n) = H p−n (X, Kp ) for 0 ≤ n ≤ 2. See [18]. 8. By a theorem of Gerdes [22], simplicial higher Chow groups CH p (k, p + q) are generated by linearly embedded cycles for q = 0 and q = 1. If k satisfies the rank conjecture of Suslin, e.g. if k is a number field, then the same holds for all q ≥ 0.

454


Example 2.1. For X = Spec(k), Zagier [49] and Goncharov [23] have constructed combinatorial complexes, involving generalizations of the Bloch groups Bm (k) of a field k as defined in loc. cit., which suggest that there exists a rational isomorphism between Bm (k)Q −→ CH m (k, 2m − 1)Q . This may be seen as a far–reaching project in order to present motivic cohomology in terms of generators and relations (a collaboration with H. Gangl). One of our ideas was to realize this isomorphism by using the parametrized cycles of Bloch/Kriz [6] and Totaro [46]: t2 tm−2 a ,1 − Ca : (t1 , . . . , tm−1 ) → t1 , . . . , tm−1 , 1 − t1 , 1 − , . . . , 1 − t1 tm−1 tm−1 for a ∈ k. For example, the boundary of Ca = (x, 1 − x, 1 − xa ) in the case m = 2 is given by the Steinberg relation: ∂Ca = (a, 1 − a). The Bloch groups Bm (k) are quotients of an inductively defined subgroup of Z[k \ {0, 1}] modulo special polylogarithmic relations. Therefore the first step in this project is to reprove all known functional equations for higher polylogarithms in higher Chow groups. This project was started in [21] for m = 2 and m = 3. There is also related work of Zhao [50] and Goncharov [23] who uses Grassmannian type complexes instead. In his thesis project O. Petras proves several integral relations in CH m (k, 2m − 1) for m = 2 and m = 3. In this case the relations are much more complicated, since the full functional equations for the complex polylogarithm have lower order terms and the map cm,2m−1 involves a bit more than just the polylogarithm [30]. S. Bloch and A. Beilinson have conjectured the existence of a filtration on higher Chow groups of smooth, projective varieties CH p (X, n)Q = F 0 ⊇ F 1 ⊇ . . . which is compatible with products and such that F 1 is given by cycles homologous to zero (which conjecturally is the same as numerically equivalent to zero). S. Saito [45], W. Raskind [44] and others have defined candidates for such filtrations which many of the required conditions. These filtrations are related to the abelian category of mixed motives by the formula GrνF CH p (X, n)Q = Extv (Q(−p), h2p−n−ν (X)) and are essentially unique, if they exist (see Jannsens article in [1]). Below we will describe examples in F 2 CH 3 (X, 2) of complex, projective surfaces [39].


455

3 Cycle Class Maps to Deligne Cohomology In this section we report about joint work with M. Kerr and J. Lewis [30]. Let X be a complex manifold. Then we define naive Deligne cohomology by √ p−1 i HD (X, Z(p)) := Hian (X, (2π −1)p Z → OX → . . . → ΩX ). For non–compact varieties this is not a functorial definition and one has to use Deligne–Beilinson cohomology [2, 19] instead which is defined using the logarithmic de Rham complex. Then we obtain abelian groups sitting in the following exact sequences if X is smooth, quasi–projective over C: i . . . → H i−1 (X, C)/F p → HD (X, Z(p)) → F p ∩ H i (X, Z(p)) → . . .

If X is furthermore smooth and projective over C, we have an exact sequence ε

2p 0 → J p (X) → HD (X, Z(p))→Hgp,p (X) → 0,

where Hgp,p (X) := {α ∈ H 2p (X, Z) | α ⊗ C ∈ F p } is the group of integral Hodge classes. Griffiths’ Abel–Jacobi map AJ p extends to a map 2p cp = cp,0 : CH p (X) → HD (X, Z(p))

and such that the composition ε ◦ cp is given by the topological cycle class ctop : CH p (X) → Hgp,p (X). p This generalizes to higher Chow groups: Bloch [4] has defined maps 2p−n cp,n : CH p (X, n) → HD (X, Z(p)). 2p−n The abelian groups HD (X, Z(p)) sit in exact sequences 2p−n 0 → J p,n (X) → HD (X, Z(p)) → F p H 2p−n (X, Z(p)) → 0,

where J p,n (X) =

H 2p−n−1 (X, C) F p + H 2p−n−1 (X, Z(p))

are generalized intermediate Jacobians. Griffiths’ intermediate Jacobians are not algebraic in general but they are compact, non–degenerate tori. The generalized intermediate Jacobians are not compact in general, but they are complex manifolds and vary holomorphically in families. They look topologically like Cm /Zn with n ≤ 2m. If we restrict cp,n to cycles homologous to zero, we get maps p AJ p,n : CHhom (X, n) → J p,n (X).

General Formulas for AJ p,n have recently been given in [30] and we describe them now. Consider the projections π1 : X × 2n → X and π2 : X × 2n → 2n and let α ∈ F d−p+1 H 2d−2p+n+1 (X, C) be a differential (test–)form.

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Theorem 3.1 ( [30]). If Z = ai Wi ∈ CH p (X, n) is a cycle homologous to zero, such that each irreducible components intersects all real faces properly, then the Abel–Jacobi image of Z is given by the following current: , 1 ai π2∗ log z1 d log z2 ∧ . . . ∧ d log zn ∧ π1∗ α −1 (2πi)d−p+n Wi \π2 [−∞,0]×2n−1 π2∗ log z2 d log z3 ∧ . . . ∧ π1∗ α − (2πi) ai

α →

−1 −1 Wi ∩π2 [−∞,0]×2n−1 \π2 [−∞,0]2 ×2n−2

+ · · · + (−2πi)n−1 n

π1∗ α

n

+ (−1) (2πi)

ai -

−1 −1 Wi ∩π2 ([−∞,0]n−1 ×21 )\Wi ∩π2 [−∞,0]n

π2∗ (log zn ) ∧ π1∗ α

,

Γ

where ∂Γ = Z ∩ π2−1 [−∞, 0]n . The existence of Γ follows from Z being homologous to zero. Remark 3.2. Deligne cohomology is related to extension groups of mixed p Hodge structures. For each cycle W ∈ CHhom (X, n), Bloch has constructed an extension [4, 17, 30] 0 → H 2p−n−1 (X) → E → Z(−p) → 0, where Z(−p) is the Tate–Hodge structure of weight 2p, i.e., an extension class in the sense of Carlson [10] e ∈ Ext1MHS (Z(−p), H 2p−n−1 (X)). It is known [17, 30] that this extension class coincides up to a constant with Bloch’s Abel–Jacobi map in this case.

4 Higher Chow Groups of Algebraic Surfaces This section is about joint work [39] with S. Saito and A. Collino. Let X be a smooth, projective surface over C. For p = 2, n = 1 we have the maps 2 (X, 1) → c2,1 : CHhom

F2

H 2 (X, C) + H 2 (X, Z(2))

which are given by integrals 1 2πi j

log(fj )α + Cj

α Γ

for test distributions α ∈ A2 (X) and auch that ∂Γ = γj , where γj is a path connecting the zeroes and poles of fj , the restriction of the projection to P1 on Cj .


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Example 4.1. In [38, 13, 12, 48] several examples of smooth, projective complex surfaces are constructed where CH 2 (X, 1) is non–torsion modulo subgroups of decomposable cycles. Note the following: Lemma 4.2. (Rigidity) [38] The classes AJ p,1 have countable image modulo classes in Hgp−1,p−1 (X)⊗C∗ . This gives some evidence for Conjecture 4.3 (Voisin). The indecomposable part of CH 2 (X, 1) is always countable, if X is a smooth, projective, complex surface. Let X be a complex smooth, projective surface and Z ⊆ X a normal crossing divisor with complement U = X \ |Z|. Then we have an exact localization sequence ∂ CH r+1 (U, r + 1)→CH r (Z, r) → CH r+1 (X, r). We are interested in the groups CH r+1 (X, r) and are looking for control over the group CH r (Z, r) which is of Milnor type, hence combinatorially easy to describe, and also give criteria when such classes in CH r (Z, r) come from CH r+1 (U, r + 1) via the coboundary map. First, there is a spectral sequence E1a,b = CH a+n (Z [−a] , 1 − b) =⇒ CH n−1 (Z, −a − b) computing of Z from its smooth components Zi . Let ! higher Chow groups ! Z [1] := i Zi and Z [2] := i<j Zi ∩ Zj . In particular for r = 1, 2, we have CH r (Z [1] , r) → CH r (Z, r) → CH r (Z, r)ind → 0, and

Hence and

' ( CH r (Z, r)ind = Ker CH r−1 (Z [2] , r − 1) → CH r (Z [1] , r − 1) . ' ( CH 1 (Z, 1)ind = Ker Z[Z [2] ] → Pic(Z [1] ) , ' ( CH 2 (Z, 2)ind = Ker CH 1 (Z [2] , 1) → CH 2 (Z [1] , 1) .

Generalizing some theorems in [38] we can prove a Noether–Lefschetz type theorem for higher Chow groups: Theorem 4.4 ( [39]). Let X ⊆ P3 be a very general hypersurface of degree d and let Z = ∪Zi be a NCD with Zi very general hypersurface sections of degrees ei . Then • If d ≥ 5, CH 2 (U, 2) → CH 1 (Z, 1)ind is surjective. • Assume d ≥ 6 and that all triples (ei , ej , ek ) = (1, 1, 2). Then CH 3 (U, 3) → CH 2 (Z, 2)ind is surjective as well.

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Proof. The methods consist of the infinitesimal method described above and a careful study of the target (Koszul complexes). In particular we look at the localization diagram: ∂

CH r+1 (U/B, r + 1) → ↓

CH r (Z/B, r) ↓

→ CH r+1 (X/B, r) ↓

∂

H 0 (B, Φ∗ (U/B)) → H 0 (B, Φ∗ (Z/B)) → H 0 (B, Φ∗ (X/B)) ↓ ↓ ↓ ∂

H 0 (B, Ψ ∗ (U/B)) → H 0 (B, Ψ ∗ (Z/B)) → H 0 (B, Ψ ∗ (X/B)) Here all objects containing (Z/B) are defined using Deligne cohomology with supports. All groups involving (U/B) are defined using relative log-complexes relative to the divisor Z. The proof then proceeds as follows: (a) Indecomposable classes in CH r (Z, r) (r = 1, 2) map injectively to H 0 (B, Φ∗ (Z/B)) (easy using above description of CH r (Z, r)). (b) For d large and r = 1, 2, the sheaves Ψ ∗ (U/B) and Ψ ∗ (Z/B) are computable and the map H 0 (B, Ψ ∗ (U/B)) → H 0 (B, Ψ ∗ (Z/B)) is surjective. Hence the image of CH r (Z, r) (r = 1, 2) in H 0 (B, Ψ ∗ (X/B)) is zero. Using the same methods one can also detect cycles in CH 2 (X, 1) and CH 3 (X, 2). We obtain the following results: Theorem 4.5. Consider the family of quintic surfaces Xu,v = {Fu,v = x50 + x1 x42 + x2 x41 + x53 + ux21 x32 + vx0 x3 K(x0 , ..., x3 ) = 0} with u, v ∈ C and K a cubic form. If u, v and K are very general, then there exist elements Zu,v in CH 3 (Xu,v , 2) which are indecomposable modulo the image of Pic(Xu,v ) ⊗ K2 (C). In an appendix to [39] A. Collino constructs further non–rigid examples on K3 surfaces: Theorem 4.6 (Collino). On every very general quartic K3 surface S, there is a 1–dimensional family of elements Zt in CH 3 (S, 2) such that, for t very general, these elements are indecomposable modulo the image of P ic(S) ⊗ K2 (C). All these cycles arise from the existence of smooth bielliptic hyperplane sections C of genus 3 on S, which means that there exists a double cover C → E onto a smooth elliptic curve E [39].

5 Inhomogenous Picard–Fuchs Equations In this section, which reports about joint work with P. del Angel, we want to study differential equations arising from families of algebraic cycles in higher


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Chow groups and to develop a theory of differential equations associated to each family Z/B of cycles in higher Chow groups over a quasi–projective base variety B. This idea is inspired by the work of Fuchs [20], Y.I.Manin [36] and Griffiths [25]. In [13, 14] we used any Picard–Fuchs operator D of the local system underlying the smooth family f : X −→ B and associated to any family Z/B and every relative smooth d–form ω a function Z/B → g(t) := DνZ/B (ω), where νZ/B is the normal function associated to Z/B. If we want to prove that this family of cycles is non–trivial, i.e., its Abel–Jacobi image is non– zero modulo torsion then it is sufficient to show that g(t) = DνZ/B (ω) is not zero for some choice of ω. If the normal function is truncated in a suitable way, i.e., for example if we evaluate only on relative holomorphic d–forms d ), then one can show that under certain assumptions g(t) ω ∈ H 0 (B, ΩX/B has moderate growth at infinity, hence is an algebraic function on a compactification B of B [16]. We do not want to present here all details from [16], so let us just give two examples in dimensions 1 and 2 and relate them to classical non–linear ODE. For dimension 1, consider a section t → (X(t), Y (t)) of the Legendre family Et : y 2 = x(x − 1)(x − t),

t ∈ P1 \ {0, 1, ∞}

in affine coordinates. The corresponding inhomogenous Picard–Fuchs differential equation can be written as X(t) dx D = g(t) y ∞ for a rational or algebraic function g(t), where D = t(1 − t)

d2 1 d + (1 − 2t) − . dt2 dt 4

Richard Fuchs [20] looked at a 4–parameter set of such equations of the form X(t) t dx (t − 1) 1 t(t − 1) t(1 − t)D = Y (t)[α + β +γ + (δ − ) ] 2 2 y X(t) (X(t) − 1) 2 (X(t) − t)2 ∞ with α, β, γ, δ ∈ C. Furthermore every solution of this equation is also a solution of the non–linear equation Painlevé VI and vice versa:

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PV I


2 1 1 1 1 dX 1 1 dX + + + + − X X −1 X −t dt t t − 1 X − t dt , X(X − 1)(X − t) t−1 t(t − 1) t + + γ + δ α + β . t2 (t − 1)2 X2 (X − 1)2 (X − t)2

d2 X 1 : = dt2 2

This last equation PV I has the Painlevé property, namely the absence of movable essential singularities and branch points in the set of solutions. For dimension 2 this correspondence can be generalized: consider a family of K3–surfaces Xt over B = P1 where the general fiber has Picard number 19. Such families were considered in [38, sect. 6.2.1] and [14]. In this case the Picard–Fuchs operator has order 3 and we assume that the cycles consist of two irreducible components. In [14] the components are a line and an elliptic curve on a surface with the equation xyz(x + y + z + tw) + w4 = 0 The transcendental part ν¯ of the normal function ν, i.e., where ω is a holomorphic 2–form, can be written as an integral b(λ(t)) d(x,λ(t)) dx F (x, y, λ(t))dy, ν¯(t) = a(λ(t))

c(x,λ(t))

where a, b, c, d are algebraic functions of two variables. Here F (x, y, λ(t))dxdy is the local expression for a chosen family of relative holomorphic 2–forms. Assuming λ(t) is locally biholomorphic we can write F as a function of λ(t) instead of t. Using the same substitution for all coefficient functions of the Picard–Fuchs operator, which is of order 3 here, we get a non–linear third order ODE of the form λ (t) = A(λ) + B(λ)λ λ + C(λ)(λ )3 which is a variant of Chazy’s equation [9, page 319]. Chazy’s equation also has the Painlevé property. The solutions λ(t) which arise from single–valued families of algebraic cycles are of course rational functions of t ∈ B, see [38, sect. 6.2.1] for an example. Non–linear ODE or PDE having the Painlevé property related to work of Hitchin and Boalch [8] about the deformation theory of irregular connections. Note that our approach using higher Chow groups leads naturally to a different way of finding such equations of higher degree.

