Vector Bundles and Complex Geometry

CONTEMPORARY MATHEMATICS 522 Vector Bundles and Complex Geometry Conference on Vector Bundles in Honor of S. Ramanan on...

Author: Oscar Garcia-Prada | Peter E. Newstead | Luis Álvarez-Cónsul | (eds.) 2010

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CONTEMPORARY MATHEMATICS 522

Vector Bundles and Complex Geometry Conference on Vector Bundles in Honor of S. Ramanan on the Occasion of his 70th Birthday June 16–20, 2008 Miraflores de la Sierra, Madrid, Spain

Oscar García-Prada Peter E. Newstead Luis Álvarez-Cónsul Indranil Biswas Steven B. Bradlow Tomás L. Gómez Editors

American Mathematical Society

Vector Bundles and Complex Geometry

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CONTEMPORARY MATHEMATICS 522

Vector Bundles and Complex Geometry Conference on Vector Bundles in Honor of S. Ramanan on the Occasion of his 70th Birthday June 16–20, 2008 Miraflores de la Sierra, Madrid, Spain

Oscar García-Prada Peter E. Newstead Luis Álvarez-Cónsul Indranil Biswas Steven B. Bradlow Tomás L. Gómez Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Dennis DeTurck, managing editor George Andrews

Abel Klein

Martin J. Strauss

2000 Mathematics Subject Classification. Primary 14H60, 14D20, 20G15, 14D07, 14D22, 58J20, 14C30, 14J60.

Library of Congress Cataloging-in-Publication Data Vector bundles and complex geometry : conference on vector bundles in honor of S. Ramanan on the occasion of his 70th birthday, June 16–20, 2008, Miraflores de la Sierra, Madrid, Spain / Oscar Garc´ıa-Prada . . . [et al], editors. p. cm. — (Contemporary mathematics ; v. 522) Includes bibliographical references and index. ISBN 978-0-8218-4750-3 (alk. paper) 1. Vector bundles—Congresses. 2. Geometry, Algebraic—Congresses. I. Ramanan, S. II. Garc´ıa-Prada, O. (Oscar), 1960– QA612.63.V418 514.224—dc22

2010 2010011114

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

15 14 13 12 11 10

To S. Ramanan on the occasion of his 70th birthday


Contents Preface

ix

The Work of S. Ramanan M. S. Narasimhan

1

Parabolic Bundles on Algebraic Surfaces II - Irreducibility of the Moduli Space V. Balaji and A. Dey 7 Finite Subgroups of PGL2 (K) Arnaud Beauville

23

Picard Groups of Moduli Spaces of Torsionfree Sheaves on Curves Usha N. Bhosle

31

On the Moduli of Orthogonal Bundles on a Nodal Hyperelliptic Curve Usha N. Bhosle

43

Hilbert Schemes of Fat r-Planes and the Triviality of Chow Groups of Complete Intersections Andr´ e Hirschowitz and Jaya NN Iyer

53

Vector Bundles and the Icosahedron Nigel Hitchin

71

Cohomology of the Toroidal Compactification of A3 Klaus Hulek and Orsola Tommasi

89

Quasi-Complete Homogeneous Contact Manifold Associated to a Cubic Form Jun-Muk Hwang and Laurent Manivel 105 Maximal Weights in K¨ ahler Geometry: Flag Manifolds and Tits Distance (with an Appendix by A. H. W. Schmitt) I. Mundet i Riera 113 Orthogonal Bundles Over Curves in Characteristic Two Christian Pauly

131

The Atiyah-Singer Index Theorem M. S. Raghunathan

141

Spin(7) Instantons and the Hodge Conjecture for Certain Abelian Four-folds: A Modest Proposal T. R. Ramadas 155

vii

viii

CONTENTS

Remarks on Parabolic Structures C. S. Seshadri

171

Iterated Destabilizing Modifications for Vector Bundles with Connection Carlos Simpson

183

Preface This volume is dedicated to S. Ramanan on the occasion of his 70th birthday. Ramanan’s influence on mathematics is the common thread running through all the articles in this volume. In some cases it is seen directly in the subject matter of the research, while in other cases it is through his association—as colleague or mentor—with the author. Ramanan has made important contributions to Differential and Algebraic Geometry. He has been a leading expert on vector bundles and moduli spaces for over 40 years. For many years a Distinguished Professor at the Tata Institute of Fundamental Research in Mumbai, he is now an honorary professor at the Institute of Mathematical Sciences in Chennai and Adjunct Professor at the Chennai Mathematics Institute. He is a Fellow of all three Indian National Science Academies and is a recipient of the Shanti Swarup Bhatnagar Prize, the Srinivasa Ramanujan Birth centenary award and the Third World Academy of Sciences Prize for Mathematics, among other things. Some of his many profound contributions are described in the article in this volume by M.S. Narasimhan. Most of the articles come from a workshop on vector bundles, held in the Miraflores conference center outside Madrid, Spain in June 2008. In addition to being a felicitation for Ramanan, the workshop served several purposes: • It was the 2008 annual workshop of the Vector Bundles on Algebraic Curves (VBAC) group. Founded in 1994 by Peter Newstead, the VBAC group has been an influential champion of vector bundles and their evolving place in mathematics. Ramanan is not only an early member of the renowned ‘Tata school’ of vector bundles, but has been a central member of VBAC from its first days. • The workshop was part of a semester-long program on Moduli Spaces, organized by CSIC (Madrid), Universidad del Pa´ıs Vasco, Universidad de Salamanca and Universitat de Barcelona. Other activities in the program included an International School on Geometry and Physics: moduli spaces in geometry, topology and physics (Cantabria, 25-29 February 2008), a workshop on Moduli spaces of vector bundles: algebro-geometric aspects (Barcelona, 12-14 March 2008), and a workshop on Bundles, gerbes and derived categories in string theory (Salamanca, 14-16 May 2008) Finally, no description of the Miraflores workshop would be complete without a mention of the evening of classical Indian song, with performances by workshop participants Kavita Ramanan, Tom´ as G´ omez, V. Balaji, C.S. Seshadri, and Ramanan himself. With its minimum of fuss but abundance of beauty and skill, the memorable evening was a perfect metaphor for Ramanan’s contribution to mathematics. ix


Contemporary Mathematics Volume 522, 2010

The Work of S. Ramanan M.S. Narasimhan Abstract. This talk gives an overview of some of the significant work of S. Ramanan in the areas of moduli problems, linear systems on flag varieties and abelian varieties, and differential geometry.

1. Introduction It is with great pleasure that I write this account of the work of S.Ramanan. I have been fortunate enough to have collaborated with him intensely over a long period. I have profited much from his mathematical insights. His mathematics is characterised by depth and breadth and covers several areas: algebra, differential geometry and algebraic geometry. In algebraic geometry he is a major figure, and his main interests have been: • moduli of vector bundles on curves, • homogeneous vector bundles on flag varieties, • abelian varieties, • geometric invariant theory, • linear systems on flag varieties in positive characteristics, • Green’s conjecture on syzygies of canonical curves, and • Higgs bundles. His contributions to the study of vector bundles on curves have been particularly extensive and profound. I will describe below some of Ramanan’s work to give an idea of the significance and the impact of his contributions. 2. Universal connections The basic theorem on the existence of universal connections was proved by Ramanan and myself in [1] and [2]. (This was our first collaboration.) Theorem 2.1. Let G be a Lie group with a finite number of connected components, and n a positive integer. Then there exists a principal G-bundle En and a connection ωn on En , such that any connection on a principal G-bundle π : P → M with dim M ≤ n is the inverse image of ωn by a G-morphism of P into En . 2010 Mathematics Subject Classification. 14D20, 14D22, 14K25, 14L24, 53C05, 20G15. c Mathematical 0000 (copyright Society holder) c 2010 American

1

2

M.S. NARASIMHAN

When G = U (k) (the unitary group), En may be taken to be the Stiefel manifold V (N, k) of unitary k-frames in Cn and ωn to be the canonical connection A∗ dA, with A = the N × k matrix representing a point of V (N, k). This theorem has been used extensively by mathematicians and physicists: in the work of Chern-Simons, in the definition of the Cheeger-Simons differential character, in Quillen’s work on super-connections and more recently in the context of stochastic differential equations. 3. Homogeneous vector bundles The work [3] constituted Ramanan’s thesis. Let G be a semi-simple algebraic group over C. Let P be a parabolic subgroup of G. (Let us assume that P does not contain a simple component of G). Let ρ be a finite-dimensional irreducible representation of G and Vρ the associated holomorphic vector bundle over G/P . Theorem 3.1. (1) If ρ1 and ρ2 are irreducible representations of P then Vρ1 is isomorphic to Vρ2 (as holomorphic bundles) if and only if ρ1 and ρ2 are equivalent representations. (2) Vρ is a simple bundle; in particular, Vρ is indecomposable. (3) In fact, for any polarisation on G/P, Vρ is a μ-stable vector bundle. The equivalence problem had been raised by Ise (1960). The statement (3) was proved first in the case of an irreducible symmetric space. Umemura observed that (3) is valid in general. Ramanan also showed that there are homogeneous vector bundles Vρ on P2 with 2 H (P2 , EndVρ ) = 0 and H 1 (P2 , EndVρ ) = 0. (These dimensions can be calculated). 4. Moduli of vector bundles on curves Ramanan has done extensive and celebrated work in this area. I shall describe below some of his contributions in this field. Let X be a smooth irreducible curve over C, of genus g ≥ 2; let ξ denote a line bundle on X of degree d. Let U (r, d) (respectively, S(r, ξ)) denote the moduli space of (semi-stable) vector bundles on X of rank r and degree d (respectively, with determinant isomorphic to ξ). We denote S(r, ξ) also by S(r, d). The following was proved in [4]: Theorem 4.1. The set of non-singular points of U (r, d) is precisely the set of stable points in U (r, d) except when g = 2, r = 2 and d even. Let S(2, 0) be the space of semi-stable bundles of rank 2 and trivial determinant. In [4] and [5] we proved: Theorem 4.2. Let g = 2. (1) S(2, 0) is isomorphic to P3 (C). (2) If ξ is a line bundle of degree 1, S(2, ξ) is isomorphic to a smooth intersection of two smooth quadrics in P5 . Let J g−1 be the space of line bundles of degree (g −1) on X and θ the canonical theta divisor on J g−1 . For any semi-stable vector bundle E of rank 2 and trivial determinant let DE denote the subset of J g−1 defined by DE = {ξ J g−1 , H 0 (X, ξ⊗ E) = 0}. Then DE is a divisor linearly equivalent to 2θ and defines a morphism f of S := S(2, 0) into the projective space |2θ|. In [4] we proved:

THE WORK OF S. RAMANAN

3

Theorem 4.3. For g = 2, f is an isomorphism. Turning to the case of genus 3, we have ([6]): Theorem 4.4. Let g = 3, and X non-hyperelliptic. Then S is isomorphic to a quartic hypersurface in P7 . (In fact the morphism f defined above gives the imbedding). This quartic had been considered earlier by Coble. In fact the above theorem helps to solve a problem posed by Coble. Note that non-stable points in S correspond to vector bundles of the form L ⊕ L−1 , where L is a line bundle of degree 0. That is, the set of non-stable points is the Kummer variety associated to the Jacobian of X. In the case g = 2 the Kummer surface is imbedded in P3 and was originally investigated by Kummer from the point of view of quadratic complexes of lines in P3 . (A quadratic complex Q is the intersection of two quadrics in P5 , where one of the quadrics is the grassmannian of lines in P3 ). The incidence correspondence between lines and points in P3 induces a diagram: H tt @@@ @@conic @@ @

P1 −bundlettt

t tt ytt Q = S(2, ξ)

bundle

P3

The P1 bundle over P3 − K does not come from a vector bundle. This implies the non-existence of a Poincaré family over the open subset of stable points in S = S(2, 0), (g = 2, r = 2, d = 0). To prove that the P1 - bundle does not come from a vector bundle one interprets the incidence correspondence in terms of vector bundles. In general one gets the Hecke correspondence, between moduli spaces. For instance when r = 2 one gets : x

P1 −bundlexxx

xx |xx S(2, 1)

HF FF Fconic FF bundle FF " S(2, 0)

The Hecke correspondence is the key to the study of the geometry of moduli spaces (and has since been used extensively by several mathematicians). Using this correspondence one proves, for example, that the number of moduli of S(r, ξ) is the same as that of the curve, when (r, d) = 1. More precisely ([7]): Theorem 4.5. (1) The group of automorphisms of S(r, ξ)((r, d) = 1) is finite and H i (S(r, ξ), T ) = 0 for i = 1, where T is the tangent sheaf. (2) dim H 1 (S(r, ξ), T ) = 3g − 3. Theorem 4.6. The canonically polarized intermediary Jacobian of S(r, ξ) ((r, d) = 1) corresponding to the third cohomology group is naturally isomorphic to the canonically polarized Jacobian of X. The Hecke correspondence was also used ([8]) to construct an explicit desingularisation of the moduli space of rank 2 bundles with trivial determinant. In the paper [9] Ramanan proved a number of important results, among them:

4

M.S. NARASIMHAN

(1) The anticanonical class of S(r, ξ) in the case (r, d) = 1 is twice the ample generator of the Picard group. (2) If r and d are not coprime, there does not exist a Poincaré family on any non-empty open subset of U (r, ε), for any rank r and genus. (3) The structure of the cohomology ring of S(2, 1), when g = 3. The paper [10] studies the Hitchin fibration with application to the generalised theta divisor. In a paper ([11]) with Desale, Ramanan proved: Theorem 4.7. Let X be an hyperelliptic curve of genus g ≥ 2 and λ0 , ......λ2g+1 be the branch points of X in P1 . Then S(2, 1) is isomorphic to the space of all (g − 2) dimensional linear subspaces of P2g+1 which are in the intersection of the two quadrics 2g+1 2 2 Σ2g+1 i=0 xi = Σi=0 λi xi = 0

Ramanan interpreted linear subspaces of other dimensions contained in these quadrics as a kind of spin bundles ([12]). 5. Ample divisors on abelian varieties Theorem 5.1. [13] Let A be an abelian surface not containing elliptic curves and L an ample bundle of type (δ1 , δ2 ), δ1 |δ2 . Then L is very ample if δ1 = 1, δ2 ≥ 5; δ1 = 2, δ2 ≥ 4 or δ1 ≥ 3 In particular L of type (1,5) embeds A in P4 (as a subvariety of codimension 2) and the Serre construction yields the Horrocks-Mumford bundle on P4 . 6. Geometric Invariant Theory: Semi-stability of bundles obtained by extension of structure group Ramanan and Ramanathan made an incisive study ([14]) of the instability flags arising in Geometric Invariant Theroy. They used this to give an algebraic proof of the following result: ¯ Let G and H be reductive groups and ρ : G → H Let Char k = 0, k = k. a homomorphism which maps the connected component of the centre of G into that of H. If E is a semi-stable G-bundle, then the extended H-bundle E(H) is semi-stable. If E is quasi-stable, so is the extended bundle. If Char k > 0, their method enables one to analyse the semistability of E(H). Coiai-Holla used this method to prove the boundedness of semi-stable bundles in positive characteristic. 7. Projective normality of flag varieties with Schubert varieties Ramanan and Ramanathan proved ([15]): Theorem 7.1. Let k be an algebraically closed field of arbitrary characteristic and G a reductive group over k. Let Q be a parabolic subgroup of G and L an ample line bundle over G/Q. Then the complete linear system of L embeds G/Q as a projective normal variety. In particular, Schubert varieties are normal.