6 Cohomology of Local Systems on Moduli Spaces In this section we explain some recent computations for the cohomology of local systems on certain Shimura varieties and their applications to motives. This is joint work with A. Miller, S. Wortmann, Y.-H. Yang and K. Zuo [37].


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Let G/K be a Hermitian symmetric space of non–compact type and X = Γ \G/K the associated locally symmetric variety, where Γ ⊂ G(R) torsion– free discrete arithmetic subgroup acting freely on G/K. An interesting and very general problem is to compute the L2 –cohomology HL∗ 2 (X, V) for V a representation of Γ . Such computations are important in algebraic K–theory for example in the work of Borel [7]. The analogue of our maps cp,n in this paper is the Borel regulator [7]. We want to show that such computations can be applied to the theory of motives through the construction of Chow–K¨ unneth decompositions. First let us recall Ragunathan’s vanishing theorem: Theorem 6.1. If Γ is cocompact and either • Lie(G) does not contain su(n, 1) and so(n, 1), or • V is regular (e.g. the adjoint representation), then H 1 (Γ, V) = 0. This theorem was later extended to the non–cocompact case. The following examples describe some Shimura varieties. Example 6.2. The Legendre family of elliptic curves Eλ : y 2 = x(x − 1)(x − λ),

λ ∈ P1 (C)

has periods dx y . They are hypergeometric functions in λ. The discriminant locus of this family is ∆ = {0, 1, ∞} and the complement P1 \∆ is uniformised by H = SL2 (R)/SO(2), the upper half plane. One has X = P1 \ ∆ = H/Γ (2) for the congruence subgroup Γ (2). Example 6.3 (Hilbert modular varieties). Let F be a totally real number field

n SL2 (R) be the product of n copies of SL2 (R). G of degree n and G = operates on Hn = H × · · · × H, i.e., n copies of upper half plane. If Γ ⊂ G is a torsion–free arithmetic subgroup, then set X = Γ \ Hn . In this case X is called a (generalized) Hilbert(–Blumenthal) modular variety. Namikawa constructed a stratified universal family p : A → X, where X, A is a projective compactification of X, A. The general fiber is an n–dimensional abelian variety with some specific extra structure.

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In this last example one has a good vanishing theorem:

n Theorem 6.4 (Matsushima/Shimura). Let G = SL2 (R), Γ torsion– free arithmetic subgroup and ρ a complex, irreducible, non–trivial representation of Γ coming from G. Then HLi 2 (X, V) = 0 for i = dim(X) = n dim(X)

and HL2

(X, V) is a space of automorphic forms.

For example, if f : A → X is an elliptic modular surface and V = R1 f∗ C is an irreducible VHS over X then only HL12 (X, V) = 0. The classical family of Picard curves is the 2–parameter family Cs,t : y 3 = x(x − 1)(x − s)(x − t),

(s, t) ∈ C2 ⊂ P2 (C).

It defines a family of genus 3 curves with an extra Z/3Z deck transformation. The discriminant locus is ∆, a union of 6 lines with 4 triple intersections, called the cusps: @ A s @ A @ A s @A ∆ = ∪∆i,j : @A @ As s A@ @ The family has 6–dimensional periods, 3 of which define a multivalued map P2 \ ∆ → B2 = {|z1 |2 + |z2 |2 < 1} ⊂ C2 ⊂ P2 (C). B2 is a homogenous space for U (2, 1). Theorem 6.5 (Picard). P2 \ cusps ∼ = B2 /Γ , where ⎛ ⎞ 100 Γ = {γ ∈ U (2, 1)(O) | γ ≡ ⎝0 1 0⎠ mod 1 − ω}. 001 Here O = Z[ω] is the ring of Eisenstein numbers, ω 3 = 1. Thus P2 is the simplest of all ball quotients parametrizing Picard curves. Let us make a definition out of such a property: Definition 6.6 (Picard modular surfaces). Let O be a ring of integers for an imaginary quadratic number field. A Picard modular surface X is a smooth, projective compactification of X = B2 /Γ , where Γ ⊆ U (2, 1; O) is an arithmetic subgroup.


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This means that X is a 2–dimensional subvariety of Hodge type in A3 , the moduli space of abelian 3–folds. Our goal is to compute the L2 –cohomology of local systems on X. To do this we use the Simpson correspondence, an equivalence of categories between: • Representations of π1 = local systems (Betti), • Holomorphic bundles with flat connection (de Rham), • (Poly)stable Higgs bundles with trivial Chern classes (Dolbeault). In the compact case this was established by C. Simpson [43] and in the non– compact case this is still open, but was shown for C–VHS (variation of Hodge structures) by J. Jost, Y.-H. Yang and K. Zuo [28]. Definition 6.7. A (log–)Higgs bundle is a vector bundle E with a holomorphic 1 (log D) such θ ∧ θ = 0. map θ : E → E ⊗ ΩX Assume now that V is a VHS on X = X \ D, where D is a normal crossing divisor with unipotent local monodromies around D. Let V ⊗ OX = F 0 ⊇ F 1 ⊇ . . . be the Hodge filtration. The Higgs bundle corresponding to V is (E = ⊕E a,b , θ) E a,b = F a /F a+1 and 1 θ : E a,b → E a−1,b+1 ⊗ ΩX

the cup product with the Kodaira–Spencer class. After extending E to X there is an algebraically defined subcomplex 0 1 2 (E) → Ω(2) (E) → Ω(2) (E) → . . . 0 → Ω(2)

of the full algebraic Higgs complex θ

θ∧θ

1 (log D) → · · · E →E ⊗ ΩX

Its cohomology is denoted by H∗L2 −Higgs (X, (E, θ)). The L2 –conditions are defined via a harmonic metric on E which is defined using the Hodge metric along the fibres and a background metric on X, which is asymptotic to the Poincaré metric at infinity [28]. This subcomplex depends only on the residue Res(θ) : E → E of θ. Example 6.8. If D = V (z1 ) is smooth and dim(X) = 2, then 0 Ω(2) (E) = Ker(Res(θ)),

1 (E) = Ω(2)

2 Ω(2) (E) =

dz1 ∧ dz2 ⊗ z1 E, z1

dz1 ⊗ z1 E + dz2 ⊗ Ker(Res(θ)). z1

The following result shows that we may compute Higgs cohomology instead of intersection cohomology or classical L2 –cohomology which are both sometimes somewhat difficult to obtain:

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Theorem 6.9 (Jost, Yang, Zuo [28]). In the general situation (D a normal crossing divisor, unipotent monodromies) we have IH ∗ (X, V) = H∗L2 −Higgs (X, (E, θ)) = HL∗ 2 (X, V), i.e., intersection cohomology, L2 –Higgs and standard L2 –cohomology are all the equal. Let us now compute such cohomologies on one explicit surface. Hirzebruch [26] constructed a series of surfaces with c21 /c2 → 3 for n → ∞ and conjectured that they are compactified ball quotients Xn with Xn = B2 /Γn , Γn ⊂ U (2, 1; O). Later Holzapfel [27] proved this and constructed more examples. Example 6.10. X = blow–up of E × E in 3 points (Pi , Pi ), where E = {y 2 z = x3 − z 3 } CM elliptic curve with automorphism x → ωx of order 3 and fixed points P1 , P2 , P3 . This is a (birational) covering of the basic Picard modular surface P2 : on P2 blow up all 4 cusp points and then blow down all 3 strict transforms of lines ∆i,4 . This results in the following situation: E1 E2 E3 @ A s @ A @ A s @A @A @ As s A@ @

E4 ∆23

↔

∆13 ∆12

Then take a branched cover along the horizontal and vertical lines. The diagonal curve splits into 3 elliptic curves which correspond to strict transforms of the 3 diagonals (z, w),

(z, ω 2 w) ⊂ E × E.

(z, ωw),

Together we get 6 elliptic cusp curves D1 , . . . , D6 : Z3 D6 Z2 D5

Z1 D1 D2 D3

D4

The exceptional curves Z1 , Z2 , Z3 are rational modular curves containing 4 cusps each. They carry a special family of Jacobians of type E × S 2 (Eλ ) [27].


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Theorem 6.11 (Holzapfel, Picard). × E is a compactified ball quotient with X = (a) Holzapfel’s surface X = E B2 /Γ , and Γ ⊂ Γ = Γ ∩ SL3 (C). (b) The universal family f : A → X has a Z/3Z eigenspace decomposition R 1 f∗ C = V1 ⊕ V2 . (c) The family has unipotent local monodromies and the Higgs bundle (E, θ) associated to the Deligne extension of V1 is of type 1 E 1,0 = ΩX (log D) ⊗ L−1 ,

E = E 1,0 ⊕ E 0,1 , L⊗3 = KX + D,

E 0,1 = L−1 ,

1 θ = id : E 1,0 → E 0,1 ⊗ ΩX (log D).

The main technical result of [37] is: Theorem 6.12. The intersection cohomology IH q (X, V1 ) vanishes for q = 2. Note that since G = SU (2, 1) we cannot expect vanishing for arbitrary Γ , hence this is a coincidence. Let us explain how the argument works. First, let us first forget about the L2 –conditions: ' ' '

(

Ω 1 (log D)⊗2 ⊗ L−1 ⊗ Ω 2 (log D) X X

⊕

(

Ω 1 (log D)⊗2 ⊗ L−1 X ↓ ( L−1 ⊗ Ω 2 (log D) . X

⊕

'

( Ω 1 (log D) ⊗ L−1 ⊕ L−1 X ↓∼ ↓ = ( L−1 ⊗ Ω 1 (log D) 0 X

'

This complex is quasi–isomorphic to a complex 1 1 2 L−1 −→S 2 ΩX (log D) ⊗ L−1 −→ΩX (log D) ⊗ ΩX (log D) ⊗ L−1 0

0

with trivial differentials. As L is nef and big, we have H 0 (L−1 ) = H 1 (L−1 ) = 0. Hence we get 1 H1 (X, (E • , ϑ)) ∼ (log D) ⊗ L−1 ). = H 0 (X, S 2 ΩX

Now, if we impose L2 –conditions and remark that [37] 1 Ω 1 (E)(2) ⊆ ΩX ⊗ E,

we conclude that 1 1 IH 1 (X, V1 ) ⊆ H 0 (X, ΩX (log D) ⊗ ΩX ⊗ L−1 ).

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This can be viewed as a kind of generalized Eichler–Shimura isomorphism. The rest of the argument consists of applying the Bogomolov–Sommese vanishing to show that it is enough to prove the vanishing of 1 (log D) ⊗ ΩZ1 ⊗ L−1 ) = 0, H 0 (Z, ΩX

where Z is the union of all 3 rational curves Zi . This can be easily computed using the splitting theorem, see [28], where this theorem is applied to show some partial results about the following conjecture of Grothendieck and its refinement by Murre [40]: Conjecture 6.13 (Grothendieck’s Standard Conjecture C). Let A be a smooth, projective variety over k of dimension d and ∆ ⊂ A × A the diagonal. Then the K¨ unneth components of [∆] ∈ H 2d (A × A) lift to a complete set of orthogonal idempotents ∆ = π0 + π1 + . . . + π2d ∈ CH d (A × A)Q . In this statement H ∗ is a Weil cohomology theory, for example singular cohomology if k = C. Conjecture 6.14 (Murre’s Refinement (simplified)). The πj are compatible with the Bloch–Beilinson filtration on Chow groups. This conjecture is essentially known for dim ≤ 2, abelian varieties, rational like varieties and some 3–folds, total spaces of Shimura type families, e.g. elliptic and Hilbert modular families. It is open for dim ≥ 3, see [40]. B. Gordon, M. Hanamura and J. Murre have worked out a strategy to prove this conjecture [24]. It uses the Conjecture 6.15 (MDC=Motivic decomposition conjecture). Let p : X → S = Sα be a stratified map in the sense of [24]. Then there exist relative Chow– K¨ unneth projectors Παj which induce a BBD(G)–decomposition Ψ : Rp∗ QX = ICSα (Vjα )[−j + dim(Sα )]. α,j

This conjecture is very likely to be true, in particular if all fibers are abelian varieties, but of course hard to prove because of degenerate fibers. Then, assuming MDC, the strategy in [24] is as follows: if one has a vanishing theorem IH k (Sα , Vjα ) = 0 for k = dim(Sα ) then relative projectors induce already absolute Chow– K¨ unneth projectors. For Hilbert modular varieties this approach goes through because of the vanishing theorem of Matsushima and Shimura. However in the case of Picard modular surfaces, there is no such vanishing theorem a priori. If the level is low, as in the case of Holzapfel’s surface we can however expect such a vanishing theorem. Our main result in [37] is:


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Theorem 6.16. The 5–dimensional total space A of the universal family over Holzapfel’s surface supports a partial Chow–K¨ unneth decomposition with projectors π0 , π1 , π2 , π3 , π7 , π8 , π9 , π10 . If the Hodge conjecture holds on A then one has a complete set of projectors πi for all 0 ≤ i ≤ 10. The assumption about the Hodge conjecture is necessary since we cannot prove that the cohomology group H 1 (X, S 2 E) = 0, but we can show that this cohomology group consists only out of pure (2, 2) classes.

References ¨ller-Stach, S. Saito and N. Yui (editors): 1. B. Gordon, J. Lewis, S. Mu The Arithmetic and Geometry of Algebraic Cycles, Nato science series, Kluwer Acad. Publ. 548 (2000). 2. A. Beilinson: Higher regulators and values of L-functions of curves, Funct. Anal. Appl. 14, 116–118 (1980). 3. S. Bloch: Algebraic cycles and higher K-theory, Adv. in Math. 61, 267–304 (1986). 4. S. Bloch: Algebraic cycles and the Beilinson conjectures. Cont. Math. 58 Part I, 65–79 (1986). 5. S. Bloch: The moving lemma for higher Chow groups, J. Algebraic Geom. 3, no. 3, 537–568 (1994). 6. S. Bloch, I. Kriz: Mixed Tate motives, Annals of Math. 140, 557–605 (1995). 7. A. Borel: Stable real cohomology of arithmetic groups, Ann. Sci. ENS 4, 235– 272 (1974). 8. Ph. Boalch: Symplectic manifolds and isomonodromic deformations, Adv. Math. 163, 137–205 (2001). 9. J. Chazy: Sur les équations diff´ erentielles du troisième ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes, Acta Math. 34, 317–385 (1910). 10. J. Carlson: Extensions of mixed Hodge structures, Proceedings Journées de Géometrie Algébrique d’Angers 1979, Sijthoff and Noordhoff, 107–127,(1980). ¨ller-Stach and Ch. Peters: Period maps and period 11. J. Carlson, S. Mu domains, Cambridge Studies in advanced math. 85, Cambridge University Press (2003). 12. A. Collino:Griffiths’ infinitesimal invariant and higher K-theory of hyperelliptic Jacobians, Journ. of Alg. Geom. 6, 393–415 (1997). ¨ller-Stach: The transcendental part of the regu13. P. L. del Angel and S. Mu lator map for K1 on a mirror familiy of K3 surfaces, Duke Math. Journal 112, 581–598 (2002). ¨ller-Stach: The transcendental part of the regu14. P. L. del Angel and S. Mu lator map for K1 on a mirror family of K3 surfaces, Version 1, preprint available on math.AG/0008207v1 (2000). ¨ller-Stach: Picard-Fuchs equations, Integrable 15. P. L. del Angel and S. Mu Systems and higher algebraic K-Theory, Fields Inst. Comm. Vol. 38, Amer. Math. Soc., 43–55 (2003). ¨ller-Stach: Differential Systems associated to 16. P. L. del Angel and S. Mu families of algebraic cycles, Preprint 2005.

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17. C. Deninger and T. Scholl: The Beilinson conjectures, in: L-functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser. 153, Cambridge Univ. Press, Cambridge, 173–209 (1991). ¨ller-Stach: Higher Chow groups, Milnor K18. Ph. Elbaz-Vincent and S. Mu theory, and applications, Inventiones Math., 148, 177–206 (2002). 19. H. Esnault and E. Viehweg: Deligne–Beilinson cohomolgogy, in M. Rapoport, N. Schappacher and P. Schneider (eds.) Beilinson’s conjectures on special values of L-functions, Academic Press, 43–91 (1990). ¨ 20. R. Fuchs: Uber lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singul¨ aren Stellen, Math. Ann. 63, 301– 321 (1907). ¨ller-Stach: Polylogarithmic identities in higher Chow 21. H. Gangl and S. Mu groups, Seattle Conference on Algebraic K-theory, Proc. of Symp. in Pure Math 67, 25–40 (1999). 22. W. Gerdes: The linearization of higher Chow groups of dimension one, Duke Math. Journal 62, 105–129 (1991). 23. A. Goncharov: Chow polylogarithms and regulators, Math. Research Letters 2, 95–112 (1995). 24. B. B. Gordon, M. Hanamura, and J. P. Murre: Absolute Chow-K¨ unneth projectors for modular varieties, Crelle Journal 580, 139–155 (2005). 25. Ph. Griffiths: A theorem concerning the differential equations satisfied by normal functions associated to algebraic cycles, Amer. J. Math. 101, 94–131 (1979). 26. F. Hirzebuch: Chern numbers of algebraic surfaces – an example, Math. Annalen 266, 351–356 (1984). 27. R.-P. Holzapfel: Chern numbers of algebraic surfaces - Hirzebruch’s examples are Picard modular surfaces, Math. Nachrichten 126, 255–273 (1986). 28. J. Jost, Y.-H. Yang and K. Zuo: The cohomology of a variation of polarized Hodge structures over a quasi–compact K¨ ahler manifold, preprint math.AG/0312145 (2003). 29. M. Kerr: Thesis, Princeton 2002. ¨ller–Stach: The Abel-Jacobi map for higher 30. M. Kerr, J. Lewis and S. Mu Chow groups, Compositio Math. 142, 374-396 (2006). ¨ller–Stach: The Milnor-Chow homomorphism revisited, 31. M. Kerz and S. Mu to appear in K-theory (2006). 32. M. Levine: Bloch’s higher Chow groups revisited, in Astérisque 226, 235–320 (1994). 33. M. Levine:Lambda operations, K-theory and motivic cohomology, Proceedings of the Fields Institute No. 16, AMS, 131–184 (1997). 34. M. Levine: K-theory and Motivic cohomology of schemes, preprint (1999). 35. M. Levine: Mixed Motives, AMS Math. Surveys 57 (1998). 36. Y. I. Manin: Sixth Painlevé equation, universal elliptic curve and mirror of P2 , Amer. Math. Soc. Translations (2), Vol. 186, 131–151 (1998). ¨ller-Stach, S. Wortmann, Y.-H. Yang and K. Zuo: 37. A. Miller, S. Mu Chow-K¨ unneth decomposition for universal families over Picard modular surfaces, Preprint math.AG/0505017 (2005), to appear in Proceedings Murre conference. ¨ller-Stach: Constructing indecomposable motivic cohomology classes on 38. S. Mu algebraic surfaces, Journ. of Alg. Geom. 6, (1997), 513–543. Journal of Alg. Geom. 1997.


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K¨ ahler Geometry of Moduli Spaces of Holomorphic Vector Bundles Georg Schumacher Fachbereich Mathematik und Informatik, Hans-Meerwein-Str., Lahnberge, 35032 Marburg [email protected]

1 Introduction In most general situations, the set of isomorphism classes of analytic objects like polarized K¨ ahler manifolds (with fixed underlying differentiable structure) or holomorphic vector bundles with fixed numerical invariants carries a natural topology. It turned out that existence theorems for moduli spaces are closely related to the Hausdorff property. In the sequel, we will display examples for this relationship. A generalized Petersson-Weil metric exists whenever the objects under consideration possess a distinguished differential metric, which can be realized as Chern form of a certain (positive) determinant line bundle, equipped with a Quillen metric, on the moduli space. Also the curvature of the Petersson-Weil metric is of interest. The curvature tensor can be computed in most cases. In this article we will focus on analytic methods.

2 Existence of Moduli Spaces Existence theorems for (coarse) moduli spaces are modeled after the classical case of quotients of Teichm¨ uller spaces of compact or punctured Riemann surfaces by the Teichm¨ uller group, whose isotropy subgroups can be identified with the automorphism groups of the corresponding fiber of the universal Teichm¨ uller family, or to be precise with the quotient of the automorphism group by the subgroup of those automorphisms, which act in a trivial way on the Teichm¨ uller spaces (in genus two). In [SCH1, SCH2] a criterion for the existence of moduli spaces (as reduced complex analytic spaces) of polarized K¨ ahler manifolds was first given, which could be applied literally to other analytic moduli problems. Subsequently we observed that the method also applies to not necessarily reduced moduli spaces, also in the category of algebraic spaces.

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The formal framework is provided by fibered groupoids p : F → A in the sense of Grothendieck (categorie cofibré en groupoides, [AR, R]). In this formal language the essence of our arguments becomes somehow more obvious. In our case A shall denote either the category Ac of complex spaces or the category As of schemes over C (separated and of finite type); Atop will denote the category of topological spaces. For any fibered groupoid there is an induced moduli functor M : A → (Sets) where M(S) = F (S) is the set of isomorphism classes from F (S), and where morphisms are defined in the obvious way. It is necessary in most cases to assign a sheafified moduli functor M to M, where the resp. topology is the classical or the étale topology depending on the choice of A). Set-theoretically, a moduli space consists of isomorphism classes of objects of a given category: Set 0 = Spec(C), and let M0 = F (0) be the set of all isomorphism classes of objects over 0. For any object a ∈ Obj(F) and S = p(a), we consider the embeddings is : {s} → S for all s ∈ S, which induce a map ϕa : S → M0 . Proposition 1. The set M0 carries a natural topology generated by all sets ϕa (p(a)) such that all such maps ϕa are continuous. We call M0 the settheoretic moduli space. 0 We consider the covariant functor hM top : A → (Sets) for A = Ac or 0 A = As , where hM top (R) is the set of all continuous maps R → M0 .

Proposition 2. The assignment a → ϕa induces a natural transformation 0 ϕ : M → hM top of functors. For the category of complex spaces Ac a coarse moduli space M is a morphism of functors Φ:M→M with the following property (i) for any complex space N and any morphism of functors F : M → N there exists a unique morphism f : M → N such that f ◦ Φ = F . (ii) the map M(Spec(C)) → M (Spec(C)) is bijective. (As usual, M is identified with hM = Hom(−, M )). We fix a fibered groupoid F → A, and suppose that any object a0 over the reduced point possesses a semi-universal deformation. Furthermore, we need the natural assumption that completeness is an open condition, which holds in most cases: if a is complete at s0 , it induces complete deformations of all a|{s}, where s is contained in an open neighborhood of s0 .

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Theorem 1 ( [SCH1, SCH2, SCH6]). Let F → A be a fibered groupoid. Assume 1. For all complex spaces S in A and all objects a, b ∈ F(S) the functor IsomS (a, b) is representable by a complex space A → S. 2. For all objects a0 over the reduced point 0 the identity component Aut0 (a0 ) of the complex Lie group Aut(a0 ) is compact. 3. For all objects a0 over the reduced point semi-universal objects exist in F. Then all semi-universal deformations are universal. Assume furthermore that 4. For all a, b ∈ F(S) the natural map IsomS (a, b) → S is proper. Then there exists a coarse moduli space. One can see immediately that the above properness condition is closely related to the Hausdorff property of the topology of M0 in Proposition 1. We are interested in not necessarily reduced moduli spaces, because these support our natural K¨ ahler structures, known as generalized Petersson-Weil forms. Applications are immediate: For families of polarized K¨ ahler manifolds, there exists a corresponding fibered groupoid with representable isomorphism functor, and a suitable deformation theory. The existence of K¨ ahler-Einstein metrics in the case of vanishing first real Chern class (and in the canonically polarized case) yield the existence of a moduli space. Also the case of canonically polarized framed manifolds (X, D), where D ⊂ X is a smooth divisor ahler-Einwith KX + D > 0, can be treated in this way using complete K¨ stein metrics on the complement [SCH5]. Finally for stable vector bundles on K¨ ahler manifolds the existence of Hermite-Einstein connections the criterion is applicable.

3 Moduli Spaces of Holomorphic Vector Bundles and Principal Bundles 3.1 Moduli of Stable Vector Bundles – Generalized Petersson-Weil Metric We will address existence theorems for moduli spaces and the aspect of Kähler geometry, starting from the Kobayashi-Hitchin correspondence between irreducible stable holomorphic vector bundles and Hermite-Einstein vector bundles on K¨ ahler manifolds. Let (X, ωX ) be a compact Kähler manifold, and E → X × S a family of holomorphic vector bundles, were a distinguished fiber carries an irreducible Hermite-Einstein connection. The standard proof involving gauge theory for integrable semi-connections and the implicit function theorem for suitable Sobolev spaces together with

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semi-continuity theorems show that neighboring fibers are equipped with unique Hermite-Einstein connections, and that these depend in a C ∞ way upon the parameter s ∈ S. All necessary axioms of deformation theory as well as the representability of the isomorphism functor hold. Let E 1 → X × S and E 2 → X × S be families of holomorphic vector bundles on X with irreducible Hermite-Einstein connections (θsj )s∈S , j = 1, 2. Let sν → s0 be a converging sequence of points in S. If ϕν : E 1 |X × ∼ {sν } −→ E 2 |X × {sν }. By uniqueness ϕ∗ν (θs2ν ) = θs1ν . Now a subsequence of the isometries ϕν converges (in any C k -topology) to an isometry of the central fibers. Hence the properness of the Isom-space over the base S is immediate for families of irreducible Hermite-Einstein bundles. Infinitesimal deformations. Let E be a holomorphic vector bundle on a complex manifold X . Denote by TX the sheaf of holomorphic vector fields on X , and by ΣX the sheaf of infinitesimal automorphisms of the pair (X , E). The corresponding short exact sequence 0 → End(E) → ΣX → TX → 0 is usually called Atiyah sequence for vector bundles. Any connection θ on E corresponds to a differential splitting of ΣX → TX . Consequently, the curvature form Ω of θ defines the edge homomorphism H 0 (X , TX ) → H 1 (X , End(E)), sending a vector field V on X to the cohomology class given by its contraction ΩV with the curvature form. For X = X × S, s0 ∈ S, E|X × {s0 } = E the resulting map Ts0 S → H 1 (X, EndE)

(1)

is exactly the Kodaira-Spencer map. Let (E, h) be a holomorphic vector bundle equipped with a Hermite-Einstein metric, over a K¨ ahler manifold (X, ωX ): ΛX Ω = λ · idE i.e. g βα Rαβ = λ · idE for some λ ∈ C. In the case of a family (Es , hs )s∈S of Hermite-Einstein bundles parameterized by S, we are given a holomorphic vector bundle E on X × S together with a hermitian metric. We denote again Ω the curvature form. We denote by si local coordinates on S, and by z α coordinates on X. We use these indices to denote components of tensors, in particular components of the curvature tensor. The definition of a generalized Petersson-Weil metric requires Hodge theory. In fact the curvature tensor Ω of a family of Hermite-Einstein bundles contains the necessary information. Proposition 3. Let v ∈ Ts S be a tangent vector. Then the harmonic representative of ρ(v) ∈ H 1 (X, EndE) equals ηv = Ωv, which equals Riβ dz β , if v = ∂/∂si .