THE WORK OF S. RAMANAN

5

8. Higgs Bundles More recently Ramanan has been interested in Higgs bundles [16]. With O. Garc´ıa-Prada he studied Higgs bundles twisted by a line bundle (of finite order) and determined the corresponding Tannaka group in terms of the pro-reductive completion of the fundamental group and the character defined by the line bundle [17]. They also study involutions on the Higgs moduli space on a curve [16] I have not gone into many other contributions of Ramanan, for example, his explanation of the mysterious Capelli identity in terms of an element of the universal enveloping algebra of the linear group or his beautiful book,“Global Analysis” ([18]) which gives an insightful perspective on basic differential analysis and geometry. References [1] M.S. Narasimhan and S. Ramanan, Existence of universal connections, Amer. J. Math. 83 (1961) 563–572. [2] M.S. Narasimhan and S. Ramanan, Existence of universal connections, II, Amer. J. Math. 85 (1963) 223–231. [3] S. Ramanan, Holomorphic vector bundles on homogeneous spaces, Topology 5 (1966) 159– 177. [4] M.S. Narasimhan and S. Ramanan, Moduli of vector bundles on a compact Riemann surface, Ann. of Math. (2) 89 (1969) 14–51. [5] M.S. Narasimhan and S. Ramanan, Vector bundles on curves, Algebraic Geometry (Internat. Colloq.), Tata Inst. Fund. Res., Bombay, (1968) 335–346 Oxford Univ. Press, London. [6] M.S. Narasimhan and S. Ramanan, 2θ-linear systems on abelian varieties, Vector bundles on algebraic varieties, pp 415–427, Bombay, 1984. [7] M.S. Narasimhan and S. Ramanan, Deformations of the moduli space of vector bundles over an algebraic curve, Ann. Math. (2) 101 (1975) 391–417. [8] M.S. Narasimhan and S. Ramanan, Geometry of Hecke cycles. I in C. P. Ramanujam – a tribute, pp. 291–345, Tata Inst. Fund. Res. Studies in Math., 8, Springer, Berlin-New York, 1978. [9] S. Ramanan, The moduli spaces of vector bundles over an algebraic curve, Math. Ann. 200 (1973) 69–84. [10] A. Beauville, M.S. Narasimhan and S. Ramanan, Spectral curves and the generalised theta divisor, J. Reine Angew. Math. 398 (1989) 169–179. [11] U.V. Desale and S. Ramanan, Classification of vector bundles of rank 2 on hyperelliptic curves, Invent. Math. 38 (1976/77) 161–185. [12] S. Ramanan, Orthogonal and spin bundles over hyperelliptic curves, Proc. Indian Acad. Sci. Math. Sci. 90 (1981) 151–166. [13] S. Ramanan, Ample divisors on abelian surfaces, Proc. London Math. Soc. (3) 51 (1985) 231–245. [14] S. Ramanan and A. Ramanathan, Some remarks on the instability flag, Tohoku Math. J. (2) 36 (1984) 269–291. [15] S. Ramanan and A. Ramanathan, Projective normality of flag varieties and Schubert varieties, Invent. Math. 79 (1985) 217–224. [16] S. Ramanan, Some aspects of the theory of Higgs pairs, “The The Many Facets of Geometry: A Tribute to Nigel Hitchin”, Editors: O.Garc´ıa-Prada, J-P. Bourguignon and S. Salamon, Oxford University Press, 2010. [17] O. Garc´ıa-Prada and S. Ramanan, Twisted Higgs bundles and the fundamental group of compact K¨ ahler manifolds, Mathematical Research Letters, 7 (2000), 1–18. [18] S. Ramanan, Global calculus. Graduate Studies in Mathematics, 65. American Mathematical Society, Providence, RI, 2005. TIFR, Centre for Applicable Mathematics, Bangalore 560065; and Department of Mathematics, IISc, Bangalore 560012, India



Parabolic Bundles on Algebraic Surfaces II - Irreducibility of the Moduli Space V. Balaji and A. Dey To Professor Ramanan on his 70th birthday

Abstract. In this paper we prove irreducibility of the moduli space of parabolic rank 2 bundles over an algebraic surface for c2 0 and with an irreducible parabolic divisor D of X. This gives parabolic analogues of theorems of O’Grady and Gieseker-Li.

1. Introduction Let X be a smooth projective surface over the ground field C of complex numbers and let D be a smooth irreducible divisor. Let H be a very ample line bundle on X which we fix throughout. We study bundles with c1 = 0 in this paper. α We denote by Mk,d the moduli space of parabolic H–stable parabolic bundles of rank 2 with parabolic structure on D together with rational weights α := (α1 , α2 ) (see (2.4) and (2.5) for the definition of the invariant d) and where k stands for the second Chern class c2 of a vector bundle. α of the moduli space In [3], the Donaldson-Uhlenbeck compactification Mk,d α α Mk,d was constructed as a projective variety by realizing it as the closure of Mk,d α in a certain projective scheme Mk,d endowed with the reduced scheme structure; it was also shown to be non-empty for large k. There are also obvious bounds on the invariant d for quasi-parabolic structures to exist. Let MH (2, c1 , c2 ) (resp. MH (2, c1 , c2 )s ) denote the moduli space of H-semistable (resp. stable) torsion free sheaves of rank 2 whose Chern classes are c1 and c2 respectively. Since the topological type of the bundles is fixed for the problem as also is the ample polarization H, we will have the following convenient notations: α α Mαs := Mk,d ; Mα := Mk,d

and Ms := MH (2, 0, c2 )s . 2010 Mathematics Subject Classification. 14D20, 14D23. Key words and phrases. vector bundle, parabolic vector bundle, moduli space, moduli stack. Research of the first author was partly supported by the J.C. Bose Research grant. c 2010 American c Mathematical 0000 (copyright Society holder)

1 7

8 2

V. BALAJI AND A. DEY

We carry the weight tuple α as a part of the notation since this parameter will be varied in the arguments and the moduli spaces will be compared for differing weights. We say a moduli space as above is asymptotically irreducible if it is irreducible for c2 0, i.e the second Chern class of the bundle underlying the parabolic bundle is large. In particular we do not quantify c2 when we address the question of asymptotic irreducibility. In this paper we prove asymptotic irreducibility of the moduli space Mα when obvious bounds are imposed on d for the existence of quasi-parabolic structures. These moduli spaces for rank 2 have been studied from a differential geometric standpoint in [12] where k = c2 stands for the “instanton number”. Our theorem generalizes the theorem of Gieseker-Li and O’Grady ([6] and [16]) to the parabolic case. The parabolic case has been of independent interest; for example, in [14] Maruyama has shown links between the parabolic moduli spaces for special parabolic weights and the moduli space of instantons. Maruyama uses these links to prove irreducibility of some of these spaces. The assumptions on the parabolic divisor, rank and full flag quasi-parabolic structure can be relaxed; one could take the parabolic divisor to be a divisor with simple normal crossings and the bundles to be of arbitrary rank and any quasiparabolic type. We have made the special choices to make the paper more readable. The choice of rational weights is the natural one and real weights are really an artifice and do not occur in the classical setting. In any case this is not a serious issue as far as the question of irreducibility of the moduli space is concerned since the “yoga of parabolic weights” allows us to deduce geometric statements for moduli spaces with real weights from those with nearby rational weights. The assumption on large second Chern class is what makes the statement an asymptotic one; the result is shown only for large c2 . Acknowledgements: We wish to thank S. Bandhopadhyay for many helpful discussions while this paper was getting prepared. The second author wishes to thank the Institute of Mathematical Sciences, Chennai and the Chennai Mathematical Institute for their hospitality while this work was being done. Finally we wish to express our grateful thanks to the diligent referee for correcting the innumerable errors in the earlier versions and helping us improve the exposition.

2. Preliminaries Our basic tool is the Seshadri-Biswas correspondence between the category of parabolic bundles on X and the category of Γ–bundles on a suitable Kawamata cover. This strategy has been employed in several papers. Most of the material written in this section is taken from §2 of [3] and the reader will find details of the Seshadri-Biswas correspondence in this reference. However in this note we are only interested in the rank 2 case, and we will give definitions in the rank 2 case alone and lay stress on those points which are relevant to our purpose. 2.1. The category of bundles with parabolic structures. Let X be a smooth projective surface over the ground field C and let D be an irreducible smooth divisor in X. Let H be a very ample line bundle on X.

PARABOLIC BUNDLES

9 3

Definition 2.2. Let E be a rank 2 torsion-free sheaf on X. A parabolic structure (with respect to D) on E is a filtration (quasi–parabolic structure) (2.1)

E∗ : E = F 1 (E) ⊃ F 2 (E) ⊃ F 3 (E) = E ⊗ OX (−D)

together with a system of weights 0 ≤ α1 < α2 < 1 where αi is the weight associated with F i (E). (See [12, Section 8] where the weights are given in [− 12 , 12 ) following the balanced convention.) We will use the notation E∗ to denote a parabolic sheaf and by E (without the subscript “∗”) when it is without its parabolic luggage. The notation E∗ therefore carries the data of the weight tuple α as well. A parabolic sheaf E∗ is called a parabolic bundle if the underlying sheaf E is a vector bundle. 2.3. Some assumptions. The class of parabolic vector bundles that are dealt with in the present work satisfy certain constraints which will be explained now. (2.2)

All parabolic weights are rational numbers.

(2.3)

F 1 (E)/F 2 (E) is torsion-free as a sheaf on D.

We need to impose these in order to have the Seshadri–Biswas correspondence (cf. [3, Remark 2.3] for details). Henceforth, all parabolic vector bundles will be assumed to have the constraints (2.2) and (2.3). Also note that the filtration (2.1) is equivalent to a filtration on E |D given by (2.4)

1 2 3 (E) ⊃ FD (E) ⊃ FD (E) = 0. E |D = FD

To see this, simply define

E |D . F (E) = ker E → i FD (E) i

α In the notation Mk,d in the introduction, the numerical invariant d is given by

(2.5)

2 d = c1 (FD (E)).D.

The slope of a rank 2 parabolic sheaf E∗ is defined as [c1 (E) + (α1 + α2 )D] · H . 2 Let PVect(X, D) denote the category whose objects are rank 2 parabolic vector bundles over X with parabolic structure over the divisor D satisfying (2.2) and (2.3), and whose morphisms are homomorphisms of parabolic vector bundles (see [3] for more detail). For an integer N ≥ 2, let PVect(X, D, N ) ⊆ PVect(X, D) denote the subcategory consisting of all parabolic vector bundles all of whose parabolic weights are multiples of 1/N . Let E∗ be a rank 2 parabolic bundle on X with parabolic weight (α1 , α2 ). Let L be a line subbundle of E, the underlying bundle of the parabolic bundle E∗ . The parabolic weights on E∗ induces a parabolic weight on L denoted by αL ; αL equals (2.6)

μα (E∗ ) =

10 4


2 α2 if L ⊂ FD (E), and is α1 otherwise. Denote this parabolic line bundle with this induced structure by L∗ . The slope of this parabolic line bundle L∗ is given by

(2.7)

μαL (L∗ ) = (c1 (L) + αL D) · H.

It is not hard to check that for the purposes of stability it suffices to worry about parabolic line subbundles L∗ of a rank 2 parabolic bundle E∗ which are obtained from a line subbundle L of E with a weight αL defined as above. The parabolic bundle E∗ is α-stable (resp. α-semi–stable) if (2.8)

μαL (L∗ ) < μα (E∗ ) (resp. ≤)

for all parabolic line subbundles L∗ of E∗ . 2.4. The Kawamata covering lemma. Let D ⊂ X be an irreducible divisor. Take any E∗ ∈ PVect(X, D) such that all the parabolic weights of E∗ are multiples of 1/N , i.e. E∗ ∈ PVect(X, D, N ). The “covering lemma” of Y. Kawamata ([11, Theorem 1.1.1], [10, Theorem 17]) says that there is a connected smooth projective variety Y over C and a Galois covering (2.9)

p :

Y

−→ X

∗

˜ := (p D)red is a normal crossing divisor of Y and such that the reduced divisor D ˜ for some positive integer k. Let Γ furthermore the pull-back p∗ D equals kN D, denote the Galois group for the covering map p (2.9). 2.5. The category of Γ–bundles. Let Γ ⊆ Aut(Y ) be a finite subgroup of ˜ the group of automorphisms of a connected smooth projective variety Y /C and H be a fixed polarization on Y . A Γ–vector bundle V on Y is a vector bundle V together with a collection of isomorphisms of vector bundles g¯ : V

−→ (g −1 )∗ V

¯ for any g, h ∈ Γ (see indexed by g ∈ Γ and satisfying the condition that gh = g¯ ◦ h §2, [3] for more detail). A Γ–homomorphism between two Γ–vector bundles is a homomorphism between the two underlying vector bundles which commutes with the Γ–actions. Let VectΓ (Y ) denote the category of Γ–vector bundles on Y with the morphisms being Γ–homomorphisms. Having fixed the parabolic divisor and the Kawamata cover together with the ramification indices, one has the concept of local type of a Γ–bundle which is described in [3, 2.4.1] (see [17] for the terminology). This is needed in order to set up the correspondence between Γ–bundles and parabolic bundles with specified parabolic datum on X. Let VectD Γ (Y, N ) denote the subcategory of VectΓ (Y ) consisting of all rank 2 Γ–vector bundles V over Y of fixed local type (see [3, 2.4.1] for details). A Γ–vector bundle V is called Γ–stable (resp. Γ–semistable) iff for all Γ– invariant line subbundles L ⊂ V the following holds ˜ ˜ < (resp. ≤) c1 (V ) · H . (2.10) c1 (L) · H 2 Note that the above definition of Γ–stability is strictly weaker than the usual definition of stability; in particular the notion of Γ– stability does not imply the stability

11 5

PARABOLIC BUNDLES

of the underlying Γ–bundle. In contrast, the notion of Γ–semistability is equivalent to the usual notion of semistability of the underlying Γ–bundle because of the uniqueness of the Harder-Narasimhan filtration. 2.6. Parabolic bundles and Γ–bundles. In [4] a categorical correspondence between the objects of PVect(X, D, N ) and the objects of VectD Γ (Y, N ) has been constructed, induced by the “invariant direct image” functor pΓ∗ . The details of this identification is also given in [2, Section 2]. ˜ denote the pullback p∗ (H). Then the above correspondence between Let H parabolic bundles on X and Γ–bundles on Y identifies the Γ–semistable (resp. Γ– stable) objects with parabolic semistable (resp. parabolic stable) objects as well. The invariant direct image functor pΓ∗ giving this equivalence of categories is moreover a “tensor functor” which sends the usual dual of a Γ–vector bundle to the “parabolic dual” of the corresponding parabolic vector bundle . 2.7. Γ–derived functors Let C be a C-linear abelian category with enough injectives. Let Γ be a finite group. Let C Γ be the category whose objects are pairs of the form (A, ρ : Γ → AutC (A)) where A ∈ C. A morphism between pairs (A, ρ : Γ → AutC (A)), (B, ρ : Γ → AutC (B)) is defined as a Γ–equivariant morphism in C, i.e. the diagram (2.11)

A

ρ(γ)

f

B

/A f

ρ (γ)

/B

is required to commute for all γ ∈ Γ. Since the ground field is assumed to be of characteristic 0, for any object (A, ρ) ∈ C Γ , we have a subobject AΓρ ⊂ A defined as follows. Given γ ∈ Γ and A ∈ C, we can define the γ-invariant subobject Aγ of A to be the kernel of the composite map: Δ

A −→ A ⊕ A

id⊕(−ρ(γ))

−→

A

where Δ is the diagonal morphism. We define the Γ-invariant subobject AΓρ of A to be the intersection of the Aγ ’s in A, γ ∈ Γ, i.e. (2.12) AΓρ := Ker((id ⊕ (−ρ(γ))) ◦ Δ). γ∈Γ

Note that the induced action of Γ on AΓρ is trivial. Therefore we will just write AΓ instead of AΓρ . Let F be a covariant left exact functor from C to an abelian category B. Any k-linear additive functor F : C → D extends uniquely to a functor F˜ : C Γ → D Γ since any action of Γ on an object A extends to an action of Γ on F (A). Let F Γ be the invariant functor which sends A to F (A)Γ . It is a subfunctor of F . We have the following useful observation. Lemma 2.8. F Γ is a direct summand of F . Consequently the right derived functors Ri F Γ are direct summands of Ri F . Proof. The fact that F Γ is a direct summand of F follows immediately from the assumption on the characteristic of the ground field.

12 6


Now we return to the case in which we are interested. Let Y be a Kawamata cover of X with a finite group Γ acting on Y such that Y /Γ = X. Note that by our previous notations if C denotes the category Vect(Y ) of vector bundles on Y then C Γ is the category VectΓ (Y ) of Γ-vector bundles on Y . The global section functor Hom(Y, −) gives rise to a left exact functor from C Γ to category of k-linear spaces. We denote the i-th right derived functor of this by Exti (Y, −). Let ExtiΓ (Y, −) be the right derived functor of the global invariant section functor HomΓ (Y, −). By Lemma 2.8 we have the following proposition. Proposition 2.9. ExtiΓ (Y, −) is a direct summand of Exti (Y, −). Hence extiΓ (Y, F ) ≤ exti (Y, F ) for all F ∈ C Γ , where “ext” denotes the dimension of the vector space “Ext”. Let us consider the category C• of filtered OY -modules whose objects are denoted by F• , i.e. sheaves F with a filtration of subsheaves F• : 0 = F0 ⊂ F1 · · · ⊂ Fn = F.

(2.13) C•Γ,−

be the category whose objects are given by Γ-filtered sheaves of OY modLet ules (as in (2.13)). For any two objects F• , G• in C•Γ,− , morphisms in C•Γ,− are defined as: HomΓ,− (F• , G• ) = {φ : F → G : φ(Fi ) ⊂ Gi for all 0 ≤ i ≤ n}.

C•Γ,+

Let be the category whose objects are the same as in C•Γ,− , and morphisms between two objects F• , G• are defined as HomΓ,+ (F• , G• ) = HomC (F, G)/HomΓ,− (F• , G• ).