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Definition 1. Let (E → X × S, h) be a holomorphic family of Hermite-Einstein vector bundles, and s ∈ S. Then the Petersson-Weil hermitian form on Ts S is defined by (u, v)P W := tr Λ(ηu ∧ ηv ) g dv; u, v ∈ Ts S. (2) X×{s}

The corresponding tensor is W GP i¯ j (s)

∂ ∂ := ( i , j )P W = tr Λ(ηi ∧ ηj ) g dv ∂s ∂s X×{s} ¯ = tr (g βα Riβ¯Rα¯j ) g dv

(3)

X×{s}

with ωP W =

√

W i j −1 GP i¯ j (s) ds ∧ ds

(4)

The above Proposition 3 allows a computation of the curvature of the generalized Petersson-Weil metric. For simple vector bundles, the kernel of Laplacian on A0 (X, End(E)) con0 sists just of constant multiples of the identity. Let A0 (X, End(E)) be the subspace of all differentiable sections σ with X trσ g dV = 0. Then the restriction of 2 to this space possesses an inverse, which is equal to the Green’s operator G on sections. We also need the Green’s operator on End(E)-valued (0, 2)-forms. Theorem 2 ([S-T]). Let (E → X × S, h) be a local universal family of Hermite-Einstein vector bundles over a smooth parameter space S. Then the curvature tensor of the Petersson-Weil metric is expressed by the following formulas: ωn PW = + tr G(Λ[ηi , ηj¯])Λ[ηk , η¯] X Ri¯ jk¯ n! X ωn + tr G(Λ[ηi , η¯])Λ[ηk , ηj¯] X n! X n−2 ωX (5) tr ([ηi , ηk ]G([ηj¯, η¯]) + (n − 2)! X i.e. PW Ri¯ jk¯ l = +

βα tr 2−1 )[Rkδ¯, Rγ ¯l ]g δγ g dv j ]g 0 ([Riβ¯ , Rα¯ ¯

X

+

¯

βα tr 2−1 )[Rkδ¯, Rγ¯j ]g δγ g dv l ]g 0 ([Riβ¯ , Rα¯ ¯

¯

X ¯

¯

¯

¯

tr [Riβ¯ , Rkδ¯]G([Rα¯j , Rγ ¯])(1/2)(g βα g δγ − g βγ g δα ) g dv (6)

+ X

where G is the Green operator for differential forms and respective skewsymmetric tensors with values in End(E).

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We see that for dim X = 1 only the first two terms in (5) and (6) resp. are present, yielding non-negative holomorphic sectional curvature. 3.2 Moduli of Principal Bundles The moduli space of polystable principal bundles of fixed topological type over a compact Riemann surface was constructed by A. Ramanathan in his thesis [RA1, RA2]. Recently T. Gómez and I. Sols constructed the moduli space of polystable principal bundles of fixed topological type over a smooth projective variety by extending the method of Ramanathan [G-S]. We will explain results about the K¨ ahler geometry of moduli spaces of principal bundles from [B-S3]. We fix a compact Kähler manifold (X, ωX ) and a complex reductive linear algebraic group G, together with a maximal compact subgroup K ⊂ G. Definition 2. A holomorphic principal G-bundle P over X is called stable (respectively, semistable), if the following holds: Take any triple (Q, U, σ), where Q ⊂ P is a maximal proper parabolic subgroup, U ⊂ X is the complement of an analytic (possibly empty) subset of codimension at least two, and σ : U → P/Q is a holomorphic reduction of the structure group on U . Then deg σ ∗ Trel is positive (respectively, semi-positive), where Trel denotes the relative tangent bundle for the projection P/Q → X. Also the notion of poly-stability was carried over to the situation of principal bundles. Hermitian connections are defined in terms of the given maximal compact subgroup K ⊂ G. We denote by θ a connection on a smooth principal G-bundle π : P → X, i.e. a differentiable G-equivariant splitting of the differential dπ : T R P → π ∗ T R X , where T R P and T R X denote the real tangent bundles, it is called a complex connection, if it commutes with the almost complex structures. If PK ⊂ P denotes a reduction of the structure group of a smooth Gbundle P to a maximal compact subgroup K ⊂ G, a connection on P that is induced by a connection on some such reduction PK is called a Hermitian connection. Observe that a connection on PK that induces a given connection on P is uniquely determined by the latter. Let PK ⊂ P be a reduction of the structure group of a holomorphic principal G-bundle P to a maximal compact subgroup K ⊂ G. Then there is a unique complex connection on P , which is induced by a connection on PK . In terms of a linear representation of ρ : G → GL(V ), together with a maximal compact subgroup K ⊂ GL(V ) containing r(K), it is possible to establish a relationship with an induced hermitian, holomorphic vector bundle.


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Denote by g the Lie algebra of G with center z. Consider the adjoint action of G on g. Since z ⊂ g is a G-submodule, we get a subbundle P ×G z ⊂ P ×G g =: ad(P ) of the adjoint bundle ad(P ). Since the adjoint action of G on z is trivial, this subbundle is canonically identified with the trivial vector bundle X × z over X with fiber z. Let θ be a Hermitian complex connection on P . The curvature of θ, which we will denote by Ω, is a (1, 1)-form on X with values in ad(P ). Let Λ be the operator that is adjoint to the exterior multiplication of forms on X with the K¨ ahler form ωX . Therefore, ΛΩ is a C ∞ -section of ad(P ). Definition 3. A Hermitian complex connection θ on a holomorphic principal G-bundle P will be called a Hermite-Einstein connection if there exists an element λ ∈ z such that the section ΛΩ of ad(P ), where Ω is the curvature of θ, has the constant value λ. A principal G-bundle admits a Hermite-Einstein connection, if and only if it is polystable. Furthermore, a polystable G-bundle admits a unique HermiteEinstein connection (cf. [R-S, A-B]). Let Ad : G → Aut(g), and ad : g → End(g) be the adjoint representations. Take a holomorphic principal G-bundle P → X together with the adjoint bundle ad(P ) = P ×G g, and the corresponding endomorphism bundle End(ad(P )). A holomorphic principal G-bundle P is polystable, if and only if ad(P ) is polystable. The Hermite-Einstein connection on P induces the unique Hermite-Einstein connection on the vector bundle ad(P ). Fix a Hermitian form on the Lie algebra g which is left invariant by the adjoint action of the maximal compact subgroup K ⊂ G. Let P be a polystable G-bundle over X. Let PK ⊂ P be a reduction of structure group of P to K such that the Hermite-Einstein connection on P is induced by a connection on PK . Since the adjoint vector bundle ad(P ) is canonically identified with PK ×K g, the vector bundle over X associated to PK for the adjoint action of K on g, the K-invariant Hermitian form on g gives a Hermitian structure on the vector bundle ad(P ). On can show that this is a Hermite-Einstein metric on ad(P ). Let Z 0 (G) ⊂ G be the connected component of the center of G containing the identity element. For a principal G-bundle P over X, let Ad(P ) := P ×G G be the fiber bundle over X associated to P for the adjoint action of G on itself. Each fiber of Ad(P ) is a Lie group isomorphic to G. The space of all smooth (respectively, holomorphic) automorphisms of the G-bundle P is identified with the space of all smooth (respectively, holomorphic) sections of Ad(P ). Since the adjoint action of G on itself preserves the subgroup Z 0 (G), we have a subbundle P ×G Z 0 (G) ⊂ P ×G G =: Ad(P )

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of the fiber bundle Ad(P ). Since G acts trivially on Z 0 (G), this subbundle is canonically identified with the trivial fiber bundle X × Z 0 (G) over X. Let P be a holomorphic principal G-bundle over X. Let Aut0 (P ) denote the connected component, containing the identity element, of the holomorphic automorphism group of P . from the above remarks it follows that Z 0 (G) is a subgroup of Aut0 (P ). Let φ be a holomorphic section of Aut0 (P ). Since deg ad(P ) = 0 and ad(θ) is a Hermite-Einstein connection on ad(P ), we conclude that φ is flat with respect to ad(θ). Using the Jordan decomposition of φ, it is possible to reduce such a section to an element of the center: Proposition 4. If P is a stable principal G-bundle over X, then Z 0 (G) coincides with Aut0 (P ). So in a holomorphic family of stable G-bundles, any automorphism of the central fiber P which lies in Aut0 (P ) can be extended to the neighboring bundles. General deformation theory now implies that semi-universal deformations are universal. Since Hermite-Einstein connections are unique and depend in a C ∞ way on the parameter, the induced analytic equivalence relation on the base spaces of universal deformations is proper. This fact implies the existence of a moduli space for stable principal G-bundles over a K¨ ahler manifold, also in the category of not necessarily reduced complex spaces (cf. Section 2). To any holomorphic family P → X ×S of principal G-bundles we associate the adjoint vector bundle ad(P) → X × S. This amounts to a morphism of deformation functors. For any s ∈ S, and P = P|X × {s}, it induces a morphism H 1 (X, ad(P )) → H 1 (X, End(ad(P ))) of spaces of infinitesimal deformations. This homomorphism coincides with the one obtained from the homomorphism of vector bundles ad(P ) → End(ad(P )) given by the Lie algebra structure of the fibers of ad(P ). Let (S, s0 ) be a pointed complex analytic space, and π ) :P →X ×S (7) a holomorphic principal G-bundle over X × S. Denote by AT (P) the sheaf on X × S defined by the space of G-invariant holomorphic vector fields on P. Since G acts transitively on the fibers of π ), this sheaf is a locally free coherent analytic one. Therefore, it defines a holomorphic vector bundle over X × S. Consequently, AT (P) is also a holomorphic vector bundle over X × S. It is easy to see that the adjoint vector bundle ad(P) corresponds to the sheaf on X × S defined by the space of G-invariant vertical holomorphic vector fields on P. Therefore, ad(P) is canonically a holomorphic subbundle of AT (P). In fact, we have an exact sequence of vector bundles 0

/ ad(P)

/ AT (P)

/ T (X × S)

/0 ,

where the projection AT (P) → T (X × S) is defined by the differential d) π.


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Let p : X × S → S be the canonical projection. Let P = P|X × {s0 }. The Kodaira-Spencer map (8) ρ : Ts0 → H 1 (X, ad(P )) can be described in a geometric way. Any complex connection θ on P gives a C ∞ -splitting σ : p∗ T S → AT (P) ×T (X×S ) p∗ T S of the Atiyah sequence over X × S. Given a tangent vector v of S at s0 , extended locally to a holomorphic vector field V , the form ∂(σ(V ))|X × {s0 } represents the element ρ(v) ∈ H 1 (X, ad(P )) in terms of Dolbeault cohomology. The form ∂(σ(V ))|X ×{s0 } is actually a (0, 1)-form with values in ad(P ). It is easy to see that this form coincides with the form given by the contraction Ω v|X × {s0 }, where Ω is the curvature of the connection θ. Proposition 5. ρ(v) = [Ω v] ∈ H 1 (X, ad(P )). We now assume the principal G-bundle P → X to be stable. We denote by Ms a connected component of a moduli space of stable G-bundles over X. This construction of Ms from local universal families P → X × S yields a global adjoint bundle on X × Ms in the orbifold sense, which comes from the adjoint bundles of local families. For short, we denote the adjoint bundle on X × Ms by ad(P). Since bundles and metrics will descend to X × Ms in the orbifold sense (from local universal families), it will be sufficient to consider local families of stable G-bundles P → X × S. We recall that the Hermite-Einstein condition reads ΛΩs = c ∈ z → C ∞ (X × {s}, ad(P|X × {s})).

(9)

Here the element c ∈ z is a topological constant, or in other words, it remains fixed for all s ∈ S, provided S is connected. The vector bundles ad(P)|X ×{s} carry Hermite-Einstein metrics, whose curvature forms ad(Ωs ) satisfy Λ ad(Ωs ) = 0 . Since ΛΩs = g αβ Rαβ (z, s) , the above condition (9) implies that −g αβ Riβ;α = −g αβ Rαβ;i = 0 .

(10)

We use here the semi-colon notation for covariant derivatives with respect to the Hermite-Einstein metric on the vector bundle ad(P) and the K¨ ahler structure on X × S, flat in S-direction. The connection form θ and the curvature tensor R have values in the vector bundle ad(P). However, when we consider the induced Hermite-Einstein structure on the vector bundle ad(P), these act as ad(θ) and ad(R) on differential forms with values in End(ad(P)).

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The decomposition g = z ⊕ [g, g] of the Lie algebra gives a decomposition of the adjoint bundle ad(P) = (P ×G z) ⊕ [ad(P), ad(P)] . In terms of Hermitian metrics on the fibers of ad(P) the above is an orthogonal decomposition. Since P ×G z is trivial, and since the action of G on z is trivial, the induced connection on P ×G z is trivial. From (9) and Proposition 5 we infer Proposition 6. The form ηi = Riβ dz β ∈ A0,1 (X × {s}, ad(P)|X×{s} ) is the harmonic representative of the Dolbeault cohomology class * + ∂ ρ ∈ H 1 (X, ad(P )) ∂si s=s0 where ρ is the homomorphism in (8). Remark 1. Like in the case of Hermite-Einstein vector bundles, the above proposition includes the case where S is singular and possibly non-reduced. On the vector bundle ad(P) we denote the inner product by parentheses. Definition 4. Let P → X × S be a local, universal family of stable vector bundles over a general complex space S ⊂ U = {(si , . . . , sN )}, equipped with a family of Hermite-Einstein metrics. Then the Petersson-Weil inner product is defined by ∂ ∂ PW PW , g αβ (Riβ , Rαj )g dV, = Gij = G ∂si s ∂sj s X×{s} and ωP W =

√

W i j −1GP ij (s)ds ∧ ds

is the corresponding Hermitian form. of Let Ω ∈ A1,1 (X × S, ad(P)) be the curvature form given by the family Hermite-Einstein connections. Again, we use fiber integrals of the form Y/S τ and write X τ for short. Let c ∈ z be the topological constant from (9). Proposition 7. The generalized Petersson-Weil form on a complex space S equals n−1 √ ωX 1 ωn +c (11) (Ω ∧ Ω) ∧ −1Ω ∧ X , ωP W = 2 X (n − 1)! n! X in particular, it is K¨ ahler, and possesses locally a ∂∂-potential.