Both these categories C•Γ,± are abelian categories with enough injectives and HomΓ,± (F• , −) are both left exact covariant functors. Let ExtiΓ,± (F• , −) be the right derived functors of HomΓ,± (F• , −). We have a long exact sequence (cf. [9, page 49]) (2.14)

···

/ ExtiΓ,− (F• , G• )

/ Exti (F• , G• ) Γ

/ ExtiΓ,+ (F• , G• )

/ ···

ss 3. Rα is irreducible for small α ss . We briefly recall the construction of semistable 3.1. A description of Rα sheaves over X. For details see ([9, Chapter 4]). Let H = OX (−m)p for some m and p. Let

(3.1)

Q := Quot(H, P )

be the Quot scheme which parametrizes quotients of H with fixed Hilbert polynomial P given by (3.2)

P (n) := n2 H 2 + n(c1 · H − KX · H) +

c21 − c1 · KX − c2 + 2χ(OX ), 2

where KX is the canonical line bundle and the ci are the Chern classes of the sheaves which we wish to parametrize. For fixed Chern classes c1 and c2 , it is known that rank 2 semistable sheaves F with ci (F ) = ci can be realized as quotients of a fixed H = OX (−m)p for suitably chosen m and p. Let Rss ⊂ Q (resp. Rs ⊂ Q) consist of points [H → F ] ∈ Q such that the quotients F are semi-stable (resp. stable) torsion-free sheaves and the

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13 7

quotient map H → F induces an isomorphism Cp = H 0 (X, H(m)) ∼ = H 0 (X, F (m)). s s ss ss Let R ⊂ R (resp R ⊂ R ) denote the open subschemes parametrizing locally free sheaves. In [13] the Donaldson-Uhlenbeck moduli space has been constructed as the closure of the moduli space M s of the moduli space of stable locally free sheaves in the scheme Mss together with the reduced scheme structure. The scheme Mss is realized as the image of a P GL(m)-invariant mapping π : Rss → Mss Furthermore, the scheme Mss is projective. Note that Mss is not a GIT quotient but it maps Rs to an open subset Ms ⊂ Mss and π |Rs is the GIT quotient Rs //P GL(m). The closure M DU of M s in Mss with the reduced scheme structure is the precise algebro-geometric analogue of the differential geometric construction due to Donaldson. The key property of the moduli space M DU is that the boundary of M s is describable in terms of locally free polystable sheaves with lower c2 and certain zero cycles. Lemma 3.2. Rss is irreducible for large c2 , for a fixed c1 . Proof. Observe that Rs is a dense open subset in Rss for large c2 ([9, Theorem 9.1.2, page 200]). So irreducibility of Rss is equivalent to the irreducibility of Rs . Now Ms is a geometric quotient of Rs for the action of P GL(m). By [9, Theorem 9.4.3, page 203] the scheme Ms is irreducible for large c2 . Since the quotient map f : Rs → Ms is an open map with both base and fibre being irreducible, it follows that Rs is irreducible (see Lemma 3.3 below) . Lemma 3.3. If f : X → Y is a morphism of schemes such that f is an open surjective morphism and each closed fibre is irreducible, then Y irreducible =⇒ X irreducible . Proof. Let U and V be two nonempty open sets in X. Since f is an open surjective map and Y is irreducible, f (U ) and f (V ) are open nonempty subsets of Y such that f (U ) ∩ f (V ) = ∅. Let y ∈ f (U ) ∩ f (V ) be a closed point and x1 ∈ U and x2 ∈ V such that y = f (x1 ) = f (x2 ). Clearly, U ∩ f −1 (y) and V ∩ f −1 (y) are two nonempty open subsets of f −1 (y). Since f −1 (y) is irreducible it implies that U ∩ V ⊃ U ∩ V ∩ f −1 (y) = ∅. Hence Y is irreducible. 3.4. The small weight case. Since our final aim in this paper is to show that the Donaldson-Uhlenbeck spaces constructed in [3] are asymptotically irreducible, we will assume for the rest of the paper that the sheaves that we consider in the Quot scheme are locally free. Recall that the scheme Rss (resp. Rs ) parametrizes semistable (resp. stable) locally free quotients. We will stick to these assumptions and notations in the paper from here onwards. We now consider bundles equipped with parabolic structures. The weight α : = (α1 , α2 ) is called small if it satisfies the condition 1 (3.3) (α2 − α1 )D · H < . 2 Now a key observation, which is easy to check, is the following. Lemma 3.5. For small weights α (3.3), for any E ∈ Rs and any quasi-parabolic structure (2.4), the parabolic bundles E∗ is α-stable and conversely any α-stable parabolic bundle E∗ has the property that its underlying bundle E is semistable.

14 8


Let T˜ be the total family parametrizing quasi-parabolic structures on rank 2 bundles. The notation T˜ is loose since it hides the topological and parabolic data of its underlying objects. However, we observe that T˜ is independent of the parabolic weights α, β etc. Since we do not need to make modifications in the topological datum to prove asymptotic irreducibility we do not carry it as a part of the notation. Again since the rank of the bundle is 2 there is not much in terms of the quasi-parabolic structure except the degree of the subbundle when restricted to the parabolic divisor D. This will figure in the discussion that follows. It will be mentioned whenever needed and should cause no confusion. α Let Rk,d be the total family for H–stable parabolic bundles with weight α. For α we direct the reader to [15]. We simplify the notation a formal definition of Rk,d and have s α Rα := Rk,d . By the definition of T˜ we have the obvious morphism, namely forget : T˜ → R which “forgets” the quasi-parabolic structure. Note however that under this map the image of Rβss or Rβs need not be contained in Rss ; similarly, the inverse image of Rs can fall outside Rβs for an arbitrary weight β. In our simple setting of a flag which is only one-step on a rank 2 bundle, when s the weights are small (3.3), the morphism forget : Rα → Rss is well-defined and the s s inverse image of R is contained in Rα (this is a consequence of Lemma 3.5). Let T s and T ss denote the inverse images of Rs and Rss in T˜ . In other words, T ss is the total family of quasi-parabolic structures on semistable bundles. The upshot is that, if the weight α is small we have open inclusions: s T s ⊂ Rα ⊂ T ss .

(3.4)

We now describe the space T ss of quasi-parabolic structures on rank 2 semistable bundles. Let F be the universal sheaf on X × Rss and let L be the Poincaré line bundle over D × Picl (D). We have a diagram of various projections: (3.5)

D × Rss × Picl (D)

nn nnn n n nn w nn n p1

D × Rss

p2

QQQ QQQ QQ p3 QQQQ Q(

Rss × Picl (D)

D × Picl (D).

Let

W := p2 ∗ (Hom(p∗1 (F |D×Rss ), p∗3 (L))), where we have assumed that l = deg(Lt ) is sufficiently large so that p2 ∗ (Hom(p∗1 (F |D×Rss ), p∗3 (L))) is locally free ([8, page 288]). Let Z = Spec Sym(W ∗ ) be the underlying geometric vector bundle. This scheme parametrizes all morphisms from (F |D ) → L for F ∈ Rss and L ∈ Picl (D). Let Zsur ⊂ Z be the open subscheme which parametrizes the surjective morphisms. It is not hard to show that by choosing l 0 we can have a non-empty set of surjective morphisms (see for example [1, Theorem 2, page 426]). By taking kernels of these

PARABOLIC BUNDLES

15 9

morphisms which give line subbundles, we get the quasi-parabolic structures. Thus, after choosing l suitably, we see that Zsur ⊂ Z is a non-empty open subset. By the definition of T ss (for suitable l) it is immediate that there is an isomorphism ∼ → T ss . Zsur − By Lemma 3.2 the scheme Rss is asymptotically irreducible, being open in Rss ; furthermore, Picl (D) is also irreducible for any integer l. Hence Z and therefore Zsur is asymptotically irreducible. This implies that T ss is asymptotically irreducible. It s follows by (3.4), that Rα is asymptotically irreducible. 2 Observe that bounds on l in turn give bounds on d = c1 (FD (F )).D, where the 2 quasi-parabolic structure FD (F ) ⊂ F |D is obtained as the kernel to F |D → L. We isolate this key result in the following proposition. s Proposition 3.6. For small α, Rα is asymptotically irreducible for suitable 2 d = c1 (FD (E)).D.

Remark 3.7. The bounds on l, which in turn give bounds on d, ensure that, irrespective of the weight α, the bundles E have enough quasi-parabolic structures on the given divisor D ⊂ X. 4. The density of Msα in Mss α ss Let Msα = Mα k,d be the moduli stack of α-stable bundles and Mα the moduli stack of α-semistable bundles on X with topological and parabolic datum as in §2. The aim of this section is to prove that the open substack Msα in the moduli stack Mss α is dense for any α. We handle the problem by converting it to the equivariant Γ-bundle setting. The general set-up is as in §2 and we use the same notation. Let Y be a Kawamata cover of X. The advantage in doing this is that the technical complications arising in handling obstruction theory in the parabolic setting is considerably simplified when we make this shift. Let H be a Γ-sheaf over Y and P1 , P2 are two fixed polynomials. Let DrapΓ (H, P1 , P2 ) denote the “generalized flag scheme” which parametrizes Γ–subsheaves of H H∗ := 0 ⊂ H3 ⊂ H2 ⊂ H1 = H

such that the Hilbert polynomial of Hi−1 /Hi is Pi−1 . These can be defined as Γ-fixed points of the usual Drap scheme (cf. [3, page 15], [9, Appendix 2.A, page 48]). Lemma 4.1. The dimension of DrapΓ (H, P1 , · · · , Pk ) at the point H∗ satisfies the following inequality ext0Γ,+ (H, H) ≥ dimH∗ (DrapΓ (H, P1 , · · · , Pk )) ≥ ext0Γ,+ (H, H) − ext1Γ,+ (H, H).

Proof. The proof of this lemma is a routine equivariant generalization of the one given in [9, Proposition 2.A.12, page 54] and we omit the details. 4.2. Parabolic Chern classes Let E∗ be a rank 2 parabolic vector bundle over X with underlying bundle E. The parabolic Chern classes are defined as (see [3, Lemma 6.1]) (4.1)

par(c1 )(E∗ ) = c1 (E) + (α1 + α2 ) · D,

(4.2)

par(c2 )(E∗ ) = c2 (E) + (α1 + α2 )c1 (E) · D + α1 α2 D2 .

16 10


Observe that when c1 (E) = 0 (which is our base assumption), both par(c1 ) and par(c2 ) differ from the usual c1 and c2 by terms which involve only the parabolic divisor. Let V be a Γ–bundle on Y . We have (see [4, Equation 3.11]) (4.3)

c1 (V ) = p∗ (par(c1 )({pΓ∗ (V )}∗ )),

where {pΓ∗ (V )}∗ is the invariant direct image of V with the canonical parabolic structure coming from the Seshadri-Biswas correspondence. For a Γ-bundle F of rank 2 of local type τ (α) (see [3, Definition 2.12] for the definition) we have the equation in the second Chern classes of the underlying bundle: (4.4)

Γ ∨ˆ Γ {p = 4 c2 ((pΓ c2 (pΓ (Hom(F, F ))) = c (F )} (F )} ⊗{p 2 ∗ ∗ ∗ (F ))), ∗ ∗ ∗

where the last equality follows by a splitting principle argument as in [3, Lemma 6.1] and the assumption that c1 ((pΓ∗ (F )) = 0. ˆ denotes the parabolic Here and elsewhere, “∨” denotes the parabolic dual and ⊗ tensor product. By the naturality of parabolic Chern classes we have (4.5)

par(ci )(E∗∨ ) = (−1)i par(ci )(E∗ ).

When we work with a Kawamata cover as in our case, then we have the following relation between the Γ–cohomology and the usual cohomology on Y /Γ = X: (4.6)

HΓi (Y, F) = H i (X, pΓ∗ (F)),

∀i.

Definition 4.3. For Γ-bundles F and G on Y , define (−1)i extiΓ (F, G). χΓ (F, G) := i

Let V be a Γ-bundle of rank r on Y . Define the Γ-discriminant of V as: ΔΓ (V ) := 2r c2 (pΓ∗ (V )) − (r − 1) c1 (pΓ∗ (V ))2 . 4.4. Γ-total families. Let RΓss (resp. RΓs ) parametrize Γ–semistable (resp. stable) bundles of type τ (α) and fixed topological datum (c1 , c2 ) over Y . In §3 and §4 of [3] we give the construction of RΓss which parametrizes Γ-torsion–free sheaves. We recall that there is an action of Γ on a suitable Quot scheme of quotients on the Kawamata cover Y of X. The scheme RΓ is the subscheme of Γ–fixed points in the Quot scheme which consists of torsion–free sheaves and RΓss is an open subscheme of RΓ . We stick to locally free sheaves in this work since we work with the DonaldsonUhlenbeck compactifications. 4.5. Cohomological computations. The following lemmas play a key role in proving that RΓs is dense in RΓss . Lemma 4.6. Let F be a Γ–vector bundle of rank 2 on Y of some type τ (α), such that (4.7)

c1 (pΓ∗ (F )) = 0 .

Then χΓ (F, F ) = −ΔΓ (F ) + 4χ(OX ).

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17 11

Proof. We have χΓ (F, F ) = χΓ (Y, Hom(F, F ))

= χ(X, pΓ ∗ (Hom(F, F ))) (because of (4.6)) (by Hirzebruch-Riemann-Roch and (4.7)) = −c2 (pΓ ∗ (Hom(F, F ))) + 4χ(OX ) (by (4.4)) = −4 c2 (pΓ ∗ (F )) + 4χ(OX ) (by Definition 4.3). = −ΔΓ (F ) + 4χ(OX )

Let F ∈ RΓss − RΓs be a rank 2 strictly Γ-semistable bundle on Y such that = 0. Let 0 → F 1 → F → F2 → 0 be the Γ-Jordan-H¨ older filtration of F . Observe that the F2 is a torsion–free Γ– sheaves of rank 1, while F1 is locally free. Let pΓ∗ (F ) = E∗ and pΓ∗ (Fi ) = Ei,∗ , i = 1, 2. Then c1 (pΓ∗ (F ))

0 → E1,∗ → E∗ → E2,∗ → 0 is the parabolic Jordan-H¨ older filtration of E∗ on X. Note that E2,∗ is a parabolic torsion-free sheaf of rank 1. Further, c1 (E) = c1 (pΓ∗ (F )) = 0. We remark that if the ∨ parabolic line bundle E1,∗ has weight α1 , then its parabolic dual E1,∗ has weight 1 − α1 (cf. [12, Section 8] where the weight will be simply −αi in the balanced convention). Lemma 4.7. Let Ei,∗ be as above with weights αi on X. Then ∨ ˆ 1,∗ χ(X, E2,∗ ⊗E ) = χ(X, E2 ⊗ E1∗ )

E1∗ being the usual dual of E1 . Proof. By the Hirzebruch-Riemann-Roch theorem (K being the canonical divisor on X), we see that ∨ ˆ 1,∗ χ(X, E2,∗ ⊗E )=

∨ 2 ∨ ˆ 1,∗ ˆ 1,∗ c1 (E2,∗ ⊗E c1 (E2,∗ ⊗E ) )·K − + χ(OX ). 2 2

∨ ˆ 1,∗ We write c1 (E2,∗ ⊗E ) for the Chern class of the underlying bundle (and not its parabolic Chern class) since it is notationally inconvenient to shed the parabolic ∨ ˆ 1,∗ luggage on the tensor product E2,∗ ⊗E , the reason being that the underlying ∨ ∗ ˆ sheaf of E2,∗ ⊗E1,∗ is not E2 ⊗ E1 . Observe that ∨ ∨ ˆ 1,∗ ˆ 1,∗ c1 (E2,∗ ⊗E ) = par(c1 )(E2,∗ ⊗E ) − (α2 − α1 )D ∨ = par(c1 )(E2,∗ ) + par(c1 )(E1,∗ ) − (α2 + 1 − α1 )D ∨ ) − (1 − α1 )D] = [par(c1 )(E2,∗ ) − α2 D] + [par(c1 )(E1,∗ ∗ ∗ = c1 (E2 ) + c1 (E1 ) = c1 (E2 ⊗ E1 )

and the result follows. Let ξ21 = c1 (E2 ) − c1 (E1 ).

Observe that, by the definition of ΔΓ , we have (4.8)

ΔΓ (F ) = 4c2 (E) = 4c1 (E1 ) · c1 (E2 ).

Now (4.9)

[c1 (E2 ) − c1 (E1 )]2 = [c1 (E2 ) + c1 (E1 )]2 − 4[c1 (E2 ) · c1 (E1 )] = −4[c1 (E2 ) · c1 (E1 )]

since c1 (E2 ) + c1 (E1 ) = c1 (E) = 0.

Hence by (4.8) and (4.9) (4.10)

2 ξ21 = [c1 (E2 ) − c1 (E1 )]2 = −ΔΓ (F ).

18 12


The Γ-Euler characteristic has the following description for the Γ-line bundles Fi . Γ ∗ χΓ (F1 , F2 ) = χ(X, pΓ ∗ (Hom(F1 , F2 ))) = χ(X, p∗ (F2 ⊗ F1 ))

(4.11)

∨ ˆ 1,∗ = χ(X, E2,∗ ⊗E )=

2 ξ21 .K ξ21 − + χ(OX ) 2 2

by the proof of Lemma 4.7. Hence χΓ (F1 , F2 ) =

(4.12)

2 ξ21 ξ21 .K − + χ(OX ). 2 2

Lemma 4.8. Let F ∈ RΓss − RΓs be as above such that c1 (pΓ∗ (F )) = 0. Then 3 Δ (F ) + B 4 Γ where B is an irrelevant number not involving the Chern classes of the bundles. ext1Γ,− (F, F ) ≤

Proof. By a Γ-equivariant version of the spectral sequence in [9, 2.A.4] we have ext1Γ,− (F, F ) ≤ ext1Γ (F1 , F1 ) + ext1Γ (F2 , F1 ) + ext1Γ (F2 , F2 ) = {ext0Γ (F1 , F1 ) + ext2Γ (F1 , F1 ) − χΓ (F1 , F1 )} +{ext0Γ (F2 , F1 ) + ext2Γ (F2 , F1 )− χΓ (F2 , F1 )} +{ext0Γ (F2 , F2 ) + ext2Γ (F2 , F2 ) − χΓ (F2 , F2 )} ≤ B1 − {χΓ (F1 , F1 ) + χΓ (F2 , F1 ) + χΓ (F2 , F2 )} (1) = B1 + χΓ (F1 , F2 ) − χΓ (F, F ) = B1 + = = ≤

2 ξ21 2

3 Δ (F ) 4 Γ 3 Δ (F ) 4 Γ 3 Δ (F ) 4 Γ

−

ξ21 .K + ΔΓ (F ) − 3χ(OX ) (by Lemma 4.6 2 2 ξ21 ξ21 .K − + B1 − 3χ(OX ) (by (4.10)) 2 4ξ21 2 K 2 − 2 + B1 − 3χ(OX ) − K4 2

and (4.12))

+ + + B.