481

Next, we interpret ad(Ω) as the curvature form for the Hermite-Einstein bundle ad(P). The orthogonal decomposition g = [g, g] ⊕ z as G-modules gives rise to a decomposition ad(P) = (P ×G [g, g]) ⊕ (X × z) . From this decomposition we obtain a decomposition of the tangent spaces of the moduli space H 1 (X, ad(P)) = H 1 (X, P ×G [g, g]) ⊕ H 1 (X, OX ) ⊗ z which is orthogonal with respect to the Petersson-Weil metric. Note that P ×G [g, g] = [ad(P), ad(P)]. Accordingly the Petersson-Weil form decomposes into ω P W = ωP W + ωPz W . It follows immediately that ωPz W is flat. Up to finite quotients, ωPz W is defined on a product of Jacobians, which we disregard in the sequel, because the description of ωPz W as a curvature form of a determinant line bundle is contained in the study of holomorphic line bundles. So we assume that G is semisimple. Now ad(P) is (relative) Hermite-Einstein with curvature form ad(Ω). Let Ad : g → End(g) be the linear representation defined by the Lie algebra structure of g. Since G is now semisimple, this homomorphism Ad is injective. We may normalize the K-invariant Hermitian form on g so that the homomorphism Ad takes the K-invariant Hermitian form on g to the Hermitian form on End(g) defined by (A, B) → tr(AB ∗ ). Lemma 1. The Petersson-Weil form for ad(P) → X × S, i.e., ad(P )

ωP W

=

1 2

tr(ad(Ω) ∧ ad(Ω)) ∧ X

n−1 ωX (n − 1)!

equals ω P W defined in Definition 4. Let r = dimC g. Then the term of lowest degree of the Chern character form of the virtual vector bundle ad(P) − (X × S) × g equals ch2 (ad(P)) = tr

1 2

√ −1 −1 Ω∧ Ω . 2π 2π

√

Next we assume that X is a projective manifold such that the K¨ ahler form ωX is the Chern form c1 (L, h) of a positive Hermitian line bundle. We consider the product of the Chern character forms and Todd forms with respect to the given metrics

482

Georg Schumacher

( ' ch (ad(P) − (X × S) × g) ⊗ (L − L−1 )⊗(n−1) td(X × S/S). The term of degree (n + 1, n + 1) in the above expression equals 1 n−1 4π 2 · 2n tr(ad(Ω) ∧ ad(Ω)) ∧ ωX . 2 The Riemann-Roch theorem by Bismut, Gillet and Soulé [BGS] implies the following theorem. Theorem 3. Let (δ, hQ ) be the determinant line bundle ' ( δ = det Rf∗ (ad(P) − g × (X × S)) ⊗ (L − L−1 )⊗(n−1) equipped with the Quillen metric hQ . Then ωP W =

1 c1 (δ, hQ ). 4π 2 · 2n · (n − 1)!

The curvature tensor of the generalized Petersson-Weil metric is accessible by similar methods (cf. [B-S3]). Finally, we address the relationship of moduli of stable principal bundles and corresponding stable vector bundles. Let G be a simple linear algebraic group defined over the field of complex numbers together with a faithful representation G → GL(V0 ). As above, we associate to any stable principal G-bundle P the stable vector bundle PV0 := P ×G V0 . This gives rise to an immersion of moduli spaces M → MV , which is an embedding in the orbifold sense, i.e. an embedding on the level of universal deformations. We equip both spaces with the Petersson-Weil metrics ωP W and ωPV W respectively. We fix a decomposition End(V0 ) = g ⊕ w

(12)

of G-modules (since G is simple, such a decomposition exists). Let P → X be a stable principal G-bundle. The decomposition of G-modules in (12) induces a decomposition of vector bundles End(PV0 ) = ad(P ) ⊕ (P ×G w) ,

(13)

which is clearly an orthogonal decomposition with respect to the HermiteEinstein connection on End(PV0 ) (the Hermite-Einstein connection on the polystable vector bundle PV0 induces a Hermite-Einstein connection on End(PV0 )). Now (13) yields a decomposition H 1 (X, ad(P )) ⊕ H 1 (X, P ×G w) = H 1 (X, End(PV0 )) ,

(14)

which is an orthogonal decomposition with respect to the Petersson-Weil inner product for moduli of polystable vector bundles, and the restriction to


483

H 1 (X, ad(P )) gives the Petersson-Weil inner product for stable principal Gbundles. Using the decomposition (14) we obtain embeddings S → V for the base spaces of universal deformations of stable principal G-bundles and polystable vector bundles, which descend to the moduli spaces. Therefore, we have the following theorem: Theorem 4. The second fundamental form of the immersion of orbifold spaces M → MV vanishes, in particular, M is totally geodesic in MV . 3.3 Moduli of Higgs Bundles A Higgs bundle over a compact K¨ ahler manifold X is a pair of the form (E, ϕ), where E is a holomorphic vector bundle over X and ϕ a holomorphic section 1 satisfying the integrability condition ϕ∧ϕ = 0. We recall that of End(E)⊗ΩX Higgs bundles over a compact Riemann surface were introduced by Hitchin in [HI] where he constructed their moduli and investigated the global, as well as the local, structures of the moduli space. One of the main results of [HI] was that a stable Higgs bundle admits a unique Hermitian-Yang-Mills connection. Simpson proved that a stable Higgs bundle admits a unique Hermitian-YangMills connection [SI1]. He also constructed the moduli space of Higgs bundles over a complex projective manifold [SI3]. In [B-S4] the local geometry of a moduli space of Higgs bundles was studied from the point of view of the generalized Petersson-Weil geometry. Some of these results will be explained below. We denote by E a holomorphic vector bundle over a K¨ ahler manifold (X, ωX ) of rank r. Definition 5. (i) A Higgs field on a vector bundle E over X is a holomorphic section 1 ) ϕ ∈ H 0 (X, End(E) ⊗OXΩX such that ϕ ∧ ϕ = 0, n i.e., [ϕα , ϕγ ] = 0 for all α, γ, where ϕ = α=1 ϕα dz α . (ii) A Higgs bundle is a pair (E, ϕ), where ϕ is a Higgs field on E.

(15)

The definition of stability of a Higgs bundle (E, ϕ) in this context is Mum1 and ford stability, where only coherent subsheaves F with ϕ(F ) ⊂ F ⊗OX ΩX 0 < rkF < rkE are considered. Polystability is defined accordingly. Polystable Higgs bundles (E, ϕ) are known to carry a unique HermitianYang-Mills connection by results of Hitchin and Simpson [HI, SI1]. Definition 6. Let (E, ϕ) be a Higgs bundle. A Hermitian-Yang-Mills connection on (E, ϕ) is a hermitian connection θE on E with curvature form ΩE such that (16) ΛωX (ΩE + ϕ ∧ ϕ∗ ) = λ · idE

484

Georg Schumacher

for some λ ∈ R, where ΛωX is the adjoint to the exterior multiplication of a form with ωX . In local holomorphic coordinates z α this equation reads ' ( g βα Rαβ + [ϕα , ϕ∗β ] = λ · idE , where ΩE = Rαβ dz α ∧ dz β . General theory provides a semi–universal deformation of pairs (E, ϕ), where ϕ is a End(E)-valued holomorphic 1-form. The integrability condition ϕ ∧ ϕ = 0 defines a complex analytic subspace of the parameter space, and thus yields a semi-universal deformation for Higgs bundles. It follows like in the classical case that stable Higgs bundles are simple in the sense H 0 (X, End(E, ϕ)) = C · idE ,

(17)

End(E, ϕ) ⊂ End(E)

(18)

where is the subsheaf that commute with ϕ. A holomorphic family (Es , ϕs )s∈S of Higgs bundles, parameterized by a complex space S, consists of a holomorphic 1 , vector bundle E on X × S and a holomorphic section Φ of End(E) ⊗ ρ∗ ΩX where ρ : X × S → X is the canonical projection, such that E|X × {s} = Es , and Φ|X × {s} = ϕs for all s ∈ S. Observe that Φ defines an End(E)-valued, holomorphic 1-form on X × S, 1 1 ⊂ ΩX×S . as ρ∗ ΩX Again any stable Higgs bundle possesses a universal deformation, and a coarse moduli space MH can be constructed by means of the methods described above. We use the previous notations for local coordinates xα on X and si on S. The K¨ ahler form ωX gives rise to a connection on X, which we will extend in a flat way to X × S. Let the hermitian connection θE on E be given locally by matrix-valued (1, 0)-forms {θα }nα=1 with respect to some local trivialization of E. Let σ be a locally defined section of End(E), which is a matrix-valued function with respect to the trivialization of E. We use ∂σ = ∂α σ = σ|α ∂z α for partial derivatives, and the semi-colon notation for covariant derivatives like σ;α = ∇α σ = σ|α + [σ, θα ]. We denote by Rαβ the components of the curvature form Ωαβ = θα|β . For tensors with values in the endomorphism bundle, we also have the contributions that arise from the K¨ ahler connection on the base. Infinitesimal deformations of Higgs bundles. For any i ≥ 0, the Higgs field ϕ gives a OX -linear homomorphism


485

i+1 i fϕ (i) : End(E) ⊗ ΩX −→ End(E) ⊗ ΩX

defined by s −→ [s, ϕ]. From the given condition that ϕ ∧ ϕ = 0 it follows immediately that fϕ (i + 1) ◦ fϕ (i) = 0 for all i. In other words, we have the cotangent complex, which is a complex of coherent OX -modules fϕ (0)

fϕ (1)

fϕ (i−1)

1 i D• : D0 := End(E) −→ End(E) ⊗ ΩX −→ · · · −→ Di := End(E) ⊗ ΩX fϕ (i)

fϕ (i+1)

fϕ (n−1)

i+1 n −→ End(E) ⊗ ΩX −→ · · · −→ End(E) ⊗ ΩX −→ 0

over X. We note that H 0 (X, End(E, ϕ)) = H0 (D• ) (see (18)). The space of all infinitesimal deformations of (E, ϕ) is parametrized by the first hypercohomology H1 (D• ), and the obstructions to deformations of (E, ϕ) are guided by H2 (D• ). The spaces (19) C p,q : = Ap,q (X, End(E)) of differentiable (p, q)-forms over X with values in End(E) xyield the Dolbeault resolution, where d is the ∂-operator, and d : C p,q → C p+1,q is defined by

(20)

d (χ) = [χ, ϕ] .

Since the ϕ is holomorphic with ϕ ∧ ϕ = 0, it follows that (C •• , d , d ) is actually a double complex. Standard arguments show that the cohomology of the associated simple complex (C • , d) with C r := C p,q and d := d + (−1)q+1 d (21) p+q=r

gives the desired tangent cohomology. Assume that (E, ϕ) admits a Hermitian-Yang-Mills connection (see Definition 6). Then any holomorphic section σ of End(E, ϕ) (defined in (18)) is parallel with respect to a Hermitian-Yang-Mills connection on (E, ϕ), i.e. σ;α = 0. From now on, we assume that the given Higgs bundle (E, ϕ) is stable. Therefore, it carries a hermitian metric satisfying the Hermitian-Yang-Mills equation (see Definition 6). This metric is unique up to a dilation by a globally constant scalar. Obviously the space H 0 (C • ) consists of those holomorphic sections of End(E) which commute with ϕ. The stability condition of (E, ϕ) implies that any such section is a constant scalar multiple of the identity automorphism of E (see (17)).

486

Georg Schumacher

As mentioned earlier, the hypercohomology H1 (D• ) = H1 (C •• ) = H 1 (C • ) is the space of all infinitesimal deformations of (E, ϕ). We denote the Kodaira-Spencer map by (22) ρ : Ts0 S → H1 (C •• ) . Now (C • , d) becomes an elliptic complex, when equipped with the inner products induced by the hermitian metric on E and the K¨ ahler metric ωX on X. In particular the formal adjoint operators to dr are in fact adjoint. Let (E, Φ) be a holomorphic family of Higgs bundles over a complex space S, and denote by {hs } any family of hermitian metrics on Es , i.e., a hermitian metric h on E over X × S., whose curvature form will be denoted by Ω.x So the contraction ∂ Ω ∂si equals Riβ dz β . One can see that the global tensors Ω and ϕ over X × S already describe the infinitesimal deformations. In other words, we have the following lemma: Lemma 2. The Kodaira-Spencer class * + ∂ ρ ∈ H1 (C •• ) ∂si s0 (the homomorphism ρ is defined in (22)) is represented by ηi = (ϕα;i dz α , Riβ dz β )|X×{s0 } .

(23)

The Kähler form ωX and the hermitian metric h together provide the above double complex C •• with a natural inner product such that the adjoint operators dj∗ are the formal adjoint operators. Now assume that for each point s ∈ S, the Higgs bundle (Es , Φs ) over X is stable. The Hermitian-Yang-Mills connections on this family of stable Higgs bundles (E, Φ) are induced by a hermitian metric h on E, whose curvature form Ω is unique up to a differential form of type idE ⊗ f ∗ ω , where ω is some (1, 1)-form on the base S and f : X × S → S is the natural projection. Indeed, this follows immediately from the fact that any two Hermitian-YangMills metrics on a stable Higgs bundle differ by multiplication with a constant scalar. Therefore, the components Riβ of the curvature tensor in Lemma 2 are uniquely determined by the the family of Hermitian-Yang-Mills connections on the Higgs bundles (Es , ϕs ). Proposition 8. The End(E)-valued 1-forms ηi = ϕα;i dz α + Riβ dz β are the harmonic representatives of the Kodaira-Spencer classes ρ(∂/∂si |s0 ).


487

Generalized Petersson-Weil metric. Now, we are in a position to introduce a generalized Petersson-Weil metric on the parameter space S for a family of stable Higgs bundles. Namely * + ∂ ∂ , (24) := Gij := ηi , ηj = GP W ∂si s0 ∂sj s0 = tr (g βα ϕα;i ϕ∗β;j )g dV + tr (g βα Riβ Rαj )g dV . X

X

We set ωP W =

√ W i j −1 GP ij ds ∧ ds .

In order to compute the induced connection, we need certain identities. Here, we need to assume that the base S is non-singular. When dealing with fiber integrals we will show the K¨ ahler property also for singular base spaces. α β Let ηi = (ϕα;i dz , Riβ dz ). Then ηi;k = ηk;i dηi;k + ηi ∧ ηk = 0 d∗ ηi;k = 0

(25) (26) (27)

ηi;j = dRij 2Rij = d∗ dRij = d∗ ηi;j

(28) (29)

d∗ ηi;j = g βα ([ϕα;i , ϕ∗β,j ] + [Riβ , Rαj ])

(30)

As a consequence of (27) and (28), we note that ηi , ηj;k = 0 .

(31)

Using the above notation, we set Gij|k = ∂sk Gij . Proposition 9. The generalized Petersson-Weil metric is K¨ ahler, more precisely, we have (32) Gij|k = ηi;k , ηj . Proof. We have ηi , ηj;k = ηi , dRjk = d∗ ηi , Rjk = 0. Curvature of the generalized Petersson-Weil metric. Since the ηi are harmonic and span the whole space H 1 (C • ), equation (32) implies that for normal coordinates {si } for the Petersson-Weil metric at the base point s0 all harmonic projections H(ηi;k |s=s0 ) vanish. The notation ηi for the harmonic representative (ϕα;i dz α , Riβ dz β )|X×{s0 } .