The last inequality with the irrelevant number B comes by the following reasoning. By the Hodge index theorem,

K ξ21 − 2 2

2 ≤

ξ21 2

2 −K ·H 2 . H2

Further, by the parabolic semistability of E∗ , since Ei,∗ are its parabolic JordanHölder terms we have par(c1 )(E1,∗ ) · H =

par(c1 )(E∗ ) · H = par(c1 )(E2,∗ ) · H. 2

Hence, (c1 (E2 ) − c1 (E1 )) · H = ξ21 · H = (α1 − α2 )D · H is an irrelevant number. The remaining terms in ( ξ221 − K 2 · H) are clearly irrelevant. 4.9. The density of RΓs in RΓss . In the rest of this section we conclude the density of RΓs in RΓss . Lemma 4.10. There is an irrelevant number B depending on the rank, X, H, the parabolic datum αi and D, such that dim(RΓss − RΓs ) ≤ endΓ (H) +

3 Δ (F ) + B 4 Γ

where F ∈ RΓss − RΓs 1 Since Fi are rank 1 Γ-torsion–free sheaves, the dimensions ext0Γ (Fi , Fj ) and ext2Γ (Fi , Fj ) are bounded ∀i, j and the irrelevant number B1 is to take care of these terms.

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19 13

Proof. Let {ρ : H → F } ∈ RΓss − RΓs with Γ–Jordan-H¨ older filtration (4.13)

0 = F0 ⊂ F1 ⊂ F2 = F

such that F1 and F2 /F1 are rank 1 torsion–free sheaves with the same μ = μ(F ) and F1 is locally free. The filtration 4.13 induces a Γ-filtration on H (4.14)

0 ⊂ H0 ⊂ H1 ⊂ H2 = H

such that F1 = H1 /H0 and F2 = H2 /H0 . Let P∗ := (P1 , P2 ) be the Hilbert polynomials of H2 /H1 and H1 /H0 . Since we fix the topological type of these quotients, we get a bounded family of sheaves with fixed μ = μ(F ) giving only finitely many choices of P∗ . Let Z be the finite union of DrapΓ (H, P1 , P2 ). There is a morphism f : Z → QΓ (the Γ–fixed points of the Quot scheme on Y ) sending H0 ⊂ H1 ⊂ H to H0 ⊂ H. It is clear that RΓss − RΓs ⊂ f (Z) (since every strictly semistable object has a Jordan-Hölder filtration). We have by Lemma 4.1 (4.15)

dim(RΓss − RΓs ) ≤ dimZ ≤ ext0Γ,+ (H, H).

The definition of Ext± gives an exact sequence 0 → Ext0Γ,− (H, H) → Ext0Γ (H, H) → Ext0Γ,+ (H, H) → Ext1Γ,− (H, H). Hence ext0Γ,+ (H, H) ≤ endΓ (H) − ext0Γ,− (H, H) + ext1Γ,− (H, H) ≤ endΓ (H) − 1 + ext1Γ,− (F, F).

The last inequality follows from the fact that a filtration of F canonically induces a filtration on H, and we also have ext1Γ,− (F, F) = ext1Γ,− (H, H). Hence, ext0Γ,+ (H, H) ≤ endΓ (H) +

3 Δ (F ) + B (by Lemma 4.8). 4 Γ

Proposition 4.11. For any {ρ : H → F } ∈ RΓss , dimρ (RΓss ) ≥ endΓ (H) + ΔΓ (F ) − 4χ(OX )

Proof. We follow the proof of O’Grady (see [16] and [9, Theorem. 4.5.8, page. 104]. Let K be the kernel of the morphism ρ. Applying HomΓ (−, F ) to 0 −→ K −→ H −→ F −→ 0, we get 0 −→ EndΓ (F ) −→ HomΓ (H, F ) −→ HomΓ (K, F ) −→ Ext1Γ (F, F ) −→ 0

Suppose that a positive integer m has been already chosen for which F is m– regular. Therefore, as we have seen earlier, H 0 (H(m)) = H 0 (F (m)). Thus we have HomΓ (H, F ) = HomΓ (H 0 (H(m)), H 0 (F (m))) = HomΓ (H, H) and we have the following equality of dimensions: homΓ (K, F ) = {endΓ (H) − endΓ (F ) + ext1Γ (F, F )}. Using this computation we get the following inequality of dimensions: dimρ (RΓss ) ≥ homΓ (K, F ) − ext2Γ (F, F ) = endΓ (H) − endΓ (F ) + ext1Γ (F, F ) − ext2Γ (F, F ) = endΓ (H) − χΓ (F, F ) (by Lemma 4.6). = endΓ (H) + ΔΓ (F ) − 4χ(OX )

20 14


Proposition 4.12. Let B be as in Lemma 4.10. If ΔΓ (F ) > 4(B + 4χ(OX )), then RΓs is dense in RΓss . Proof. We have the following inequalities: dim(RΓss − RΓs ) ≤ {endΓ (H) + 34 ΔΓ (F ) + B} < {endΓ (H) + ΔΓ (F ) − 4χ(OX )} ≤ min{dim(Xi ) : Xi a component of RΓss }.

Note that the above inequalities show that the dimension of any component is at least (endΓ (H) + ΔΓ (F ) − 4χ(OX )). Hence RΓs intersects all components of RΓss . Hence RΓs is dense in RΓss . Corollary 4.13. If the second Chern class c2 of the underlying bundles is large, then Msα is dense in Mss α for any weight α. Proof. By the Seshadri–Biswas correspondence we see that the moduli stack s ss Mss Γ of Γ–semistable bundles (resp. MΓ of Γ–stable bundles) is isomorphic to Mα s s (resp. Mα ). Hence by Proposition 4.12, since c2 0 it follows that RΓ is dense in s RΓss . Since RΓss (resp. RΓs ) is the atlas of the Artin stack Mss Γ (resp. MΓ ) we have ss ss a surjective morphism RΓ → MΓ and the result follows. 5. Variation of Mss α and the Main Theorem s Let Mss α (resp. Mα ) be the moduli stack of α-semistable (resp. α-stable) bundles with first Chern class 0. Here we tacitly assume that d is chosen as in Proposition 3.6 (see Remark 3.7). Now we study these moduli stacks as we vary weights. Let N0 = D · H,

W = {(α1 , α2 ) : 0 < α1 < α2 < 1} and δW :=

(α1 , α2 ) : 0 < α1 < α2 < 1 such that | α1 − α2 |=

k , 1 ≤ k ≤ 2N0 . 2N0

Let W ◦ = W − δW A connected component of W ◦ is called a chamber. Observe s that, if α is a weight within a chamber then Mss α = Mα . Moreover if α and β are ss ss s s in same chamber then Mα = Mβ and also Mα = Mβ . We have the following lemma: Lemma 5.1. If α is in a chamber and ω is on an adjacent wall, then any ω-stable bundle is α-stable and any α-stable bundle is ω-semistable. Proof. For 0 ≤ t ≤ 1, let αt denote the vector tα + (1 − t)ω in W . Then, αt is also in the chamber for t = 0. Suppose that E∗ is ω-stable and suppose that E∗ is not α-stable. Then, there exists a subbundle E∗ of E∗ such that μα (E∗ ) ≥ μα (E∗ ). The continuous function t → μαt (E∗ ) − μαt (E∗ ) assumes a negative value at t = 0 and is non-negative at t = 1 and hence takes the value 0 for some 0 < t0 ≤ 1. But then E∗ is strictly semistable with respect to the weight αt0 contradicting the fact that αt0 is within the chamber. Similarly, if E∗ is α-semistable (therefore in fact α-stable) and E∗ is not ωsemistable, then there exists a subbundle E∗ of E∗ such that μω (E∗ ) > μω (E∗ ). Thus, μαt (E∗ ) − μαt (E∗ ) is negative at t = 0 and non-negative at t = 1; this would imply that with respect to some αt0 within the chamber, E∗ is strictly semistable, again a contradiction. We have the following:

PARABOLIC BUNDLES

21 15

Corollary 5.2. If α is in a chamber and ω is in a adjacent wall then we have the following inclusions: ss Msω ⊂ Msα = Mss α ⊂ Mω .

Theorem 5.3. Mβ is asymptotically irreducible for all β. s is an atlas for the Artin stack Msα and we Proof. Recall that the scheme Rα s have a canonical surjective morphism Rα → Msα . Hence by Proposition 3.6, Msα is asymptotically irreducible for small α. For any ω in an adjacent wall, by Corollary 5.2 we see that Msω is asymptotically irreducible being an open substack of Msα . Now by Corollary 4.13 it follows that Mss ω is asymptotically irreducible. Now taking β in any chamber with ω in an adjacent wall and different from the “small” chamber, we see again by Corollary 4.13 that Mss β is asymptotically irreducible. We proceed similarly to reach all weights in W using the connectedness of W and finiteness of the number of walls; since Msβ surjects onto Mβs it follows that Mβs is asymptotically irreducible. Now recall that Mβ is by definition the closure of Mβs (with the reduced scheme structure) in a certain Mβk,d . This implies that Mβ is also asymptotically irreducible and the theorem follows.

Remark 5.4. The subtle point is that even though we finally need to prove that Mαs is asymptotically irreducible, we are forced to go to the semistable bundles since we need to go over various weight chambers. Remark 5.5. Observe that the arguments in this paper automatically give as a consequence the generic smoothness and asymptotic non-emptiness of the moduli space Mα . In specific situations, one can use the techniques of this paper to conclude rationality of certain parabolic moduli. References [1] M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414–452. [2] V. Balaji, I. Biswas, D. S. Nagaraj, Principal bundles over projective manifolds with parabolic structure over a divisor, Tˆ ohoku Math. J. (2) 53 (2001), no. 3, 337–367. [3] V. Balaji, A. Dey, R. Parthasarathi, Parabolic bundles on algebraic surfaces, I. The Donaldson-Uhlenbeck compactification, Proc. Indian Acad. Sci. Math. Sci. 118 (2008), no. 1, 43–79. [4] I. Biswas, Parabolic bundles as orbifold bundles, Duke Math. J. 88 (1997), no. 2, 305–325. [5] H. U. Boden, Y. Hu, Variations of moduli of parabolic bundles, Math. Ann. 301 (1995), no. 3, 539–559. [6] D. Gieseker, J. Li, Irreducibility of moduli of rank-2 vector bundles on algebraic surfaces, J. Differential Geom. 40 (1994), no.1, 23–104. [7] A. Grothendieck, Sur quelques points d’alg` ebre homologique, Tˆ ohoku Math. J. (2) 9 (1957), 119–221. [8] R. Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics 52. Springer-Verlag, New York-Heidelberg, 1977. [9] D. Huybrechts, M. Lehn. The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31. Friedr. Vieweg und Sohn, Braunschweig, 1997. [10] Y. Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), 253–276. [11] Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model problem, Algebraic Geometry, Sendai, 1985, 283–360, Adv. Stud. Pure. Math. 10, North-Holland, Amsterdam, 1987.

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[12] P. B. Kronheimer, T. S. Mrowka, Gauge theory for embedded surfaces II, Topology, 34 (1995), No. 1, 37–97. [13] J. Li, Algebraic geometric interpretation of Donaldson’s polynomial invariants, J. Differential Geom. 37 (1993), no.2, 417–466. [14] M. Maruyama, Instantons and parabolic sheaves, Geometry and analysis (Bombay, 1992), 245–267, Tata Inst. Fund. Res., Bombay, 1995. [15] M. Maruyama, K. Yokogawa, Moduli of parabolic stable sheaves, Math. Ann. 293 (1992), no.1, 77–99. [16] K. G. O’Grady, Moduli of vector bundles on projective surfaces: some basic results, Invent. Math. 123 (1996), no.1, 141–207. [17] C. S. Seshadri, Moduli of π-vector bundles over an algebraic curve, 1970 Questions on Algebraic Varieties (C.I.M.E, III Ciclo, Varenna, 1969), pp. 139–260. Edizione Cremonese, Rome. Chennai Mathematical Institute, Plot H1, SIPCOT IT Park Padur PO, Siruseri 603103, India E-mail address: [email protected] School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India E-mail address: [email protected]


Finite Subgroups of PGL2 (K) Arnaud Beauville To Ramanan on his 70th birthday

Abstract. We classify, up to conjugacy, the finite subgroups of PGL2 (K) of order prime to char(K).

Introduction The aim of this note is to describe, up to conjugacy, the finite subgroups of PGL2 (K), for an arbitrary field K. Throughout the paper, we consider only subgroups whose order is prime to the characteristic of K. When K = C, or more generally when K is algebraically closed, the answer is well known: any such group is isomorphic to Z/r, Dr (the dihedral group), A4 , S4 or A5 , and there is only one conjugacy class for each of these groups. If K is arbitrary, the group PGL2 (K) is contained in PGL2 (K), so the subgroups of PGL2 (K) are among the previous list; it is not difficult to decide which subgroups occur for a given field K, see §1. So the only question left is to describe the conjugacy classes in PGL2 (K) of the subgroups in the list. In §2 we give a general answer for subgroups of G(K), for an algebraic group G, in terms of (non-abelian) Galois cohomology. We illustrate the method on one example in §3, and apply it to the case G = PGL2 in §4. The motivation for looking at this question was to understand the appearance of the Brauer group in the case of (Z/2)2 considered in [B]. The result is somewhat disappointing, as it turns out that this case (which could be treated directly, as in [B]) is the only one where some second Galois cohomology group plays a role. At least our method explains this role, and hopefully may be useful in other situations. 1. The possible subgroups We repeat that whenever we mention a finite group, we always assume that its order is prime to the characteristic of K. The following is classical (see [S2], 2.5). Proposition 1.1. 1) PGL2 (K) contains Z/r and Dr 1 if and only if K contains ζ + ζ −1 for some primitive r-th root of unity ζ. 2010 Mathematics Subject Classification. 20G15. 1We denote by D the dihedral group with 2r elements. r c 2010 American c Mathematical 0000 (copyright Society holder)

1 23

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ARNAUD BEAUVILLE

2) PGL2 (K) contains A4 and S4 if and only −1 is the sum of two squares in K. 3) PGL2 (K) contains A5 if and only if −1 is the sum of two squares and 5 is a square in K. ∼ Proof. One way to prove this is to use the isomorphism PGL2 (K) −→ SO(K, q), where q is the quadratic form q(x, y, z) = x2 + yz on K 3 ([D], II.9). If a group H embeds into SO(K, q), we have a faithful representation ρ of H in K 3 , which preserves an indefinite quadratic form. • Case H = Z/r : let g be a generator; the existence of q forces the eigenvalues of ρ(g) in K to be of the form (ζ, ζ −1 , 1), with ζ a primitive r-th root of 1. This implies ζ + ζ −1 ∈ K. Conversely, if λ := ζ + ζ −1 is in K, the homography z → (λ + 1)z − 1 is an element of order r of PGL2 (K). z+1 • Case H = Dr : by the previous case, if Dr ⊂ PGL2 (K), λ := ζ + ζ −1 is in K. (λ + 1)z − 1 generate a Conversely if λ ∈ K, the homographies z → 1/z and z → z+1 subgroup of PGL2 (K) isomorphic to Dr .

• Cases H = A4 , S4 or A5 . The representation ρ must be irreducible. Each of the groups A4 and S4 has exactly one irreducible 3-dimensional representation with trivial determinant, which is defined over the prime field; the only invariant quadratic form (up to a scalar) is the standard form q0 (x, y, z) = x2 + y 2 + z 2 . Thus A4 and S4 are contained in PGL2 (K) if and only if q0 is equivalent to λq for some λ ∈ K ∗ , which means that q0 represents 0. √ Since A5 contains elements of order 5, the √ condition 5 ∈ K is necessary. Suppose this is the case, and put ϕ = 12 (1 + 5); the subgroup of SO(K, q0 ) preserving the icosahedron with vertices {(±1, 0, ±ϕ) , (±ϕ, ±1, 0) , (0, ±ϕ, ±1)} is isomorphic to A5 . It follows as above that A5 embeds in SO(K, q) if and only if q0 represents 0. 2. Some Galois cohomology 2.1. In this section we consider an algebraic group G over K, and a subgroup H ⊂ G(K). We choose a separable closure Ks of K, and put g := Gal(Ks /K). We are interested in the set of embeddings H → G(K) which are conjugate in G(Ks ) to the natural inclusion i : H → G(K), modulo conjugacy by an element of G(K). We denote this (pointed) set by Embi (H, G(K)). We will use the standard conventions for non-abelian cohomology, as explained for instance in [S3], ch. I, §5. We will also use the notation of [S3] for Galois cohomology: if G is an algebraic group over K, we put Hi (K, G) := Hi (g, G(Ks )). Proposition 2.2. Let Z be the centralizer of H in G(Ks ). The pointed set Embi (H, G(K)) is canonically isomorphic to the kernel of the natural map H1 (K, Z) → H1 (K, G). Proof. Let X ⊂ G(Ks ) be the subset of elements g such that g −1 σg ∈ Z for all σ ∈ g. The group G(K) (resp. Z) acts on X by left (resp. right) multiplication. By [S3], ch. I, 5.4, cor. 1, the kernel of H1 (K, Z) → H1 (K, G) is identified with the (left) quotient by G(K) of the subset of g-invariant elements in G(Ks )/Z; but this

FINITE SUBGROUPS OF PGL2 (K)

25 3

subset is by definition X/Z, so we can identify our kernel to the double quotient G(K)\X/Z. For every g ∈ X, the conjugate embedding gig −1 belongs to Embi (H, G(K)). Any element j ∈ Embi (H, G(K)) is of the form gig −1 for some g ∈ G(Ks ); for σ ∈ g, the element σg again conjugates i to j, hence g −1 σg ∈ Z and g ∈ X. Thus the map g → gig −1 from X to Embi (H, G(K)) is surjective. Two elements g and g of X give the same element in Embi (H, G(K)) if and only if g belongs to the double coset G(K)gZ. Therefore the above map induces a canonical bijection ∼ G(K)\X/Z −→ Embi (H, G(K)). 2.3. Let us write down the correspondence explicitly: a class in our kernel is represented by a 1-cocycle g → Z which becomes a coboundary in G, hence is of the form σ → g −1 σg for some g ∈ X; we associate to this class the embedding gig −1 . 2.4. We are actually more interested in the set Conj(H, G(K)) of subgroups of G(K) which are conjugate to H in G(Ks ), modulo conjugacy by G(K). Associating to an embedding its image defines a surjective map im : Embi (H, G(K)) → Conj(H, G(K)). The normalizer N of H in G(Ks ) acts on H by automorphisms, hence also on Embi (H, G(K)). Two embeddings with the same image differ by an automorphism of H, which must be induced by an element of N if the embeddings are conjugate under G(Ks ). It follows that im induces an isomorphism ∼ Embi (H, G(K))/N −→ Conj(H, G(K)). 2.5. Let us translate this in cohomological terms. Let H1 (K, Z)0 denote the kernel of the map H1 (K, Z) → H1 (K, G). An element n of N acts on Embi (H, G(K)) by j → j ◦ int(n−1 ); if j = gig −1 , this amounts to replace g by gn, hence the 1-cocycle ϕ : σ → g −1 σg by n−1 ϕ σn. This formula defines an action of N on H1 (K, Z) which preserves H1 (K, Z)0 ; the map g → gHg −1 induces ∼ an isomorphism of pointed sets H1 (K, Z)0 /N −→ Conj(H, G(K)). 3. An example 3.1. In this section we fix an integer r ≥ 2, prime to char(K), and we assume that K contains a primitive r-th root of unity ζ. We consider the matrices A, B ∈ Mr (K) defined on the canonical basis (e1 , . . . , er ) of K r by A · ei = ei+1

,

B · ei = ζ i ei

for 1 ≤ i ≤ r, with the convention er+1 = e1 . The matrices A and B generate the K-algebra Mr (K), with the relations Ar = B r = I

,

BA = ζAB .