488

of

Georg Schumacher

ρ(∂/∂si |s0 ) ∈ H1 (C •• )

fits into the general concept of the curvature of a generalized PeterssonWeil metric. The computations are rather involved. Based upon the identities (25,26,28,29,30) of the previous section, we compute the curvature tensor in terms of the harmonic Kodaira-Spencer tensors (cf. [B-S4]). Its holomorphic sectional curvature for moduli of bundles on curves is semi-positive. Fiber integral formula. We will show a fiber integral formula involving both the global curvature form for a family of Hermite-Einstein metrics and the Higgs fields. It also implies the existence of a local ∂∂-potential for the generalized Petersson-Weil metric. We encounter a certain symmetry between curvature form and Higgs field. We consider a moduli space of stable Higgs bundles MH . Although, in general there is no universal holomorphic vector bundle E globally on X ×MH , the bundle End(E) exists in the orbifold sense over all of X × MH , since the non-zero scalars act trivially on End(E). The function χ on MH defined by s → χ(s) = g βα tr (ϕα ϕ∗β )g dV X×{s}

is a function of class C ∞ on MH . In a similar way, the curvature form Ω of the Hermitian-Yang-Mills connections is a well-defined End(E)-valued (1, 1)-form over X × MH . In the notation of fiber integrals, we replace X × S/S by just X. Proposition 10. Let Ω be the curvature form of (E, h). Then the following fiber integral formula holds: n−1 ωX 1 + (33) tr (Ω ∧ Ω) ∧ ωP W = 2 X (n − 1)! n−1 n √ √ ωX ωX 1 +λ + −1∂∂ . tr ( −1Ω) ∧ tr (ϕ ∧ ϕ∗ ) ∧ n! 2 X (n − 1)! X Here λ is determined by n−1 n √ ωX ωX =λ tr ( −1Ω) ∧ (n − 1)! X X n! which is independently of s ∈ S over any connected component of S. From now on, we assume that X is a Kähler manifold whose K¨ ahler form is the Chern form ωX = c1 (L, hL ) of a positive hermitian line bundle (L, hL ).


489

Given a proper, smooth holomorphic map f : X → S and a locally free sheaf F on X , the determinant line bundle of F on S is by definition det Rf∗ F [BGS]. We apply the generalized Riemann-Roch theorem by Bismut, Gillet and Soulé [BGS]. X /S

ch(F , h)td(X /S, ωX ) .

The above Proposition 10 is translated into a formula involving Chern character forms of hermitian bundles. Theorem 5. 1 ωP W = 4π 2

( ' 2 1 ch (End(E) − Or ) ⊗ (L − L−1 )⊗(n−1) n 2 r(n − 1)! ( ' 1 − n+2 ch (Λr E − (Λr E)−1 )⊗2 ⊗ (L − L−1 )⊗(n−1) 2 r(n − 1)! 1 λ + ch Λr E − (Λr E)−1 ⊗ (L − L−1 )⊗n n+1 2π 2 n! n−1 ωX 1 √ . tr (ϕ ∧ ϕ∗ ) ∧ + 2 −1∂∂ 8π (n − 1)!

−

Using Theorem 5, we will express the generalized Petersson-Weil Kähler form as the curvature form of a holomorphic Hermitian line bundle. Let q : X × MH → MH be the canonical projection. We introduce the following determinant line bundles δj , equipped with Quillen metrics hQ j : ' ( 2 δ1 = det Rq∗ (End(E) − Or ) ⊗ (L − L−1 )⊗(n−1) ' ( ⊗2 ⊗ (L − L−1 )⊗(n−1) δ2 = det Rq∗ Λr E − (Λr E)−1 δ3 = det Rq∗ Λr E − (Λr E)−1 ⊗ (L − L−1 )⊗n . Setting

χ=

tr (ϕ ∧ ϕ∗ ) ∧

n ωX (n − 1)!

we equip the trivial bundle OMH with the hermitian metric eχ . Combining Theorem 5 and [BGS, Theorem 0.1] we have the following theorem: Theorem 6. The generalized Petersson-Weil K¨ ahler form is a linear combi), j = 1, 2, 3, and c1 (OMH , eχ ). nation of the (1, 1)-forms c1 (δj , hQ j

490

Georg Schumacher

A holomorphic closed 2-form on a moduli space of Higgs bundles Using the previous notation, we introduce a holomorphic two-form π on the base of a universal deformation S with dπ = 0. Let ηi = ϕα;i dz α + Riβ dz β ∈ C 1,0 (X × S, End(Es )) ⊕ C 0,1 (X × S, End(Es )). Then a two-form π = πik (s)dsi ∧ dsk on S is given by

πik = X×{s}

tr (g βα (ϕα;i · Rkβ − ϕα;k · Riβ ))g dV .

Proposition 11. The two-form π is holomorphic, and furthermore, it is of the form π = dν for a certain holomorphic 1-form ν on S. The forms ν and π descend to the moduli space of Higgs bundles as holomorphic forms. Denote by MH a moduli space of stable vector bundles on X. As any stable vector bundle E defines a Higgs bundle (E, ϕ) with Higgs field ϕ = 0. The Hermite-Einstein connection on the stable vector bundle E coincides with the Hermitian-Yang-Mills connection on (E, 0). We have an embedding i : M → MH into the corresponding moduli space of stable Higgs bundles. Let MsH ⊂ MH denote the Zariski open subset defined by all Higgs bundles (E, ϕ) with E stable. Therefore, we have a retraction f : MsH → M that sends any (E, ϕ) to E. Proposition 12. The forms ν and π vanish on the fibers of the above map f : MsH → M. The form ν does not descend under f unless it is zero on MsH . 3.4 Non-Abelian Hodge Symmetry, Symplectic and Hyper-K¨ ahler Structure on the Moduli Space of Higgs Bundles Recently, hyper-Kähler structures were constructed, which arose from moduli spaces [OGR, LE]. In this section, we will construct such structures in a different context, following [B-S4]. It was shown in [B-S4] that the holomorphic 2-form π constructed in the previous section is non-degenerate under some assumptions on the Higgs bundles. This was done by constructing an involution ι on the first hypercohomology. This involution ι is a non-abelian analogue of Hodge symmetry and it gives a hyper-Kähler structure on the moduli space of Higgs bundles. We note that in [S-S], a hyper-K¨ ahler structure on a moduli space of flat connections was constructed by following Hitchin’s original approach. Let (E, ϕ) be a stable Higgs bundle over X equipped with a HermitianYang-Mills connection. We first provide the space of infinitesimal deformations of (E, ϕ), namely H1 (C •• ), with a quaternionic structure under an assumption on H2 (C •• ), where C •• is the complex defined in (19).


491

Lemma 3. The End(E)-valued (1, 1)-form ωX · idE defines a non-zero element in H2 (C •• ). In other words, there is a natural embedding of the line C · ωX · idE → H2 (C •• ) . Assumption. For the rest of this section we restrict ourselves to stable Higgs bundles (E, ϕ) satisfying the following two conditions: A The rational characteristic class c2 (End(E)) vanishes. This is equivalent to the condition that the Hermitian-Yang-Mills connection on (E, ϕ) is projectively flat, i.e., Rαβ + [ϕα , ϕ∗β ] = λ · gαβ · idE

(34)

for some λ ∈ C. B dim H 2 (C •• ) = 1. In view, of Lemma 3 this is equivalent to the condition that (35) H2 (C •• ) = C · ωX · idE . The above assumption B implies that deformations of (E, ϕ) are not obstructed. We have an involution of the space of End(E)-valued 1-forms defined by ι : C1 → C1 (a, b) → (−b∗ , a∗ )

(36)

where C 1 is defined in (21). Obviously ι2 = −idC 1 . We shall see this involution descends to the space of infinitesimal deformations of (E, ϕ). Proposition 13. Let η ∈ C 1 be harmonic (so η gives an infinitesimal deformation of (E, ϕ)). Then ι(η) is harmonic. Let η = 0. Since deformations are not obstructed, we can assume that there is a coordinate system on the base S of a universal deformation so that η is of the form ηi in the sense of (23). We just indicate the essential steps and omit the technical proofs. The first observation is d∗ (ιηi ) = 0. However, in general ιηi need not be d-closed, but d(ιηi ) = ξ;ı , where ξ = (ϕα;γ dz α ∧ dz γ , (Rαβ + [ϕα , ϕ∗β ])dz α ∧ dz β , −ϕ∗β;δ dz β ∧ dz δ ). is harmonic. Now, by our above assumption (B), the harmonic form ξ is of the form c(s) · ωX idE . From the definition of ξ and (34) we know that c(s) = λ is independent of s ∈ S. Hence d(ιηi ) = 0. This completes the proof of the proposition.

492

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Corollary 1. The d-closed holomorphic 2-form π is non-degenerate at. In particular, the dimension of H1 (C •• ) is even. Proof. In the above notation, πik = ηi , ι(ηk ). As ι takes harmonic elements of C 1 to harmonic elements of C 1 , it induces a bijective map of H1 to itself. This shows the non-degeneracy. It may be noted for X = CP2 , there are examples of stable Higgs bundles where dim H1 (C •• ) is odd. (They do not satisfy the two assumptions.) Now the quaternionic structure can be given explicitly: For η = (a, b) ∈ H1 (C •• ) we have ι(a, b) = (−b∗ , a∗ ), and for a tangent vector η we set √ √ I(η) = −1 · η, J(η) = ι(η), K(η) = −1 · ι(η). By definition the equations for an almost quaternionic structure are verified. For complex tangent vectors η, ϑ of the base space have Iη, ϑ = ω P W (η, ϑ), Jη, ϑ = π(η, ϑ), √ Kη, ϑ = −1 · π(η, ϑ), where π denotes the conjugate of π. The corresponding differential forms are closed (and non-degenerate), so that together with [HI, Lemma (6.8)] the following holds: Theorem 7. Consider the Zariski open subset W of the moduli of stable Higgs bundles over which the condition H2 = C · ωX idE holds. Then W carries a natural hyper-K¨ ahler structure, related to the Petersson-Weil structure ω P W and the holomorphic symplectic form π.

3.5 Coupled Vortex Equations and Moduli We use our basic notation concerning complex manifolds, holomorphic vector bundles and coordinates. The K¨ ahler geometry of moduli spaces for coupled vortex equations was introduced in [B-S5]. We will describe our approach below. Let X be a compact, connected Kähler manifold of dimension n equipped √ with a K¨ ahler form ωX . We will write ωX = −1gαβ dz α ∧ dz β with respect to local holomorphic coordinates (z 1 , . . . , z n ), and we will always use the summation convention. In the sequel, we identify locally free coherent sheaves with holomorphic vector bundles. Let σ be a locally defined section of E, which is a vector-valued function with respect to the trivialization of E. Again, we use


493

∂σ = ∂α σ = σ|α ∂z α and set σ;α = ∇α σ = σ|α + θα ◦ σ. Hence σ;αβ = σ;βα − Rαβ ◦ σ, where Rαβ denote the components of the curvature form Ωαβ = θα|β . For tensors with values in the endomorphism bundle, we also have the contributions that arise from the K¨ ahler connection on the base. For any differentiable homomorphism ψ : E 2 → E 1 , where (E i , hi ) are hermitian vector bundles, i , we have with curvature tensors Rαβ 1 2 ψ;αβ = ψ;βα − Rαβ ◦ ψ + ψ ◦ Rαβ .

Deformations. Let E1 , E2 be holomorphic vector bundles and φ : E2 → E1 a morphism. An automorphism of a triple T = (E1 , E2 , φ) consists by definition of a pair of automorphims ψj of Ej ; j = 1, 2 with φ ◦ ψ2 = ψ1 ◦ φ. A holomorphic family of such triples over a complex space S consist of a triple T = (E1 , E2 , Φ) on X ×S. For any s ∈ S the fiber Ts is just the restriction of (E1 , E2 , Φ) to X ×{s} X. From the notion of a holomorphic family, we can derive the notion of a deformation of an object T over a space (S, s0 ) with a distinguished base point in the usual way, i.e. by fixing an isomorphism ∼ T −→ Ts0 . Isomorphism classes of deformations of such triples (E1 , E2 , φ) satisfy the Schlessinger condition, and semi-universal deformations exist by general theory. An isomorphism class of an infinitesimal deformation of (E1 , E2 , φ) over the double point D = (0, C ⊕ εC) with ε2 = 0 can be identified with the equivalence class of an extension of the map φ : E2 → E1 by itself, i.e. the equivalence class of a diagram 0

/ εE2

0

/ εE1

φ

/ E2 Φ

/ E1

/ E2

/0

(37)

φ

/ E1

/0

In order to describe infinitesimal deformations and infinitesimal automorphisms resp. of such triples (E1 , E2 , φ) we use the following complex of OX modules. C • : 0 → EndOX (E1 ) ⊕ EndOX (E2 ) −→ HomOX (E2 , E1 ) → 0, ∆

where ∆(ψ1 , ψ2 ) = ψ1 ◦ φ − φ ◦ ψ2 . The groups H (C ) and H1 (C • ) resp. stand for automorphisms and deformations over D of (E1 , E2 , φ) resp. whereas obstructions are in H2 (C • ). 0

•

494

Georg Schumacher

Let T → S, T = (E1 , E2 , Φ) be a holomorphic family of triples Ts , s ∈ S. Suppose that hi , i = 1, 2 are hermitian metrics on Ei such that the restrictions to X × {s} are solutions of the coupled vortex equations (40), (41). Let Ω i be the curvature forms of the hermitian connection for hi on E i over X × S with curvature tensors Ri , i = 1, 2. Stability, vortex equations and Kobayashi-Hitchin correspondence. Given a triple T = (E1 , E2 , φ), a subtriple T consists of coherent, torsion free subsheaves Ei ⊂ Ei ; i = 1, 2 such that φ = φ|E2 maps E2 to E1 . A subtriple is called proper, if it is neither equal to T nor equal to the zero triple. For a real number α, the the α-degree and α-slope of T (and in the same way for a triple of subsheaves) are defined to be degα (T ) = deg(E1 ) + deg(E2 ) + α · rk(E2 ) degα (T ) . µα (T ) = rkE1 + rkE2

(38) (39)

Here the degree is measured in terms of the Kähler metric ωX . Definition 7. A triple (E1 , E2 , φ) is called α-stable, if for any proper subtriple T µα (T ) < µα (T ) holds. Corresponding to α-stability there exist solutions of certain HermiteEinstein type equations, known as vortex equations. Let hi be hermitian metrics on Ei with curvature forms Ω i for i = 1, 2. Assume that the ωX -volume of X is normalized to 1. Denote by ΛX the operator dual to the multiplication of differential forms with values in vector bundles by ωX . Definition 8. The vortex equations for ((E1 , h1 ), (E2 , h2 ), φ) read √ −1ΛX Ω 1 + φφ∗ = τ1 · IdE1 √ −1ΛX Ω 2 − φ∗ φ = τ2 · IdE2

(40) (41)

or equivalently 1 g βα Rαβ + φφ∗ = τ1 · IdE1

(42)

2 − φ∗ φ = τ2 · IdE2 g βα Rαβ

(43)

for real numbers τ1 , τ2 . The Kobayashi-Hitchin correspondence asserts the unique existence of solutions of the vortex equations (up to unitary gauge transformations, under


495

necessary numerical conditions for the above constants, which follow from integrating the traces: Namely deg E1 + deg E2 = τ1 rkE1 + τ2 rkE2

(44)

must hold, where the sum of the degrees can be expressed in terms of the α-degree using (38). It is established in various situations: For dim X = 1 it is stated in [B-C2, Theorem 5.1], whereas for K¨ ahler manifolds of arbitrary dimension, it is shown for rkE2 = 1 in [G-P, Theorem 4.33]. Elliptic complex. In order to use the theory of elliptic operators, we observe that the Dolbeault complexes provide a resolution C •• of C • : / EndOX (E1 ) ⊕ EndOX (E2 )

0

/ A0,• (EndO (E1 ) ⊕ EndO (E2 )) X X

∆

/ HomOX (E2 , E1 )

0

∆

/ A0,• (HomO (E2 , E1 )) X

/ ...