¯ B ¯ in PGLr (K) commute; we consider the embedding i : (Z/r)2 → Their classes A, ¯ The image H of i is its PGLr (K) which maps the two basis vectors to A¯ and B. own centralizer; in particular, H is a maximal commutative subgroup of PGLr (K). By the Kummer exact sequence (and the choice of ζ), the group H1 (K, Z/r) is identified with K ∗ /K ∗r ; the pointed set H1 (K, PGLr ) can be viewed as the set of isomorphism classes of central simple K-algebras of dimension r 2 ([S1], X.5). ¯ , β¯ be their images in K ∗ /K ∗r . The map Lemma 3.2. Let α, β ∈ K ∗ , and let α 1 2 1 ¯ the class of the cyclic H (i) : H (K, Z/r) → H (K, PGLr ) associates to (α, ¯ β) 1

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K-algebra Aα,β generated by two variables x, y with the relations xr = α, y r = β, yx = ζxy. Proof. We choose α , β in Ks with αr = α and β r = β. The Kummer isomorphism associates to (α, β) the homomorphism (a, b) : g → (Z/r)2 defined by σ

α = ζ a(σ) α

σ

β = ζ b(σ) β

for each σ ∈ g . ¯ b(σ) . Its image in H (K, PGLr (Ks )) is the class of the 1-cocycle σ → A¯a(σ) B Now let us recall how we associate to the algebra Aα,β a cohomology class [Aα,β ] in H1 (K, PGLr ) (loc. cit.). We choose an isomorphism of Ks -algebras u : ∼ Mr (Ks ) −→ Aα,β ⊗K Ks . For each σ ∈ g, u−1 σu is an automorphism of Mr (Ks ), hence of the form int(gσ ) for some gσ in PGLr (Ks ). Then [Aα,β ] is the class of the 1-cocycle σ → gσ . In our case we define u on the generators A, B by u(A) = β y −1 , u(B) = α−1 x. Then the automorphism u−1 σu multiplies A by ζ b(σ) and B by ζ −a(σ) , which gives ¯ b(σ) as above. gσ = A¯a(σ) B 1

3.3. The exact sequence 1 → Gm → GLr → PGLr → 1 gives rise to a coboundary homomorphism ∂r : H1 (K, PGLr ) → H2 (K, Gm ) = Br(K) which is injective (loc. cit.). The class ∂r [Aα,β ] ∈ Br(K) is the symbol (α, β)r ; it depends only on the classes of α and β (mod. K ∗r ). The map ( , )r : (K ∗ /K ∗r )2 → Br(K) is bilinear and alternating. Since ∂r is injective, we find: Proposition 3.4. The set Embi ((Z/r)2 , PGLr (K)) is isomorphic to the set of couples (α, β) in (K ∗ /K ∗r )2 such that (α, β)r = 0. We will describe the correspondence more explicitely in the case r = 2 in the next section. 4. Conjugacy classes in PGL2 (K) Proposition 4.1. Assume that K is separably closed. Two finite subgroups of PGL2 (K) which are isomorphic (and of order prime to char(K)) are conjugate. Proof. Again this is certainly well-known; we give a quick proof for completeness. The possible subgroups are those which appear in Proposition 1.1. An element of order r of PGL2 (K) comes from a diagonalizable element of GL2 (K), hence is conjugate to the homothety z → ζz for some ζ ∈ μr (K) 2; thus a cyclic subgroup of order r of PGL2 (K) is conjugate to the group Hr of homotheties z → λz, λ ∈ μr (K). There is only one group Dr containing Hr , namely the subgroup generated by Hr and the involution z → 1/z; it follows that all dihedral subgroups of order 2r are conjugate to this subgroup. ∼ For the three remaining groups, we use again the isomorphism PGL2 (K) −→ SO3 (K). The groups A4 and S4 have exactly one irreducible representation of dimension 3 with trivial determinant, while A5 has two such representations which differ by an outer automorphism: this is elementary in characteristic 0, and the general case follows by [I], ch. 15. Therefore two isomorphic subgroups H and H of SO3 (K) of this type are conjugate in GL3 (K). The only quadratic forms 2As usual we denote by μ (K) the group of r-th roots of unity in K. r

27 5


preserved by H or H are the multiple of the standard form; thus the element g of GL3 (K) which conjugates H to H must satisfy tg g = λI for some λ ∈ K. Replacing g by ±μg, with μ2 = λ−1 , we have g ∈ SO3 (K), hence our assertion. Recall that the determinant induces a homomorphism K ∗ /K ∗2 .

det : PGL2 (K) →

Theorem 4.2. 1) PGL2 (K) contains only one conjugacy class of subgroups isomorphic to Z/r (r > 2), A4 , S4 or A5 . 2) The conjugacy classes of cyclic subgroups of order 2 of PGL2 (K) are parametrized by K ∗ /K ∗2 : to α ∈ K ∗ (mod. K ∗2 ) corresponds the involution z → α/z. 3) The homomorphism det : PGL2 (K) → K ∗ /K ∗2 induces a bijective correspondence between: • conjugacy classes of subgroups of PGL2 (K) isomorphic to (Z/2)2 ; • subgroups G ⊂ K ∗ /K ∗2 of order ≤ 4, such that (−α, −β)2 = 0 for all α, β in G (see (3.3)). 4) Assume that μr (K) has order r. The conjugacy classes of subgroups Dr of PGL2 (K) are parametrized by K ∗ /K ∗2 μr (K). The subgroup corresponding to α ∈ K ∗ (mod. K ∗2 μr (K) ) consists of the homographies z → ζz and z → αη/z, for ζ, η ∈ μr (K). Proof. Using Proposition 4.1 we can apply the method of §3. We give the list of the subgroups of PGL2 (Ks ) and their centralizers:

H

Z/2

Z/r (r > 2)

Z/2 × Z/2

Dr (r > 2)

A4

S4

A5

Z

Gm Z/2

Gm

Z/2 × Z/2

Z/2

1

1

1

In case 1), we have H1 (K, Z) = {1} (using H1 (K, Gm ) = {1}). The result follows from (2.5). Case 2): This is the case where a direct approach is definitely simpler than our method, so we follow the former and leave the latter to the reader. Let s be an involution of PGL2 (K), and let α ∈ K ∗ such that α ≡ −det(s) (mod. K ∗2 ). Then 2 2 s is represented by amatrix A ∈ GL2 (K) with A = α I. In a basis (v, Av) of K , 0 α , hence s is conjugate to the involution z → α/z. This implies we have A = 1 0 2). Case 3): Let i : (Z/2)2 → PGL2 (K) be the embedding which maps the basis vectors e1 and e2 to the involutions z → 1/z and z → −z. By Proposition 3.4 the set Embi ((Z/2)2 , PGL2 (K)) is canonically identified to the set of couples (α, β) in (K ∗ /K ∗2 )2 with (α, β)2 = 0. We make the correspondence explicit following (2.3). Let α, β ∈ K ∗ with (α, β)2 = 0. This means that the conic x2 − αy 2 − βz 2 = 0 is isomorphic to P1K , thus there exists λ, μ in K with λ2 − α − βμ2 = 0. We choose α and β in Ks such that α2 = α and β 2 = β; as above we define the homomorphisms a and b : g → Z/2 by σ

α = (−1)a(σ) α

and

σ

β = (−1)b(σ) β

for each σ ∈ g .

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z−θ βμ λ − α ; let g ∈ PGL2 (Ks ) be the homography z → α . = λ+α βμ z+θ An easy computation gives

Put θ :=

g −1 σg = i(a(σ), b(σ)) . Thus the embedding of (Z/2)2 associated to (α, β) is gig −1 ; it maps e1 to the λu − α , and e2 to h2 : z → α/z . Note that det(h1 ) = −β homography h1 : z → z−λ and det(h2 ) = −α. Now we have to take into account the action of the normalizer N of H in PGL2 (Ks ). This is the subgroup S4 generated by H and the homographies z+1 , n2 : z → ιz , n1 : z → z−1 where ι is a square root of −1. We apply the recipe of (2.5). Since n1 ∈ PGL2 (K), it acts on H1 (K, H) through its action on H, which permutes e1 and e2 ; thus it maps (α, β) ∈ (K ∗ /K ∗2 ) × (K ∗ /K ∗2 ) to (β, α). The action of n2 on H fixes e2 and exchanges e1 with e1 + e2 ; to get the action on H1(K, H) we have to multiply by σ n2 , that is, σ → i (σ(ι)/ι) e2 . Hence n2 acts on the class of the cocycle σ → n−1 2 H1 (K, H) by n2 · (α, β) = (α, −αβ) . Let Gα,β be the subgroup of K ∗ /K ∗2 generated by −α and −β; it is the image of H by the homomorphism det : PGL2 (K) → K ∗ /K ∗2 . If Gα,β ∼ = (Z/2)2 , the ∗ ∗2 ∗ ∗2 orbit N · (α, β) in (K /K ) × (K /K ) has 6 elements, which are the couples (−x, −y) with x, y ∈ Gα,β , x = y. If Gα,β ∼ = (Z/2), the orbit has 3 elements, which are the couples (−x, −y) with x, y ∈ Gα,β , (x, y) = (1, 1). Finally if Gα,β is trivial the orbit consists only of (−1, −1). Thus the conjugacy classes of subgroups (Z/2)2 in PGL2 (K) are parametrized by the subgroups G ⊂ K ∗ /K ∗2 of order ≤ 4, with the property (−α, −β)2 = 0 for each α, β in G. Case 4): The group Dr is generated by two elements s, t with the relations s2 = tr = 1 and sts = t−1 . We choose a primitive r-th root of unity ζ and consider the embedding i : Dr → PGL2 (K) such that i(s) is the involution z → 1/z and i(t) the homothety z → ζz. The centralizer is Z/2, generated by the involution z → −z. As in case 2) it follows that Embi (Dr , PGL2 (K)) is isomorphic to H1 (K, Z/2). Also the previous argument shows that the embedding corresponding to α ∈ K ∗ is the conjugate of i by the homography z → α z, with α2 = α, so it maps s to z → α/z and t to z → ζz. To complete the picture we have to take into account the action of the normalizer N of i(Dr ) in PGL2 (Ks ). This is the subgroup D2r generated by i(s) : z → 1/z and the homothety n : z → ηz, where η ∈ Ks is a primitive 2r-th root of unity. The action of i(s) is trivial, and n acts by multiplication by the cocycle σ → n−1 σn, which corresponds to the class of η 2 in K ∗ /K ∗2 . Since η 2 generates μr (K), the assertion 4) follows. References [B] A. Beauville: p-elementary subgroups of the Cremona group, J. of Algebra 314 (2007), 553– 564. [D] J. Dieudonné: La g´ eom´ etrie des groupes classiques. Springer-Verlag, Berlin-G¨ ottingenHeidelberg, 1955. [I] I. Isaacs: Character theory of finite groups. AMS Chelsea Publishing, Providence, RI, 2006.


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[S1] J.-P. Serre: Corps locaux. Hermann, Paris, 1962. [S2] J.-P. Serre: Propri´ et´ es galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331. [S3] J.-P. Serre: Galois cohomology. Springer-Verlag, Berlin, 1997. ´ UMR 6621 du CNRS, Universit´ Laboratoire J.-A. Dieudonne e de Nice, Parc Valrose, F-06108 Nice cedex 2, France E-mail address: [email protected]



Picard Groups of Moduli Spaces of Torsionfree Sheaves on Curves Usha N. Bhosle Abstract. This article is a short survey of results on the Picard groups of the moduli spaces of semistable torsionfree sheaves on irreducible projective complex curves with at most ordinary nodes as singularities.

1. Introduction In this article we present some old and new results on the Picard groups of the moduli spaces of torsionfree sheaves on curves. X and Y will always denote reduced, irreducible, projective, complex curves. A singular curve will be often denoted by Y and its normalisation by X. Let U (n, d) be the projective moduli space of S-equivalence classes of semistable torsionfree sheaves of rank n, degree d on X and U (n, d) its open subset corresponding to locally free sheaves (vector bundles). The superscript s will denote the open subsets corresponding to stable points. After a short discussion of the case n = 1, we deal with the case n ≥ 2 in the rest of the paper. The first computations of Picard groups of U (n, d), n ≥ 2, were carried out by C.S.Seshadri for n and d coprime and X a smooth curve. We present here a proof by Ramanan [R] expanding on the ideas of Seshadri. These results were extended to the noncoprime case by Drezet and Narasimhan [DN]. Generalizations to singular curves were carried out by us in a series of papers [B3], [B4], [B5]. We state the main results and briefly sketch their proofs. For details, the reader may refer to the original proofs. 2. The rank 1 case The moduli spaces U (1, d) and U (1, d) are respectively the generalized Jacobian d and the compactified Jacobian of degree d of the curve X and are denoted by JX d 0 and J X respectively. JX = JX is the component Pic0 (X) of the Picard group Pic X of X containing the identity. 2010 Mathematics Subject Classification. 14H60, 14D20, 14F05. Key words and phrases. Vector bundles, torsionfree sheaves, moduli spaces, Picard groups. I would like to thank the referee and P.E. Newstead for a very careful reading of the first version and useful suggestions. c Mathematical 0000 (copyright Society holder) c 2010 American

1 31

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When X is smooth, U (1, 0) = U (1, 0) is the Jacobian JX of X, it is an Abelian variety. Fixing a line bundle L of degree 1 on X, there is the Abel map AL : X → JX , defined by x → L(−x). The induced pull-back map on line bundles gives an isomorphism A∗ : Pic0 (JX ) ∼ = JX L

independent of the choice of L. Thus JX is a selfdual Abelian variety (see [L, Theorem 3, p.156]; [M, Proposition 6.9, p.118]). For a singular curve X, JX is not an Abelian variety, nor is the projective variety J X , the compactified Jacobian of X. The Abel map is defined as the map AL : X → J X , defined by x → Ix ⊗ L, where Ix is the ideal sheaf of x ∈ X. Esteves, Gagné and Kleiman have the following generalization of the above result on the selfduality of the Jacobian of a smooth curve [EGK]. Theorem 2.1. (1) Suppose that a singular curve Y has surficial singularities, that is, singularities of embedding dimension 2. Then A∗L : Pic0 (J Y ) → JY has a right inverse independent of the choice of L i.e. there is f : JY → Pic (J Y ) such that A∗L ◦ f = 1JY . A∗L is itself independent of the choice of L. (2) Let Y be a curve with double points as singularities, i.e. with nodes or cusps. Then A∗L is an isomorphism. In fact a relative version of the theorem is proved for flat, projective families of geometrically integral curves over an arbitrary locally Noetherian base scheme [EGK]. The compactified Jacobian of a nodal curve is a seminormal variety. Recall that a variety is called seminormal if all its local rings are seminormal. If B is the normalisation of a ring A, the ring A is said to be seminormal if it contains each b ∈ B such that b2 , b3 ∈ A. The following general result is very useful for computing Picard groups of seminormal varieties. Proposition 2.2. [B5, Proposition 2.5] Let U be a seminormal variety (not necessarily projective) and W the non-normal locus of U , i.e. the scheme of non˜ → U be the normalisation of U and W ˜ the inverse normal points of U . Let π : U ˜ ˜ image of W in U . Define f1 : Pic U → Pic U ⊕ Pic W by f1 (L) = (π ∗ L, L|W ). ˜ ⊕ Pic W → Pic W ˜ by f2 (L1 , L2 ) = L ˜ ⊗ (π ∗ L2 )∗ , for Define f2 : Pic U 1|W ˜ , L2 ∈ Pic W . Then: L1 ∈ Pic U (1) There is an exact sequence f1 f2 ˜ ⊕ Pic W → ˜. Pic U → Pic U Pic W

˜ has p connected (2) Assume further that U is projective, U, W are connected and W components. Then one has an exact sequence f1 f2 ˜ ⊕ Pic W → ˜. 0 → ×p−1 Gm → Pic U → Pic U Pic W

We now apply Proposition 2.2 to compute the Picard groups of JY , the generalized Jacobian, and J Y , the compactified Jacobian, of a nodal curve Y with one node y. J Y is a seminormal projective variety containing the smooth subvariety JY .

PICARD GROUPS OF MODULI SPACES OF TORSIONFREE SHEAVES ON CURVES

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It has a natural desingularisation h : J˜Y → J Y . The map h induces an isomorphism of JY with h−1 (JY ), we identify JY with h−1 (JY ) ⊂ J˜Y . Let X be the normalisation of Y and x, z the points of X lying over y. The variety J˜Y ∼ = P(Px ⊕ Pz ) is a P1 -bundle over JX [AK], Px , Pz being line bundles on JX . Let p : J˜Y → JX be the canonical map. J˜Y − JY = D1 ∪ D2 , where Di are divisors such that p |Di : Di → JX are isomorphisms. h maps each of D1 and D2 isomorphically onto the smooth Weil divisor J Y − JY . One has ∗ Pic J˜Y ∼ = Pic JX ⊕ Z. = p (Pic JX ) ⊕ Z OP(Px ⊕Pz ) (1) ∼

We shall write the operation in Pic additively. Lemma 2.3. [B5, Lemma 3.1] One has Pic JY ∼ = ( Pic JX )/(Pz − Px ), where (Pz − Px ) denotes the subgroup of Pic JX generated by Pz − Px . Proof. Since J˜Y is smooth and JY is open in J˜Y the restriction map Pic J˜Y → Pic JY is surjective. Hence we have an exact sequence 1 → H → Pic J˜Y → Pic JY → 0. Since J˜Y − JY is the union of two irreducible disjoint divisors D1 , D2 the kernel H of the restriction map is generated by O(D1 ) = O(1) − Px , O(D2 ) = O(1) − Pz in Pic JX ⊕ Z ∼ = Pic J˜Y . Let pZ denote the projection to the Z-factor. Then pZ (aO(D1 ) + bO(D2 )) = 0 if and only if a + b = 0. Hence Ker (pZ |H ) is the subgroup generated by (Px − Pz ) in Pic JX . Also pZ |H is surjective (with each generator O(Di ) mapping to 1 ∈ Z). It follows that H∼ = (Px − Pz ) ⊕ Z and

Pic JY ∼ = Pic JX /(Px − Pz ).

This completes the proof of the lemma.

For u ∈ X let tu denote translation by OX (u) in JX . Denote by HJ the subgroup of Pic JX ⊕ Z defined by HJ = {(L, m) | L ∈ Pic JX , m ∈ Z, t∗z−x L − L = m(Px − Pz )}. Proposition 2.4. [B5, Proposition 3.2] There is an exact sequence 1 → Gm → Pic J Y → HJ → 0. Proof. We sketch the idea of the proof. J Y is a seminormal variety with J Y − JY its non-normal locus. By Proposition 2.2, we have an exact sequence f1 f2 1 → Gm → Pic J Y → Pic J˜Y ⊕ Pic (J Y − JY ) → Pic D1 ⊕ Pic D2 .

We have

∗ Pic J˜Y = p (Pic JX ) ⊕ ZO(1),

Pic D1 = (p |D1 )∗ Pic JX , Pic D2 = (p |D2 )∗ Pic JX . The map p ∗ is an injection and (p |D1 )∗ , (p |D2 )∗ are both isomorphisms. We show that Ker f2 is isomorphic to the subgroup HJ of Pic JX ⊕ Z.

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Remark 2.5. The connected component Pic0 J Y of Pic J Y containing the identity is isomorphic to JY (Theorem 2.1; [EGK]). Under these identifications, the restriction to Pic0 J Y of the sequence in the statement of Proposition 2.4 is the exact sequence 1 → Gm → JY → JX → 0, where the map JY → JX is the pull-back π ∗ under the normalisation map π. 3. Moduli of vector bundles of rank ≥ 2 Let X be a nonsingular, projective, complex curve of genus g ≥ 2, n, d integers, n ≥ 2. Let UL (n, d) ⊂ UX (n, d) be the subvariety consisting of vector bundles with fixed determinant L. The first result on the Picard groups of moduli spaces of vector bundles of rank ≥ 2 is due to Seshadri. We sketch here a proof of this result given by Ramanan [R] following Seshadri’s ideas. Theorem 3.1. Let (n, d) = 1. Then Pic UL (n, d) ∼ = Z. Proof. (Proof sketch following the proof of [R, Proposition 3.4].) Take F ∈ UL (n, d) and E ∈ UL (n − 1, d ), d an integer. Choose a line bundle M of sufficiently high degree such that there is an injective homomorphism i : E → F ⊗ M , M satisfying certain properties [R, Lemma 3.1]. Note that F ⊗ M/i(E) ∼ = M n ⊗ L ⊗ L−1 . Then Y := PH 1 (X, M −n ⊗ L ⊗ L−1 ⊗ E) parametrizes a family of vector bundles of rank n, determinant L and F occurs in this family. Let Y s ⊂ Y be the subset corresponding to stable bundles in this family, then F ∈ Y s . By the universal property of moduli spaces, there is a morphism λ : Y s → UL (n, d). Since Pic Y → Pic Y s is surjective, rank Pic Y s ≤ 1. Let F be the universal bundle on UL (n, d) × X. Define V = pUL ∗ (Hom(p∗X (E), F ⊗ p∗X (M )) and let Z ⊂ P(V ) be the subset consisting of injective homomorphisms. Then there is an isomorphism Y s → Z and λ factors through Z. Hence ∼ Pic Y s . Pic UL (n, d) → Pic Z = One shows that P(V ) − Z is irreducible [R, Lemma 3.5], so that rank Pic Z + 1 ≥ rank Pic P(V ) = rank Pic UL (n, d) + 1, i.e. rank Pic UL (n, d) ≤ rank Pic Z = rank Pic Y s ≤ 1. Since UL (n, d) is projective, rank Pic UL (n, d) ≥ 1. Hence rank Pic UL (n, d) = rank Pic Z = rank Pic Y s = 1. Then it follows that Pic UL (n, d) ∼ = Pic Z ∼ = Z.

This result was extended to the noncoprime case by Drezet and Narasimhan [DN]. Theorem 3.2. [DN, Theorems A,B and C]. Let X be a smooth, projective curve. (1) Pic UL (n, d) ∼ = Z.


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d (2) Pic UX (n, d) ∼ ⊕ ZLX , where LX is the determinant line bundle = Pic JX on UX (r, d). (3) UX (n, d) and UL (n, d) are locally factorial.

We shall give the proof of Theorem 3.2 as a part of the proof of a more general result on nodal curves. More generally, we have the following result on any irreducible reduced projective curve. Theorem 3.3. [B3, Proposition 2.3]. Let Y be any irreducible projective curve of arithmetic genus gY ≥ 2. Let n ≥ 2 be an integer. Let ULs (n, d) denote the moduli space of stable vector bundles on Y of rank n and determinant isomorphic to a fixed line bundle L. Then Pic ULs (n, d) ∼ = Z or Z/mZ.

Proof. The proof is on similar lines to that of [DN] in the case of a nonsingular curve, one needs some modifications in the singular case. We briefly sketch the proof. By tensoring by a line bundle, we may assume that d >> 0. If E is a semistable vector bundle of high degree and determinant L, then E is globally generated and is given by an exact sequence 0 → OYn−1 → E → L → 0. Let P = P(H 1 (L∗ ⊗ Cn−1 )). There is a family E of vector bundles of rank n, degree d on P × Y . Let Ps = {p ∈ P | Ep stable}. By the universal property of moduli spaces, there is a canonical surjective morphism fE,L : Ps → ULs (n, d). One shows that the induced map Pic ULs (n, d) → Pic Ps (∼ = Z or Z/mZ) is an injection by using an alternative construction of Ps and the construction of the moduli space as follows. The moduli space UYs (n, d) is a GIT quotient of an irreducible nonsingular open subset Rs of a suitable quot scheme by P GL(q) (see [N1], Remark, p.167, Chapter (F0 ) is a 5, §7). Let F0 be the universal quotient bundle on Rs × Y, F0 := pRs∗ vector bundle. Let Gr0 be the Grassmannian of subspaces V of dimension n − 1 of F0 . There exists a geometric quotient Γ0 = Gr0 /P GL(q) ([DN], 7.3.1). One has Pic Gr0 = Pic Rs ⊕ ZOGr0 (1) and hence Pic Γ0 = Pic UYs (n, d) ⊕ Z.OΓ0 (a), where a = gcd(n, d) and OΓ0 (a) is a line bundle on Γ0 ([DN], Proposition 7.2). Let Gr0 ⊂ Gr0 be the open subset corresponding to vector bundle injections OX ⊗V → F0,r , r ∈ Rs , V ⊂ H 0 (F)0,r and Gr0 /P GL(q) = Γ0 . Then Γ0 − Γ0 is an irreducible hypersurface [DN, Corollary 7.4]. Computing its ideal sheaf one shows that

0 → Pic UYs (n, d) → Pic Γ0 → Z/(d/a)Z → 0 is exact. Let UGr0 be the universal (relative) subbundle on Gr0 . Let T = Isom(Gr0 ⊗ Cn−1 , UGr0 ), T = T /GL(q) = P(T )/P GL(q). T is an open subset of T where T = P(Hom(Gr0 ⊗ Cn−1 , UGr0 ))/P GL(q), a locally trivial projective bundle over Γ0 . Since the ideal sheaf of T − T in T is OT (n − 1) ⊗ p∗Γ M , where M is a line bundle on Γ0 , one has 0

0 → Pic Γ0 → Pic T → Z/(n − 1)Z → 0

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USHA N. BHOSLE

exact ([DN], 7.8). Let TL be the inverse image (in T ) of UL (n, d) ⊂ UY (n, d). Then there is an exact sequence d 0 → Pic ULs (n, d) → Pic TL → Z/(n − 1) Z → 0. a The final observation is that TL ∼ = PsL ([DN], Proposition 7.9). 3.1. Notation. Henceforth Y denotes an irreducible reduced curve with ordinary nodes as its only singularities (unless otherwise stated) and p : X → Y its normalisation. Let g be the genus of the normalisation X and gY the arithmetic genus of Y . Let UY (n, d) (resp. UY (n, d)) be the moduli space of torsionfree semistable sheaves (resp. semistable vector bundles) of rank n and degree d on Y . Let UL (n, d) be the closed subset of UY (n, d) corresponding to vector bundles with fixed determinant L. UL (n, d), UY (n, d) are normal quasiprojective varieties. Let UYs (n, d) ⊂ UY (n, d) and ULs (n, d) ⊂ UL (n, d) be the open subsets corresponding to stable vector bundles. For a line bundle L on Y , let UL (n, d) be the closure of UL (n, d) in UY (n, d). Theorem 3.4. [B3, Theorem I] Assume that g ≥ 2 and if g = n = 2 then d is odd. Then one has the following. (1) Pic ULs (n, d) ∼ = Z. (2) Pic UL (n, d) ∼ =Z. Proof. In view of Theorem 3.3, to prove Part (1) we need only to show that Pic ULs (n, d) has rank ≥ 1. Suppose that Y is nonsingular. Then UL (n, d) is projective, hence its Picard group has rank at least 1. Under the assumptions of the theorem, one has codimUL (n,d) (UL (n, d) − ULs (n, d)) ≥ 2. Since UL (n, d) is normal, this implies that Pic UL (n, d) ⊂ Pic ULs (n, d) and hence the theorem follows from Theorem 3.3. This is essentially the proof of [DN] in the nonsingular case. If Y is nodal, UL (n, d) is not projective, so one has to work harder. Moreover, it is impossible to do the above codimension computations on Y (since tensor products of semistable torsionfree sheaves on Y are not semistable, etc.). Hence we use the normalisation ML (n, d) of UL (n, d) [B1]. ML (n, d) is the moduli space of generalized parabolic bundles of rank n, degree d and determinant isomorphic to L = p∗ (L) on the normalisation X of Y . There is a rational map ML (n, d) → UX,L (n, d), it induces an isomorphism MLs (n, d) ∼ = ULs (n, d). Under the assumptions of the theorem, it is proved that the generator of Pic UX,L (n, d) gives a nontorsion element in Pic MLs (n, d) and hence in Pic ULs (n, d)). Thus Pic ULs (n, d) has rank ≥ 1 [B3, Proposition 2.2]. This proves Part (1). Now if we can show that (3.1)

codimUL (n,d) (UL (n, d) − ULs (n, d)) ≥ 2,

the normality of UL (n, d) will imply that Pic UL (n, d) ⊂ Pic ULs (n, d) and hence Pic UL (n, d) has rank 1, proving Part 2 (in fact the restriction map Pic UL (n, d) → Pic ULs (n, d) can be shown to be an isomorphism [B3, Proposition 3.6]). Using s ML (n, d) and the fact that codimUX,L (n,d) (UX,L (n, d) − UX,L (n, d)) ≥ 2 for g ≥ 2 (except for g = n = 2, d even), one can show that the inequality (3.1) holds under these conditions ([B3], Corollary 1.7).


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Theorem 3.5. [B4, Theorem 3A] With the same assumptions as in Theorem 3.4, let JYd be the generalized Jacobian of Y of degree d i.e. the moduli variety of degree d line bundles on Y . Then (a) Pic UYs (n, d) ∼ = Pic JYd ⊕ Z, ∼ (b) Pic UY (n, d) = Pic JYd ⊕ Z, (c) UY (n, d) and UL (n, d) are locally factorial. Proof. (a) and (b) are proved using the fact that the determinant map induces an injection on Picard groups and the fibres of the determinant map are isomorphic to UL (n, d). (c) Since ULs (n, d) is smooth, its Picard group and class group coincide. The normality of UL (n, d) and the inequality (3.1) imply that the class groups of UL (n, d) and ULs (n, d) are isomorphic. Moreover, Pic UL (n, d) → Pic ULs (n, d) is injective. Hence it is easy to see that to show that UL (n, d) is locally factorial, it suffices to show that the restriction map Pic UL (n, d) → Pic ULs (n, d) is surjective. This is proved using the fact that the determinant line bundle generates Pic ULs (n, d) and comes from Pic UL (n, d) ([B3], p. 262). It is easy to prove the local factoriality of UY (n, d) by similar arguments using (a) and (b). 4. Moduli of torsionfree sheaves on nodal curves. We start with the simple case of curves with lower genera. For low genera and ranks, there are explicit descriptions of moduli spaces of torsionfree sheaves. This makes the computation of their Picard groups much easier. 4.1. Case gY ≤ 2. . Any irreducible nonsingular (respectively nodal) curve Y with gY = 1 is a nonsingular (respectively nodal) Weierstrass curve. If Y is nonsingular, it is an elliptic curve. If Y is nodal, it has normalisation P1 and a unique singular point which is an ordinary node. If gcd(n, d) = h, the moduli space U (n, d) is isomorphic to S h (Y ), the hth symmetric product of Y (see [At], [Tu] for the nonsingular case; [BBH, Corollary 6.47], [HLST, Remark 1.27] for the singular case). In particular, for (n, d) = 1, UY (n, d) is isomorphic to Y . For Y nodal, UY (n, d) is isomorphic to the variety of nonsingular points of Y i.e. to the affine variety A1 − 0. Proposition 4.1. Let gY = 1. (i) If (n, d) = 1, then Pic UY (n, d) ∼ = Pic Y . If Y is singular, then Pic UY (n, d) ∼ = Gm ⊕ Z and Pic UY (n, d), Pic UL (n, d) are trivial. (ii) If h = (n, d) > 1 then Pic UY (n, d) ∼ = Pic S h (Y ). ∼ For L ∈ JY , Pic UL (2, 0) = Z. If Y is singular, then Pic UL (2, 0) and Pic UY (2, 0) are trivial. Proof. By the explicit description, Pic UY (n, d) ∼ = Pic S h (Y ). If (n, d) = 1, then UL (n, d) is a point, so its Picard group is trivial. If Y is nodal, then Pic Y ∼ = Gm ⊕ Z = Gm ⊕ Pic P1 ∼