(45) It follows immediately that ∆ ◦ ∂ = ∂ ◦ ∆. Now the cohomology of C • can be identified with the cohomology of the • associated to C •• . single complex C Now we have the Kodaira-Spencer mapping •• ) ρs0 : Ts0 S → H1 (C be the Kodaira-Spencer mapping, and ' ( 1 2 dz β , Riβ dz β ) ∈ µi = Φ;i , (Riβ (A0,0 (X, HomOX (E2 , E1 )), A0,1 (X, EndOX (E1 ) ⊕ EndOX (E2 )) represents the class ρs0 (∂/∂si )|s0 . Given hermitian metrics hi on Ei , together with ωX , the single complex associated to (45) is elliptic, namely 0

1

d d • ) : 0 → C 0,0 (X) −→ C 1,0 (X) ⊕ C 0,1 (X) −→ C 1,1 (X) ⊕ C 0,2 (X) → . . . Γ (C 0

d • ) : 0 → A0,0 (X, EndO (E1 ) ⊕ EndO (E2 )) −→ i.e. Γ (C X X d1

A0,0 (X, HomOX (E2 , E1 )) ⊕ A0,1 (X, EndOX (E1 ) ⊕ EndOX (E2 )) −→ d2

A0,1 (X, HomOX (E2 , E1 )) ⊕ A0,2 (X, EndOX (E1 ) ⊕ EndOX (E2 )) −→ . . . with 0

d0 (f ) = (∆f, ∂ f ) ; 1

f = (f1 , f2 ) 1

d (a, b) = (∂ a − ∆b, ∂ b) ; 1

/ ...

b = (b1 , b2 )

The (formal) adjoint operators d1∗ , and d2∗ can be computed explicitly.

(46) (47)

496

Georg Schumacher

Proposition 14. The forms µi are the harmonic representatives of the Kodaira-Spencer classes ρ(∂/∂s|s0 ). Moduli space and Hermitian structure. From the view of deformation theory, the moduli space of solutions of coupled vortex equations can be treated as follows. ahler manifold (X, ωX ) We consider a stable triple T = (E1 , E2 , φ) over a K¨ and assume the existence of a solution of the coupled vortex equation for a fixed pair (τ1 , τ2 ). General deformation theory provides a semi-universal deformation given by a family T over a base space (S, s0 ) and a an isomorphism ∼ T −→ Ts0 . Since the dimension of the automorphism group is constant near the distinguished base point, the deformation is universal. Now we use (exactly like in the Hermite-Einstein case) the implicit function theorem for appropriate Sobolev spaces (plus regularity theorems) to extend the solution of the coupled vertex equation to neighboring fibers. In this way, after replacing S by a neighborhood of s0 , if necessary, we get a family of hermitian metrics his on the bundles E|X×{s} , which amounts to hermitian metrics hi on Ei . We indicate, how to proceed in the case of singular base spaces S. Here, the strategy is, to embed S in a smooth space W , say a polydisk, and extend all elliptic operators to the ambient space, 2 but disregard the integrability equations ∂ = 0. The the implicit function theorem is applicable, and in the last step solutions of the coupled vortex equations are restricted to S. This deformation theoretic approach only depends of the uniqueness of the solutions and is independent of the validity of the Kobayashi-Hitchin correspondence. In this way solutions of coupled vortex equations can be studied from the viewpoint of K¨ ahler geometry (cf. [B-S5]).

References [A-B]

[A-V] [AR] [A-S] [BGS]

[B-S1]

Anchouche, B.; Biswas, I.: Einstein-Hermitian connections on polystable principal bundles over a compact K¨ ahler manifold. Amer. Jour. Math. 123 (2001), 207–228. Andreotti, A., Vesentini, E.: Carleman estimates for the Laplace-Beltrami equation on complex manifolds. Publ. Math. IHES 25, 81–130 (1965) Artin, M.: Versal deformations and algebraic stacks. Invent. math. 27, 165–189 (1974) Axelsson, R., Schumacher. G.: K¨ ahler Geometry of Douady Spaces, to appear in manuscripta math. Bismut, J.M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles I, II, III. Commun. Math. Phy. 115 49–78 (1988); 115 79–126 (1988); 115 301–351 (1988). Biswas, I.; Schumacher, G.: Determinant bundle, Quillen metric, and Petersson-Weil form on moduli spaces, GAFA, Geometric and Functional Analysis 9, 226-256 (1999)

K¨ ahler Geometry of Moduli Spaces of Holomorphic Vector Bundles [B-S2]

[B-S3] [B-S4] [B-S5] [B-C2] [F-S]

[G-P] [G-S] [HI] [LE]

[M] [OGR] [RA1] [RA2] [R-S]

[R]

[S-S]

[SCH1] [SCH2] [SCH3]

[SCH4] [SCH5]

497

Biswas, I.; Schumacher, G.: Generalized Petersson-Weil metric on the Douady space of embedded manifolds. Complex analysis and algebraic geometry, 109–115, de Gruyter, Berlin, 2000. Biswas, I.; Schumacher, G.: K¨ ahler structure on moduli spaces of principal bundles, to appear in J. of Math. Analysis and Appl. Biswas, I.; Schumacher, G.: Geometry of moduli spaces of Higgs bundles, to appear in Commun. Anal. Geom. Biswas, I.; Schumacher, G.: Vortex equation and Moduli – Deformation theoretic Approach and K¨ ahler Geometry, preprint 2005/06 Bradlow, S., Garc´ıa-Prada, O.: Stable triples, equivariant bundles and dimensional reduction. Math. Ann. 304, 225–252 (1996). Fujiki, A., Schumacher, G.: The moduli space of compact extremal K¨ ahler manifolds and generalized Petersson-Weil metrics. Publ. RIMS, Kyoto Univ. 26 101–183 (1990). Garc´ıa-Prada, O.: Dimensional reduction of stable bundles, vortices and stable pairs, Internat. J. Math. 5, 1–52 (1994). G´ omez, T.L.; Sols, I.: Moduli space of principal sheaves over projective varieties. Ann. of Math. 161 (2005), 1033–1088. Hitchin, N.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55, 59-126 (1987) Lehn, M.: Symplectic moduli spaces. Arbarello, E. (ed.) et al., School and conference on intersection theory and moduli. Trieste: ICTP Lecture Notes 19, 139-184 (2004) Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, 3 rd. edtn. Springer-Verlag, Berlin 1994. O’Grady, K. G.: A new six-dimensional irreducible symplectic variety. J. Algebr. Geom. 12 (2003), 435-505 Ramanathan, A.: Moduli for principal bundles over algebraic curves: I. Proc. Ind. Acad. Sci. (Math. Sci.) 106 (1996), 301–328. Ramanathan, A.: Moduli for principal bundles over algebraic curves: II. Proc. Indian Acad. Sci. (Math. Sci.) 106 (1996), 421–449. Ramanathan, A.; Subramanian, S.: Einstein-Hermitian connections on principal bundles and stability. J. Reine Angew. Math. 390 (1988), 21– 31. Rim, D.S.: Formal deformation theory, Groupes de monodromie en géométrie algébrique, exposé 6. Lect. notes in math. 288, Berlin, Heidelberg, New York, Tokyo: Springer 1972 Scheinost, P.; Schottenloher, M.: Metaplectic quantization of the moduli spaces of flat and parabolic bundles. Jour. Reine Angew. Math. 466, 145– 219 (1995) Schumacher, G.: Construction of the coarse moduli space of compact polarized manifolds mit c1 = 0., Math. Ann. 264, 81–90 (1983). Schumacher, G.: Moduli of polarized K¨ ahler manifolds, Math. Ann. 269, 137–144 (1984). Schumacher, G.: The curvature of the Petersson-Weil metric on the moduli space of K¨ ahler-Einstein manifolds, in “Complex Analysis”, Trento 1993 (Plenum Publ.) Schumacher, G.: Non-reduced moduli spaces, Proceedings, Hayama 1998. Schumacher, G.: Moduli of framed manifolds, Invent. math. 134 229–249 (1998).

498 [SCH6] [S-T]

[SI1]

[SI2] [SI3] [SIU]

[T-Y]

[VA1] [VA2] [VA2]

Georg Schumacher Schumacher, G.: Moduli as algebraic spaces. Oka 100 conf., Adv. Stud. Pure Math. 2001 Schumacher, G.; Toma, M.: On the Petersson-Weil metric for the moduli space of Hermite-Einstein bundles and its curvature. Math. Ann. 293 101–107 (1992). Simpson, C.T.: Constructing variations of Hodge structure using YangMills theory and applications to uniformization. Jour. Amer. Math. Soc. 1, 867–918 (1988) ´ Simpson, C.T.: Higgs bundles and local systems. Publ. Math. I.H.E.S. 75, 5–95 (1992) Simpson, C.T.: Moduli of representations of the fundamental group of a ´ smooth projective variety. II. Publ. Math. I.H.E.S. 80, 5–79 (1994) Siu, Y.-T.: Curvature of the Weil-Petersson metric in the moduli space of compact K¨ ahler-Einstein manifolds of negative first Chern class, Complex Analysis, Papers in honor of Wilhelm Stoll, P.-M Wong and A. Howard edts. Tian, G., Yau, S.-T.: Existence of K¨ ahler-Einstein metrics on complete K¨ ahler manifolds and their applications to algebraic geometry, Adv. Ser. in Math. Phys. Vol. 1, Mathematical aspects of string theory, S.T. Yau ed., San Diego, 1986 Varouchas, J.: Stabilité de la classe des variétés K¨ ahlériennes par certains morphismes propres. Invent. Math. 77 (1984), 117–127. Varouchas, J.: K¨ ahler spaces and proper open morphisms. Math. Ann. 283 (1989), 13–52. Viehweg, E.: Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin Heidelberg 1995.

Index

1-form

133

(a,b,c)-surfaces 45 a.q.e.d.-equivalence 49 Abel–Jacobi map 455 Abundance Conjecture 321, 324 adapted complex structure 257 additive Chow groups with higher modulus 69 additive higher Chow groups 67, 68 adjoint representation 230 affine symmetric space 235 algebraic approximation 329 algebraic cycle 451 algebraic group 229 Almost Fano 340, 346, 348, 350 Almost holomorphic map 365 ample Cartier structure 246 ample cone 313 analytic family of cycles 245 analytic type 178 anti-Zariski pair 198 Anticanonical map 340, 342 Anticanonical model 340 Arakelov inequality 418, 420 Arakelov, Sergei J. 405 Artin’s braid group 37 Artin–Schreier 75 asymptotically optimal condition 173 asymptotically proper condition 173 automorphic representation 281 cuspidal, 282 Ax-Katz 63, 64

bands and strings 104, 105, 107, 110 base cycle 248 Base points 341, 343, 344 BDPP, Theorem of Boucksom, Demailly, Peternell, P˘ aun 359, 361 Beauville surfaces 31 Bend-and-Break 368, 370 Bloch’s decomposition of the diagonal 62, 63, 65 Bloch–Beilinson filtration 466 Bogomolov, Fedor A. 362, 364, 366 Borel subalgebra 232 Borel subgroup 232 Borel’s Fixed Point Theorem 238 Borel, Armand 398 Boucksom, Sébastien 359, 361, see BDPP bunches of chains 83, 102, 104, 107, 117 Calabi-Yau threefold 285, 303 rigid, 281, 285, 291 Calabi-Yau variety 271, 284, 302 Campana, Frédéric 364 Cartan decomposition 211–213 Cartan involution 221 ´ Cartan, Elie 360, 379, 387, see also Cartan-Fubini type theorem Cartan-Fubini type theorem 387 for varieties covered by lines, 360 Cauchy characteristic of a distribution 386 Cauchy subvariety 387

500

Index

cDV singularity 338, 344, 353 Characterization of Pn 386 Chazy’s equation 460 Chevalley’s theorem 229 Chow scheme 245 Chow–K¨ unneth decomposition 461 Co-adjoint orbit 220–222 coadjoint orbit 240 completely reducible 231 complex multiplication 271, 283, 289, 291 complex radial index 137 cone-like singularity 133 congruences 60, 66, 67 connecting family 322, 324 Contact manifold 388, 390 uniqueness of contact structure, 359, 388 crystalline cohomology 60, 79 cusp 173 cusp form 277 cycle space 243 de Jong’s alteration 65 de Rham-Witt complex 70 def-equivalence 43 Deformation of a surjective morphism 360, see Deformation space of a surjective morphism deformation pattern 189 Deformation rigidity of certain varieties, 388 of Hermitian symmetric spaces, 359 Deformation space of a surjective morphism 398–404 description of, 399–401 dimension of, 399 relative over the rationally connected quotient, 400 when target is not uniruled, 398–399, 401, 404 Degree of a Fano threefold 340, 341 Dehn twists 39 Deligne cohomology 455 Deligne’s integrality theorem 61, 65 Demailly, Jean-Pierre 359, 361, see BDPP determinant line bundle 481, 482, 489 de Rham complex 73

diffeomorphism 133 Differential system 386 Divisibility 60, 62 Divisorial contraction 342 du Val double planes 14 Elementary transform 394 Emerton 76 Emerton–Kisin correspondence equianalytic family 177 equisingular family 171, 178 equisingularity stratum 189 étale cohomology 72, 274 existence of curves 171 extrasymmetric matrices 27

76

fake projective planes 23 Faltings, Gerd 405 Faltings-Serre-Livné method 287, 292 Family of canonically polarized varieties 360, 375, 405–411, see also Shafarevich conjecture, Viehweg conjecture isotrivial, 405 of maximal variation, 405 over C∗ are isotrivial, 409 theorem relating variation and log. Kodaira dimension of the base, 406 variation of, 360, 405 family of curves 177 Family of rational curves 376–379, see also Space of rational curves on a variety kth-spanning dimension of, 389 dominating, 377 dominating of minimal degrees, 376–379 are generically unsplit 378 has few singular members 376, 379–383 generically unsplit, 378 maximal, 377 unsplit, 378 Fermat’s Last Theorem 282 fibrations of surfaces to curves 6 flag manifold 238 Flenner, Hubert 362, 403