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and since UY (n, d) ∼ = A1 − 0, it has a trivial Picard group. For the results in rank 2 note that for L = O, every E ∈ UL (2, 0) is S-equivalent to N ⊕ N ∗ for N ∈ J Y . Hence UL (2, 0) is isomorphic to the quotient of Y by Z/2Z which is isomorphic to P1 . Moreover, UL (2, 0) ∼ = A1 and UY (2, 0) is fibred over 1 ∼ JY = A − 0 with fibres UL (2, 0). For more details, see [B4, Proposition 2.3]. For gY = 2, Y is a hyperelliptic curve. If Y is singular, then Y has one or two nodes, the normalisation is an elliptic curve or a projective line. Suppose that d is odd. If Y is nonsingular, then UL (2, d) is isomorphic to a nonsingular intersection Q of two quadrics ([N], [NRa, Theorem 4]). If Y is nodal, then the study of the extension of the determinant map U (2, d) → JYd to UY (2, d) [B1] shows that the singular set UL (2, d) − UL (2, d) consists of direct images of stable vector bundles of rank 2 and a suitable fixed determinant on the (partial) normalisations of Y . Note that the (partial) normalisations are curves of arithmetic genus 1 and the projective line. By Grothendieck’s theorem, there are no stable vector bundles on a projective line. On a curve of arithmetic genus 1, there is a unique stable vector bundle of rank 2 and a fixed determinant of odd degree. It follows that UL (2, d) − UL (2, d) consists of one or two singular points according as Y has one or two nodes. If d is even, one has UL (2, d) = UL (2, d) ∼ = P3 and the subset UL (2, d)−ULs (2, d) is isomorphic to the Kummer variety in P3 corresponding to Q considered as a quadratic complex of lines in P3 ([NRa, Theorem 2], [B2, Corollary 3.5]). Lemma 4.2. The Kummer variety K is an irreducible quartic surface in P3 . Proof. The curve Y is associated to a pencil of quadrics in k6 with the following standard equation (k = C in our case, in general k can be an algebraically closed field of characteristic = 2). q1

=

I i=1

q2

=

I i=1

2Xi Yi +

Xj2

j≥I+1

(λi Xi Yi + Xi2 ) +

λj Xj2

j≥I+1

Here I = 0, 1 or 2 according as Y is nonsingular, has one node or two nodes. The intersection of quadrics can be considered as a quadratic complex of lines in P3 of type [111111], [21111], [2211] in the three cases. For the first type, the singular surface K is the classical Kummer surface, which is an irreducible quartic surface with 16 nodes. In the other two cases, the equation of K may be found, for instance, in Article 271 on p.211 and article 186 on p.219 of [J]; in both cases K is a quartic which is easily checked to be irreducible. Proposition 4.3. Let gY = 2. Then (i) Pic UL (2, d) = Pic UL (2, d) ∼ = Z for d even. (ii) Pic ULs (2, d) ∼ = Z/4Z for d even. (iii) Pic UL (2, d) = Pic ULs (2, d) ∼ = Z for d odd. (iv) Pic UY (2, d) ∼ = Pic JY ⊕ Z for d odd. (v) Pic UY (2, d) ∼ = Pic JY ⊕ Z for d even and Y nonsingular. Proof. (i) follows immediately from the fact UL (2, d) = UL (2, d) ∼ = P3 . 3 ∼ (ii) Since UL (2, d) = P is nonsingular there is a surjective homomorphism of Picard


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groups Pic UL (2, d) → Pic ULs (2, d). As the complement of ULs (2, d) in UL (2, d) is a divisor of the line bundle OP3 (4) (by Lemma 4.2), the kernel of the surjective homomorphism is 4Z. Hence Pic ULs (2, d) ∼ = Z/4Z for d even. (iii) In case Y is nonsingular, Q is a nonsingular complete intersection in P5 of dimension 3. By [H, Corollary 3.2, p.179], Pic Q ∼ = ZOQ (1), proving the result. Alternatively, the result also follows from Theorem 3.3 as ULs (2, d) = UL (2, d) is a projective variety. ˜L (2, d) → UL (2, d) be a normalisation. Since Suppose that Y is nodal. Let p : U UL (2, d) is normal (in fact nonsingular), p is an isomorphism over UL (2, d). As the singular set UL (2, d) − UL (2, d) consists of one or two points, it follows that ˜L (2, d) − p−1 UL (2, d) = codim UL (2, d) − UL (2, d) ≥ 3. codim U ˜L (2, d) is normal, this implies that Since U ˜L (2, d) → Pic p−1 UL (2, d) ∼ Pic U = Pic UL (2, d). ˜L (2, d) and hence Since UL (2, d) is projective, so is U ˜L (2, d) ≥ 1 . rank Pic UL (2, d) ≥ rank Pic U ∼ Z. Now it follows from Theorem 3.3 that Pic UL (2, d) = (iv) This follows from (iii) as in the following Part(v) (or as in [B4, Theorem 3A]). (v) The morphism det : UY (2, d) = UY (2, d) → JYd is a P3 fibration (not locally trivial). Hence for any line bundle M on JYd , one has det∗ (det∗ M ) = M , so that det∗ is injective on line bundles. Let N be a line bundle on UY (2, d). The restriction of N to the fibre over L is O(mL ), mL ∈ Z. JYd being irreducible, mL = m must be constant. The allowable values of m form a non-trivial subgroup of Z, so restriction to a fibre determines a surjection Pic UY (2, d) → Z. By the seesaw theorem, the kernel of this surjection is isomorphic to Pic JYd . This completes the proof. 4.2. Case gY ≥ 3. . In the rest of this section, we assume that Y is a nodal curve of genus gY ≥ 3 with one node y and x, z points of the normalisation X lying over y. Our aim is to compute the Picard groups of UY (2, d), UL (2, d) and UL (2, d). For gY ≥ 3, the variety UY (2, d) is not normal, it is seminormal [NR]. Theorem 4.4. [B4, Proposition 2.8] For gY ≥ 3, one has: (1) ULs (2, d) ∼ = Z. (2) UL (2, d) ∼ = Z. ˜L → UL be a Proof. We write ULs = ULs (2, d), UL = UL (2, d). Let p : U normalisation, it is an isomorphism over UL (since UL is normal). Then codim ˜L is normal, this ˜L − p−1 U = codim UL − U ≥ 3 [B4, Lemma 2.5]. Since U U L L −1 ∼ ˜ implies that Pic UL → Pic (p UL ) = Pic UL . Since UL is projective, so is ˜L ) ≥ 1, rank (Pic U ) ≥ 1. Since U is normal and ˜L and hence rank (Pic U U L L codim (UL − ULs ) ≥ 3 [B4, Lemma 2.4], we have Pic UL → Pic ULs . Thus rank (Pic ULs ) ≥ 1. Now it follows from Theorem 3.3 that Pic ULs ∼ = Z and hence Pic UL ∼ = Z.

The variety UY (2, d) has a filtration U := UY (2, d) = W2 ⊃ W1 ⊃ W0

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USHA N. BHOSLE

where Wi are seminormal, the non-normal locus of Wi is Wi−1 for i = 1, 2 and W0 ∼ = UX (2, d − 2) is smooth. Let U = W2 − W1 , U1 = W1 − W0 , U0 = W0 be the strata (for more details, see [NR] or [B5, section 4.2]). Denote by U1,L (2, d) the subvariety of U1 (2, d) corresponding to torsionfree sheaves with a fixed (not locally free) determinant [B4, 2.10]. Theorem 4.5. [B4, Theorem 2 and Theorem 3B] Let gY ≥ 2 and let L be a torsionfree sheaf of rank 1 on Y which is not locally free. Then (a) Pic U1,L (2, d) ∼ = Z. Moreover, assume that d is odd if gY = 2. Then s (b) Pic U1,L (2, d) ∼ = Z, Pic U1s (2, d) ∼ = Pic JX ⊕ Z. ∼ (c) Pic U1 (2, d) = Pic JX ⊕ Z. (d) U1 (2, d) is locally factorial. Proof. This is proved as in the case of UL (2, d), U (2, d).

To compute the Picard group of U using our Proposition 2.2, we need to know the Picard group of the normalisation P of U and that of the non-normal locus W1 . The normalisation P was constructed by Narasimhan and Ramadas [NR] as the moduli space of GPS (generalized parabolic sheaves) on the normalisation X of Y . A GPS is a sheaf E, which is torsionfree outside {x, z} ⊂ X, together with a 2-dimensional quotient Q of Ex ⊕ Ez . A GPS is called a GPB (generalized parabolic bundle) if E is a vector bundle. The moduli space P is a normal projective (irreducible) variety with rational singularities. It has a filtration P = P 2 ⊃ P 1 ⊃ P 0. We have P 1 = D1 ∪ D2where D 1 ,D2 are irreducible and normal Cartier divisors [NR]. P 0 = (D1 ∩ D2 ) D1 (0) D2 (0) where Di (0) are closed subsets of Di , i = 1, 2. Let P = P − P 1 , P1 = P 1 − P 0 , P0 = P 0 . The normalisation map p : P → U maps P i onto Wi . Its restriction to P is an isomorphism onto U . Moreover, for each j, the restriction p |Dj : Dj → W1 is a normalisation map with D j = Dj − (D1 ∩ D2 ) − Dj (0) mapping isomorphically onto U1 = W1 − W0 . Theorem 4.6. [B5, Theorem I] For gX ≥ 2, rank 2 and odd degree d, one has the following. (1) P − P0 is nonsingular. (2) Pic (P − P0 ) ∼ = Pic UX (2, d) ⊕ Z. d (3) Pic P ∼ ⊕ Zp∗ LY , = Pic JX LY being the determinant line bundle on U and p : P → U the normalisation map. The divisor class group of P (and of P − P0 ) is isomorphic to Pic UX (2, d) ⊕ Z. Idea of Proof. Part(1) is proved by using the smooth determinant map [B1] detp : P − P0 → J˜Y and proving that all its fibres are smooth [B5, Proposition 4.6].


41 11

For (2), let E → UX (2, d) × X be the universal bundle. Define Ex = E|UX (2,d)×x , Ez = E|UX (2,d)×z and let pr : (G := Gr(Ex ⊕ Ez )) → UX (2, d) be the Grassmannian bundle of two dimensional quotients of Ex ⊕ Ez . Then G parametrizes a family of GPBs giving a morphism G s → P . One shows that it induces an isomorphism from a big open subset of G onto a big open subset of P − P0 , here big means the complement has codimension ≥ 2 [B5, Proposition 4.6]. The proof of (3) is far more complicated [B5, Proposition 4.14]. Since P is normal and P0 is of codimension ≥ 2, Pic P → Pic(P − P0 ), but computation of the image is hard. We show that d Pic Di ∼ ⊕Z∼ = Pic JX = Pic Di (0)

and study the restriction maps from Pic P to these schemes.

Using the fact that W1 is seminormal with normalisation D1 and non-normal locus W0 , by applying Proposition 2.2, we compute the Picard group of W1 (for odd degree d). We partially succeed in computing the Picard group of U similarly. Theorem 4.7. [B5, Theorem II]. ∼ Pic JX ⊕ Z. (1) Pic W0 = (2) Pic W1 is given by an exact sequence ⊕Z→0 1 → Gm → Pic W1 → HW 1 d where HW is isomorphic to the subgroup G of Pic JX defined by 1 d | (⊗OX (z − x))∗ L = L}. G = {L ∈ Pic JX

(3) One has an exact sequence 1 → Gm → Pic U

resW1

→

Pic W1 .

Proof. (1) is proved using the isomorphism W0 ∼ = UX (2, d − 2) [B5, Corollary 4.11]. (2) is proved in [B5, Proposition 4.15] using (1) and Proposition 2.2. (3) is deduced from Proposition 2.2 using (2) and a computation of Pic D1 [B5, Proposition 4.17]. Remark 4.8. Computing the image HU of Pic U in Pic W1 explicitly seems difficult as we do not have enough information about Pic P1 . One has HU = {B ∈ Pic W1 | p∗ (B) = (Det∗ L + m LY ) |P1 , L ∈ G, m ∈ Z}. Here LY = p∗ LY , LY being the determinant line bundle on Y . Det denotes the d determinant morphism P → JX which associates to a GPB (E, Q) the determinant of the underlying vector bundle E. Note that every GPS is S-equivalent to a GPB, so there is a well-defined morphism Det [Su].

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USHA N. BHOSLE

References Altman A., Kleiman S. The presentation functor and the compactified Jacobian, The Grothendieck Festschrift, Vol. I, Progr. Math., 86, Birkhauser Boston, (1990) 15–32. [At] Atiyah M. F. Vector bundles over an elliptic curve, Proc. London Math. Soc. 7 (1957), 412–452. [BBH] Bartocci, C., Bruzzo, U., Hernandez Ruiperez D. Fourier-Mukai and Nahm transforms in geometry and mathematical physics. Progress in Mathematics 276, Birkhauser (2009). [B1] Bhosle, Usha N. Generalised parabolic bundles and applications to torsion free sheaves on nodal curves, Arkiv f¨ or Matematik 30 (1992), No.2, 187–215. [B2] Bhosle Usha N. Vector bundles of rank two and degree zero on a nodal curve, Proc. Barcelona-Catania Conferences. 1994–1995. Lecture notes in pure and applied Mathematics, Vol.200, Marcel-Dekker (June 1998). [B3] Bhosle Usha N. Picard groups of the moduli spaces of vector bundles, Math. Ann. 314 (1999), 245–263. [B4] Bhosle Usha N. Picard groups of the moduli spaces of semistable sheaves, Proc. Indian Acad. Sci. (Math. Sci.) 114 (2004), no. 2, 107–122. [B5] Bhosle Usha N. Seminormal varieties, torsionfree sheaves, and Picard groups, Communications in Algebra, 36 (2008), no. 3, 821–841. [DN] Drezet J.-M., Narasimhan M.S. Groupe de Picard des varietes de modules de fibres semistables sur les courbes algebriques, Invent. Math. 97 (1989), 53–94. [EGK] Esteves E., Gagn´ e M. and Kleiman S. Autoduality of the compactified Jacobian, J. London Math. Soc. (2) 65 (2002), no. 3, 591–610. [H] Hartshorne R. Ample subvarieties of Algebraic varieties, Lecture Notes in Mathematics 156, Springer-Verlag (1970). [HLST] Hernandez Ruiperez D., Lopez Martin A.C., Sanchez Gomez D., Tejero Prieto C. Moduli spaces of semistable sheaves on singular genus one curves, arXiv:0806.2034v2. [J] Jessop C.M. A treatise on the line complex, Cambridge University Press, Cambridge, (1903). [L] Lang S. Abelian varieties Interscience Tract 7, John Wiley, New York, (1959). [M] Mumford D. Geometric invariant theory, Ergebnisse der Mathematik 34, Springer, Berlin (1965). [NR] Narasimhan M.S., Ramadas T. Factorisation of generalised theta functions I, Invent. Math. 114 (1993), 565–623. [NRa] Narasimhan M.S., Ramanan S. Moduli of vector bundles on a compact Riemann surface, Ann. Math. 89 (1969), no. 1, 19–51. [N] Newstead P.E. Stable bundles of rank 2 and odd degree over a curve of genus 2, Topology 7 (1968), 205–215. [N1] Newstead P.E. Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics, Springer-Verlag (1978). [R] Ramanan S. The moduli spaces of vector bundles over an algebraic curve, Math. Ann. 200 (1973), 69–84. [Su] Sun X. Degeneration of moduli spaces and generalized theta functions, J. Algebraic Geom. 9 (2000), No. 3, 459–527. [Tu] Tu L. Semistable vector bundles over elliptic curves, Adv. Math. 98 (1993), 1–26. [AK]

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India E-mail address: [email protected]


On the Moduli of Orthogonal Bundles on a Nodal Hyperelliptic Curve Usha N. Bhosle Dedicated to Prof. Ramanan on his 70th birthday.

Abstract. Let X be a complete irreducible hyperelliptic curve of arithmetic genus g with an ordinary node as its only singularity. We find explicit descriptions of the moduli spaces of rank 4 orthogonal bundles with a Z/2Z-action (and of a certain fixed topological type) on X in terms of spaces associated to the singular pencil of quadrics determined by X.