Index Foliation 362, 363, 372, 373, see also Leaf of a foliation in positive characteristic, 362 special in the sense of Miyaoka, 372 Fountain-Mazur conjecture 280, 281, 285 Fourier-Mukai transform 83, 86, 94, 95, 98, 123–125 Frankel, Theodore 361 Frobenius 60, 75 Frobenius morphism 274, 276, 285, 297 Fubini, Guido 360, 379, 387, see also Cartan-Fubini type theorem Fubini-Study metric 212 G–bundle 238 Galois representation 271, 276, 278, 303 coming from geometry, 280 geometric, 280 Gauss-Manin connection 299 Gelfand problem 83, 87, 102 General complete intersection curve 362, 374, 402 General elephant 346 generalized hyperelliptic surfaces 18 generic coverings 42 generically nef 318, 319 Geometric Invariant Theory 212 GHS Theorem, the base of the MRC fibration is not uniruled 359, 365, 367 Gr¨ ossencharacter 271, 290, 295 Graber, Tom 359, 365, see GHS Theorem gradient vector field 138 Griffiths-Yukawa coupling 420 Grothendieck’s generalized Hodge conjecture 64 groupoid, fibered 472 GSV index of a 1-form 140 GSV index of a vector field 138 Hard Lefschetz Theorem 314 Harder-Narasimhan filtration ample terms in, 372, 373

501

of the tangent bundle, 363, 372, 374, see Partial rationally connected quotients Harris, Joe 359, 365, see GHS Theorem Hartshorne, Robin 361, 367, 370, 372 characterization of ampleness, 370, 372 Hartshorne-Frankel conjecture 361 Hecke curves 360, 393–395 are free and smooth, 395 are rational, 395 form a dominating family, 396 have minimal degree, 395–397 Hecke eigenform 278 Hermite-Einstein connection 477 hermitian semi-positive 313, 334 Higgs bundle 420, 424, 463, 483, 484, 486, 490, 492 Higgs bundle, stable 424 Higgs field, strictly maximal 420, 435 higher Chow group 452 Hilbert functor 178 Hilbert quotient 225 Hodge level 60 Hodge theory 451 holomorphically convex 333 holonomic DX –module 74 Homf (X, Y ) see Deformation space of a surjective morphism homogeneous manifold 234 homological index of a 1-form 141 homological index of a vector field 141 H¨ oring, Andreas 364 Hwang, Jun-Muk 361, 376, 385–388, 395–397 hyper-K¨ ahler structure 490, 492 Hyperbolicity problems 375 Hyperelliptic 341, 343, 346, 348 incidence variety 245 index of a collection of 1-forms 159 index of a collection of vector fields 159 Index-one-cover 410 integrable connection 73 integrality theorem 61 intermediate extension 77 intersection cohomology 463

502

Index

isolated special point 162 isotropy representation 214, 215, 221 Iwasawa–envelope 258 Iwasawa–Schubert variety 250 Iwasawa-Borel subgroup 249 Jacobi identity Jahnke, Priska

230 359

(k, l)-stable sheaf 394 K3 surface 285, 288 singular, 288, 290, 302 K¨ ahler cone 313, 315, 326 K¨ ahlerian form 211–213, 220 K¨ ahlerian manifold 211, 212, 214, 220 Keel, Se´ an 360, 363, 375, 405, see KMcK Killing form 240 Kisin 76 KMcK, rational curves on quasiprojective varieties 360, 375 Kobayashi-Hitchin correspondence 473, 494 Kodaira dimension of a quasi-projective variety see Logarithmic Kodaira dimension Kodaira-Spencer map 396 Koll´ ar, J´ anos 364, 405 Kov´ acs, S´ andor J. 405 Kummer construction 289, 294 Kummer variety 317, 327 -adic cohomology 60, 65 L-function 275 Lang-Manin conjecture 60 Langlands’ programme 281 Lazarsfeld, Robert 402 Negativity theorem, 402, 403 Leaf of a foliation 363 algebraicity, 362–364, 366, 367, 374 rational connectivity, 362–364, 367, 374 Lefschetz fibrations 40 Lefschetz trace formula 274, 282, 297 Length of a Fano manifold, 376 of a uniruled manifold, 388–393 Levi-Malcev factor 229 Lie’s flag theorem 238

link 133 lock (x), locus of chains of rational curves through a point 389 local Chern obstruction 163 local cohomology 78 local Euler obstruction of a 1-form 145 local Euler obstruction of a function 144 local system 74 Log minimal model of a surface 406, 407 Logarithmic abundance 407 Logarithmic Kodaira dimension 360, 363, 375, 405 of the base of a family, see Family of canonically polarized varieties, Shafarevich conjecture, Viehweg conjecture L ojasiewicz’s inequality 219 Lyubeznik 78 Mr

see Moduli space of vector bundles on a curve manifold with isolated singularities 133 mapping class group 38 matrix problem 83, 86, 98, 101–103, 119 maximal Albanese dimension 9 maximal compact subgroup 233 maximal torus 231 Maximally étale factorization 400 characterization, 400 stability under deformations, 400 universal property, 400 Maximally ample subbundle 371, 403 Maximally rationally chain connected fibration 365, see also Rationally connected quotient universal property, 365, 367 McKernan, James 360, 363, 375, 405, see KMcK McQuillan, Michael L. 362, 364, 366 measurable orbit 252 Mehta, Vikram B. 362, 402, 403, see also Metha-Ramathan theorem Mehta-Ramanathan theorem 362, 403 Mellin transform 279

Index Mirror Symmetry 299, 305 Miyanishi, Masayoshi 375, 405 conjecture on affine lines on quasi-projective varieties, 375, 405 Miyaoka, Yoichi 360, 362, 364, 371, 372, 398 criterion of uniruledness, 360, 362, 371–373, 398, 402, 403, 409 foliations in positive characteristic, see Foliation modular curve 277 modular form 271, 277, 283 moduli for coupled vortex equations 492 moduli of Higgs bundles 483 moduli of holomorphic vector bundles 473 moduli of principal bundles 476 Moduli space of vector bundles on a curve 360, 385, 388, 393–398 automorphism group, 397 is sometimes Fano, 394 rational curves of minimal degree, 396 stability of the tangent bundle, 397 tangent map for Hecke curves, 396 Torelli-type theorem, 397 variety of minimal rational tangents, 396 secant defect 397 moduli spaces 471 moduli spaces, existence theorems 471 Mok, Ngaiming 376, 385–388 moment map 240 momentum map 211–214, 216, 217, 220, 225 momentum norm 215, 217, 221 Monodromy group 437 Monsky-Washnitzer cohomology 297 Mordell conjecture 405 Mori, Shigefumi 359, 361, 363, 364, 368, 370 Bend-and-Break argument, see Bend-and-Break Reduction argument modulo p, 368 motive 466 motivic cohomology 59 movable cone 315 movable curve 315

503

MRCC fibration see Maximally rationally chain connected fibration multiplier ideal sheaf 314 Mumford, David 361, 363 conjecture relating uniruledness and Kodaira dimension, 361, 363 Mumford-Tate group 436 Nadel, Alan M. 388, 390 product theorem, 390 generalization 390, 397 Narasimhan, Mudumbai S. 394–396 Narasimhan, Raghavan 398 nef dimension 335 nef line bundle 313, 314, 321, 326 nef reduction 322, 336 newform 278, 302 node 173 non-degenerate singular point 162 non-degenerate special point 162 normal function 459 O’Neill tensor 373, 386 orbit duality 241 order of T-existence 192 Painlevé property 460 parabolic 239 Parshin, Alexei N. 405 Partial crepant resolution 339 Partial rationally connected quotients 373, 374 (in)dependence of polarization, 374 Partial resolution of singularities 380 patchworking method 190, 193 P˘ aun, Mihai 359, 361, see BDPP Peternell, Thomas 359, 361, see BDPP Petersson-Weil metric 475, 480, 481, 487 Petersson-Weil metric, curvature 476 Picard modular surface 462 Picard–Fuchs equation 459 Picard-Fuchs equation 299 Poincaré 74 potential 212, 217, 224 pseudo-effective cone 313, 315 pseudo-effective line bundle 313–315, 318, 319, 321, 324

504 Purity

Index 62

radial index of a 1-form 134, 136 radial index of a vector field 134, 135 radical 229 Radloff, Ivo 359 Ramanan, S. 394–396 Ramanathan, Annamalai 362, 402, 403, see also Mehta-Ramathan theorem rank of a real form 237 RatCurvesn (X) see Space of rational curves on a variety Rational chain connectivity 363, see also Maximally rationally chain connected fibration is not a birational invariant, 365 Rational connectivity 363, see also Rationally connected quotient criterion, 364 criterion for the leaf of a foliation, see Leaf of a foliation implies simple connectivity, 364 openness of, 364 Rational curve see Family of rational curves, Space of rational curves on a variety existence conjecture of Mumford, see Mumford existence if KX is not nef, 361 existence on Fano manifolds, 361 existence result of Miyaoka, see Miyaoka, criterion of uniruledness free, 378, 409 immersed, 379 of minimal degree is determined by a point and a tangent direction 385 is determined by two points 385 is hardly ever singular see Family of rational curves singular, 379–384 splitting type, 378 standard, 378 Rational homogeneous space 385 rational points 60, 62 Rational quotient see Rationally connected quotient rational quotient 325

rationally connected 312, 320, 323, 332, 336 Rationally connected quotient 363– 365, 367, 371, 374, 400 non-uniruledness, see GHS Theorem relative deformations over, see Deformation space of a surjective morphism subsheaf vertical with respect to, 371 real form 211, 213, 214, 220, 221, 233 real GSV index 140 real structure 233 reduced Euler characteristic 145 reductive 235 Reeb stability theorem 364 Refinement of Stein factorization 360, 400 Remmert–Van de Ven / Lazarsfeld problem 388 representation 230 restricted roots 237 Riemann singularity theorem 360, 397 Riemann–Hilbert correspondence 59, 75 Riemannian symmetric space 234 rigid cohomology 60, 65, 66 ring of big Witt vectors 59, 69 root spaces 231 scale 244 Schoen quintic 291, 299 Schubert cell 240 Schubert envelope 253 Schubert incidence variety 253 Schubert slice 250 Schubert variety 240 Semipositive vector bundle 362 semiquasihomogeneous hypersurface singularity 185 semisimple 229 semistable 212, 216 Severi variety 176 Shafarevich conjecture 419 Shafarevich, Igor R. 360, 405 Finiteness conjecture, 405 Hyperbolicity conjecture, 360, 405 Shimura curve 421, 445 Shimura variety 437, 460 simple variety 317, 327

Index Simpson correspondence 425, 463 Simpson correspondence, unitary local systems 425 Simpson correspondence, variation of Hodge structures 425 Simultaneous resolution 350 singular metric 313 singular point of a 1-form 133 singular point of a manifold 133 singular point of a vector field 133 slice 236 slice model 265 Slice Theorem 223, 226 slope 60, 65, 66 Small contraction 341, 342 Small set avoidance theorem 409, 411 Smoothing 342, 350 Space of rational curves on a variety 377, see also Family of rational curves relation with the Hom-scheme, 377 through a given point is proper if degree is minimal, 378 Spanning dimension see Family of rational curves special point 161 spherical embedding 248 spherical variety 239 Stability of the tangent bundle see Tangent bundle standard conjecture 466 Starr, Jason 359, 365, see GHS Theorem Stein factorization see Refinement of Stein factorization, Maximally étale factorization strongly rigid varieties 30 Sun, Xiaotao 395–397 surfaces isogenous to a product 18 Tangent bundle of the moduli space of vector bundles on a curve, see Moduli space of vector bundles on a curve of the variety of minimal rational tangents, see Variety of minimal rational tangents positivity, 361

505

stability and partial rationally connected quotients, see Partial rationally connected quotients Tangent map 383, see also Tangent morphism for Hecke curves in the moduli space of vector bundles on a curve, see Moduli space of vector bundles on a curve generic injectivity, 385 global, 384 Tangent morphism 376, 383–387, see also Tangent map birationality, 376, 385–387 existence, 376 finiteness, 376 Taniyama-Shimura-Weil conjecture 271, 283 Tate conjecture 304 Tate twist 280 Teichm¨ uller curve 422, 442, 444, 445 Teichm¨ uller curve, characterization 422 Teichm¨ uller curve, field of definition 422 Teichm¨ uller disc 442 Teichm¨ uller space 441 theta divisor 66 topological type 178 trace transfrom 245 transcendental lattice 288, 295 triangle curves 31 T-smooth 172 Tsunoda, Shuichiro 375 Uniqueness of contact structures see Contact manifold uniruled 312, 318, 319 Uniruledness criterion of Miyaoka see Miyaoka, Yoichi unit OX,F –module 76 universal cover 333, 334 universal domain 256 Vanishing result in characteristic p 363, 369, 370 Vanishing threshold 369, 370 Variation of a family see Family of canonically polarized varieties

506

Index

Variety of minimal degree 347, 348 Variety of minimal rational tangents 360, 376, 379, 383–393, 397, 399 determines the variety, see CartanFubini type theorem example applications, 388 has smooth normalization, 385 importance, 387 of the moduli space of vector bundles on a curve, see Moduli space of vector bundles on a curve secant defect, 388 for the Moduli space of vector bundles on a curve see Moduli space of vector bundles on a curve secant variety and spanning dimension, 390 for varieties covered by lines 390 short explanation, 360 tangent space for varieties covered by lines, 385 total, 384 vector field 133 versality of deformation 200 Vertical subsheaf with respect to the rationally connected quotient see Rationally connected quotient

Viehweg, Eckart 360, 405–407, 411, see also VZ Theorem conjecture relating variation and log Kodaira dimension, 360, 405, 406 generalization of 406 proof in dimension two see Family of canonically polarized varieties VMRT 387, 388, 391, 393, 397, see variety of minimal rational tangents vortex equation 492, 494 VZ Theorem, existence of pluri-log differentials on the base of a family 405, 407–408, 411 Weierstraß curve 86, 87, 98, 100–102, 107, 112, 114–117, 123, 124, 126 Weil conjectures 273 Witt complex 70 Witt vector cohomology 60, 65–67 Yang-Mills connection

483, 485, 486

Zhang, De-Qi 375 Zuo, Kang 405, 407, 411, see also VZ Theorem

Global Aspects of Complex Geometry

Global Aspects of Complex Geometry

Global Aspects of Complex Geometry