1. Introduction The moduli spaces U2L of semistable vector bundles of rank 2 and a fixed determinant L of degree d on a smooth hyperelliptic curve X of genus g have beautiful explicit descriptions. P.E. Newstead showed that, for g = 2 and d odd, the moduli space U2L is isomorphic to a nonsingular intersection Q of two quadrics in P5C ([N], see also [NRa, Theorem 4]). Narasimhan and Ramanan proved that, for g = 2 and d even, U2L is isomorphic to P3C = PH 0 (J, θ 2 ) where θ 2 denotes the square of the theta line bundle on the Jacobian J of X. In my maiden work with Ramanan, we generalized these results to smooth hyperelliptic curves of any genus g ≥ 2 defined over an algebraically closed field of characteristic = 2 [DR]. Generalizations to characteristic 2 were carried out by me later ([B3], [B5]). It was soon realized that the explicit descriptions for U2L in fact follow from the explicit descriptions for the moduli spaces of rank 4 orthogonal bundles with Z/2Z-action of certain types ([R], [B1]). A generalization of the results of [B1] to a hyperelliptic curve with one ordinary node was carried out in [B4]. The aim in this paper is to generalize the results in [R] to a hyperelliptic curve with one ordinary node. Let k be an algebraically closed field of characteristic 0. Let X be a complete irreducible reduced hyperelliptic curve of arithmetic genus g ≥ 2 over k with a single ordinary node as its only singularity. To such a curve we associate a singular 2010 Mathematics Subject Classification. 14H60, 14D20, 14F05. Key words and phrases. Pencils of quadrics, nodal curves, torsionfree sheaves, moduli spaces. The initial part of this work was done during my visits to Liverpool University. I would like to thank P.E. Newstead for numerous discussions and the Liverpool University for hospitality. c Mathematical 0000 (copyright Society holder) c 2010 American

1 43

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USHA N. BHOSLE

pencil of quadrics with Segre symbol [21 · · · 1] given by q1 =

2g i=1

Xi2 + 2X0 Y0 , q2 =

2g

ai Xi2 + (X02 + 2a0 X0 Y0 ),

i=1

with ai distinct scalars. Let R be the scheme of (g − 1) -dimensional subspaces of k2g+2 which are isotropic for this pencil. Let R0 be the subscheme of R consisting of those subspaces which contain the unique singular point of the intersection of quadrics of the pencil. Our main result is the following. Theorem 1.1. The moduli space M of semistable orthogonal bundles of rank 4 with a Z/2Z -action (and of a certain fixed topological type) on X is isomorphic to the quotient of R − R0 by (Z/2Z)2g . In section 2, we study torsionfree sheaves with Z/2Z-actions on X. The pencil of quadrics associated to X is described and some results related to it are given in section 3. The main theorem is proved in section 4. 2. Torsionfree sheaves with Z/2Z-action Let k be an algebraically closed field of characteristic 0. Let X be an irreducible complete reduced curve over k with a single node as the only singularity. Let g denote the arithmetic genus of X. It is well known that X is Gorenstein and hence has a locally free dualising sheaf ωX of rank 1 and degree 2g − 2, where deg(.) = χ(.) − χ(OX ). → X be the normalisation map. Let π : X Lemma 2.1. Let E be a torsionfree coherent sheaf on X. (a) If E is semistable (respectively stable and of rank ≥ 2) and χ(E) > (rank E)(g − 1) (respectively ≥), then H 1 (E) = 0. (b) If E is semistable (respectively stable and of rank ≥ 2) and χ(E) < (rank E)(1 − g) (respectively ≤), then H 0 (E) = 0. Proof. (a) Suppose that E is semistable with χ(E) > (rank E)(g − 1) and H 1 (E) = 0 i.e. H 0 (Hom(E, ωX )) = 0. Then there exists a nonzero homomorphism E → ωX . Since E is semistable, χ(ωX ) ≥ χ(E)/rank E, i.e. χ(E) ≤ (rank E)(g − 1), a contradiction. (b) Suppose that E is semistable with χ(E) < (rank E)(1 − g). If H 0 (E) = 0, then there exists a nonzero homomorphism OX → E. The semistability of E implies that χ(E)/ rank E ≥ χ(OX ) = 1 − g, giving a contradiction. In the case E is stable, the result follows similarly. 2.1. Henceforth we assume that X is hyperelliptic, i.e. there exists a finite degree two map p : X → P1 . Let i denote the hyperelliptic involution on X. The unique node w0 of X must be i-invariant and hence it is a ramification point of p. Let W0 = {w0 , w1 , · · · , w2g } be the set of ramification points of X. Let OX (1) denote the line bundle p∗ OP1 (1). We define an i-action on a torsionfree coherent sheaf E on X as a Z/2Z-action on E which lifts the involution i. We call E i-invariant if E ≈ i∗ E.

ORTHOGONAL BUNDLES ON A NODAL CURVE

45 3

Lemma 2.2. (a) There is a one-one correspondence between i-invariant line bundles of even degree (respectively odd degree) d and the set of partitions of W0 = {w0 , w1 , · · · , w2g } into two subsets T1 , T2 such that the sets T1 ∩ {w1 , · · · , w2g } and T2 ∩ {w1 , · · · , w2g } have even cardinality (respectively odd cardinality). (b) There is a 1 − 1 correspondence between i-invariant torsionfree sheaves of rank 1 and even degree (respectively odd degree) which are not locally free and the partitions of W0 − w0 = {w1 , · · · , w2g } into two subsets T1 , T2 of odd (respectively even) cardinality. Proof. This lemma can be proved as in the case when X is nonsingular [DR], so we omit the details. (a) The correspondence is given by ⎧ d 1 ⎪ OX (w) ⊗ OX (− #T ⎪ 2 + 2) ⎨ 1 L = w∈T 1 ⎪ OX (w) ⊗ OX ( d+1−#T )⊗M ⎪ 2 ⎩

if w0 ∈ T1 if w0 ∈ T1 ,

w∈T1 −w0

where M is the unique square root of OX which becomes trivial on pulling The set T1 is characterised as the set of back to the normalisation X. points in W0 such that i acts on the fibre of L at w by −Id. (b) Let T (X) be the set of torsionfree sheaves of rank 1 on X which are → T (X) defined by L → π∗ L is an not locally free. The map Pic X isomorphism; moreover L is i-invariant if and only if π∗ L is so. Thus are in bijective correspondence with i-invariants in i-invariants in Pic X T (X). Using this and the correspondence between i-invariants in Pic X and partitions of W0 − w0 , we get the result. In case deg L = 0, one has −1−#T1 OX (w) ⊗ OX ( ) ⊗ π∗ (OX ), L= 2 w∈T1

for T1 ⊂ {w1 , · · · , w2g }, #T1 odd. T1 is the set of points w in W0 − w0 such that i acts on the fibre of (π ∗ L/ torsion) at w by −Id. 2.2. The coordinate ring of an affine neighbourhood of w0 in X can be written as B = A[y]/(y 2 − a), where A is the coordinate ring of an affine neighbourhood of p(w0 ) and a ∈ A. The element y is a generator of the maximal ideal at w0 and defines a Cartier divisor Y such that OY is supported at w0 . Since w0 is a node, a ∈ m2 , m = maximal ideal at p(w0 ), and OY ≈ (OP1 )/m2 . Let I denote the ideal sheaf OX (−Y ); it has an i-action given by y → −y. The i-action on a torsionfree coherent sheaf F induces i-actions on its fibres over the ramification points, on its cohomology and on the Euler characteristic χ(F ). Let ( )− denote the space of anti-invariants for the i-action, i.e. the eigenspace corresponding to the eigenvalue (−1) for the i-action. We define rw0 (F ) := dim (F/IF )− and for w ∈ W0 − w0 , rw (F ) := dim (F ⊗ OX /mw )− , where mw is the maximal ideal at w. Clearly, rw , w = w0 , are additive for exact sequences of torsionfree coherent sheaves on X. [B4, Lemma 2.2] shows that this is true for rw0 also.

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Proposition 2.3. [B4, Proposition 2.3] Let F be a torsionfree coherent sheaf with an i-action on X. Then

2g 1 − χ(F ) = χ(F ) − (rank F )(g + 1) + rwi . 2 i=0

3. The singular pencil 3.1. The singular pencil in P2g+1 . Let w0 , w1 , · · · , w2g be distinct closed points in P1 . Consider the divisor W = 2w0 + w1 + · · · + w2g in P1 . We have OW = OP1 /m2w0 ⊕

2g

OP1 /mwi . The multiplication in OP1

i=1

induces quadratic maps Ai on OP1 /mwi and A0 on OP1 /m2w0 given by Ai (Xi ) = Xi2 , A0 (X0 , Y0 ) = (X02 , 2X0 Y0 ), with Xi ∈ OP1 /mwi , X0 ∈ OP1 /mw0 , Y0 ∈ mw0 /m2w0 . We interpret Ai to have values in (OP1 (2g + 1)) ⊗ OP1 /mwi and A0 to have values in OP1 (2g + 1) ⊗ OP1 /m2w0 . The Ai , i ≥ 0 combine to form a quadratic map A with values in OP1 (2g + 1) ⊗ OW . The canonical map Ψ : H 0 (OP1 (2g + 1)) → OP1 (2g + 1) ⊗ OW can easily be checked to be an isomorphism. Let e denote the evaluation map e : P1 × H 0 (OP1 (2g + 1)) → OP1 (2g + 1) and let ex denote the evaluation map H 0 (OP1 (2g + 1)) → (OP1 (2g + 1))x . We define a quadratic form Q on P1 × k2g+2 with values in OP1 (2g + 1) by Q = e ◦ Ψ−1 ◦ A, 2g where A = Ai . For every x in P1 , let Qx = ex ◦ Ψ−1 ◦ A be the ‘quadratic form i=0

at x’. Remark 3.1. Notice that for every x, the one-dimensional subspace D0 = {(0, Y0 , 0, · · · , 0) | Y0 ∈ k} is isotropic for Qx . The quadratic form Qx is nonsingular if x ∈ supp W , i.e. the forms Qwi , i = 0, · · · , 2g are the only singular members. In fact, Q is dual to the singular pencil (I) of quadratic forms with Segre symbol [211 · · · 1] given by (I)

q1 =

2g

Xi2 + 2X0 Y0 ,

i=1

q2 =

2g i=1

with distinct scalars ai .

ai Xi2 + (X02 + 2a0 X0 Y0 )


47 5

3.2. The variety S. Let S be the variety of (g − 1)-dimensional vector subspaces V0 of k2g+2 which are isotropic for the pencil (I). Let D (resp. D0 ) be the two-dimensional (respectively one-dimensional) subspace of k2g+2 with coordinates X0 , Y0 (respectively Y0 only ); thus D0 is the unique subspace of D isotropic for the pencil and determines the unique singular point of the intersection of quadrics of the pencil (I). In particular, if V0 ∈ S, then V0 ∩ D = {0} if and only if V0 ⊃ D0 . We have an orthogonal decomposition k2g+2 = D ⊕ C with respect to the pencil (I), {Xi }i>0 being the coordinates in C. The pencil (I) induces the dual pencil Q on (k2g+2 )∗ ≈ OW . Under the natural correspondence between Grassg−1 (k2g+2 ) and Grassg+3 (k2g+2 )∗ which maps a subspace V0 to the subspace V of linear forms on k2g+2 vanishing on V0 , the variety S corresponds to the variety R ⊂ Grassg+3 (k2g+2 )∗ consisting of subspaces V such that the orthogonal complement of V with respect to a general quadratic form in the pencil is an isotropic subspace of dimension g − 1. Using the isomorphism k2g+2 → (k2g+2 )∗ given by a generic member of the pencil, say q1 , V gets identified with the orthogonal complement of V0 with respect to q1 . Let S0 be the closed subvariety of S consisting of V0 such that V0 ⊃ D0 , or equivalently V0 ∩ D = {0}. Let R0 be the corresponding subvariety of R. Let ⊥ denote the orthogonal complement. Now V0 ∩ D = {0} if and only if V0⊥ + D⊥ = k2g+2 , i.e. V0⊥ + C = k2g+2 , i.e. dim V0⊥ ∩ C = g + 1. But g + 3 ≥ dim V0⊥ ∩ C ≥ dim V0⊥ − 2 = g + 1. Hence for V0 ⊃ D0 , dim V0⊥ ∩ C = g + 2 or g + 3. Now dim V0⊥ ∩ C = g + 3 ⇒ V0 ⊂ V0⊥ ⊂ C, contradicting the assumption V0 ⊃ D0 . Thus dim V0⊥ ∩ C = g + 2 for V0 ⊃ D0 . Hence, if V ∈ R, then V ∈ R0 if and only if dim V ∩ C = g + 2 and V ∈ R − R0 if 2g+2 and only if dim V ∩C . Also for i = 0, V0 ∩ Di = {0}.

= g + 1, i.e. V + C = k ⊥ 2g+2 Dj = k , i.e. Hence V0 + j=i

V +

Dj = k2g+2 , V + C = k2g+2

j=i

for V ∈ R − R0 . Lemma 3.2. The group G of transformations leaving the pencil (I) invariant is isomorphic to (Z/2Z)2g+1 . Proof. For i > 0, let Di denote the subspace of k2g+2 with coordinate Xi . Any transformation which keeps the pencil invariant must keep the spaces D and Di invariant as k2g+2 = D ⊕ ( i Di ) is a decomposition orthogonal with respect to the pencil. Hence G = Π2g i=0 Gi where G0 is the group of transformations which keep the restriction of the pencil to D invariant and for i ≥ 1, Gi denotes the group of transformations keeping the pencil restricted to Di invariant. Clearly, Gi ≈ Z/2Z for i > 0. It is easy to check that the only elements of GL(2) which keep both the forms X0 Y0 and 2a0 X0 Y0 + X02 invariant are ±Id, i.e. G0 ≈ Z/2Z. Remark 3.3. The group G acts on S (or R) with ±Id acting trivially, hence the group acting on S (or R) is P(G) ≈ (Z/2Z)2g . In view of Lemma 2.2, there is a 1-1 correspondence between elements of this group and elements of the set

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USHA N. BHOSLE

{i-invariant line bundles of a fixed even degree } ∪ {i-invariant line bundles of a fixed odd degree}. Proposition 3.4. Let M be a vector space, M = w Mw . Let Q1 = Qw w and Q2 = Qw be nondegenerate quadratic forms on M with Qw , Qw nondew

generate quadratic forms on Mw . Let Gk (respectively Gk ) ⊂ Grassn (M ) be the subvariety of V ⊂ M such that Q1 (respectively Q2 ) restricted to V has rank exactly k. If V is such that either V ∩ Mw = {0} or V → Mw is onto, then Gk and Gk intersect transversely at V . Proof. We just remark that the proof of [B1, Proposition 3.2] goes through in this more general set-up also. Corollary 3.5. R − R0 is nonsingular. Proof. In the above notations, take M = k2g+2 , Mw0 = D, Mwi = Di for i > 0. Then S = G0 ∩ G0 and, if V ∈ S − S0 , then V satisfies the conditions V ∩ Mw = 0 for all w. Proposition 3.4 implies that R − R0 ≈ S − S0 is nonsingular. 4. The moduli spaces We keep the notations of sections 2 and 3. Fix a line bundle α of degree 2g + 1 with i-action on X; note that α is a square root of OX (2g+1). Let F be a semistable orthogonal bundle of rank 4 with i-action. We regard F as a vector bundle with a nondegenerate quadratic form with values in OX . Let E = F ⊗ α, then E is a vector bundle of rank 4 and slope 2g + 1 with a nondegenerate quadratic form with values in OX (2g + 1). Our aim in this section is to prove the main theorem in the following form. Theorem 4.1 (Theorem 1.1). Let M be the moduli space of semistable vector bundles E of rank 4 and slope 2g + 1 on the nodal hyperelliptic curve X with a nondegenerate quadratic form with values in OX (2g + 1) and with an i-action such that rw (E) = 1 for w = w0 and rw0 (E) = 2. Then M is isomorphic to the quotient of R − R0 by (Z/2Z)2g . The proof of the theorem is on similar lines to those of the main theorems in [B1], [B2], so we omit some details of the proofs. Most of the modifications needed in the nodal curve case have been worked out in sections 2 and 3. In the case when the curve is nonsingular, every ramification point w determines (uniquely) a Cartier divisor. In the nodal case, we have to choose a suitable Cartier divisor Y at the node w0 with OY ≈ OP1 /m2w0 (section 2.1). We shall often use the same notation 2g wi ; thus W is for a ramification point in X and its image in P1 . Let W = Y + i=1

a Cartier divisor on X with i-action. Let E ∈ M. Recall that a superscript ‘−’ denotes the eigenspace corresponding to the eigenvalue (−1) for the i-action. Consider the evaluation map h : H 0 (E)− → (E ⊗ OW )− .


49 7

The kernel of this map is contained in H 0 (E(−W ))− . Since E(−W ) is semistable and χ(E(−W )) = (−4g), by Lemma 2.1(b) we have H 0 (E(−W )) = 0. Thus h is an injection. Recall that in Remark 3.1 we had the orthogonal decomposition 2g

k2g+2 = D ⊕ Di with respect to the pencil (I) as well as the dual pencil. i=1

The quadratic form on E induces a quadratic map q0 on (E ⊗ OY )− with values in OX (2g + 1) ⊗ OY ≈ OY . On D we have the quadratic map A0 with values in OP1 /m2w ≈ OY . Let n0 denote an isomorphism from (E ⊗ OY )− onto D which preserves q0 on the left and A0 on the right. Let For i > 0, let ni denote an − isomorphism from Ew to Di which preserves qwi on the left and Ai on the right. i 2g ni . The composite n ◦ h embeds H 0 (E)− in k2g+2 . Given (E, n), let Let n = i=0

f (E, n) denote the image of H 0 (E)− in k2g+2 under n ◦ h. Since χ(E) = 4(g + 2), by Lemma 2.1(a), h1 (E) = 0. As rw0 (E) = 2, rw (E) = 1, w = w0 , by Proposition 2.3, we have h0 (E)− = g + 3. It can be seen that E → f (E, n) modulo (Z/2Z)2g gives a morphism from the moduli space M to the quotient by (Z/2Z)2g of the Grassmannian of (g + 3) dimensional subspaces of k2g+2 . We shall see below that this morphism in fact maps into R − R0 /(Z/2Z)2g as f (E, n) ∈ R − R0 . Lemma 4.2. (a) H 0 (E(−x − ix))− is the orthogonal complement of H 0 (E)− with respect to Qx for all x in X − (w0 , · · · , w2g ). In particular, H 0 (E(−x − ix))− is totally isotropic and Qx has rank 4 on 0 H (E)− . Thus f (E, n) ∈ R. (b) f (E, n) ∈ R0 . Proof. (a) This follows as in [B1, Proposition 2.1 (b) and (c)], in fact the proof is slightly simpler in our case. In the notations of [B1], Cw = k2g+2 , h = O(1). 1 − In our case, H (E(−x − ix)) = 0 as χ(E(−1)) > 4(g − 1) (Lemma 2.1(a)), so h0 (E(−1))− can be computed using Proposition 2.3. (b) In view of section 3.2, it suffices to show that 2g

dim f (E, n) ∩ Di = g + 1. i=1

Now the L.H.S. is isomorphic to the space H 0 (E ⊗ mw0 )− , where mw0 denotes the ideal sheaf of the point w0 . Using the facts that χ(E ⊗ mw0 ) = 4g, rw (E ⊗ mw0 ) = rw (E) = 1, w = w0 , and rw0 (E ⊗ mw0 ) = 2, Lemma 2.1 and Proposition 2.3 give h0 (E ⊗ mw0 )− = g + 1. 4.1. The inverse of f . We now want to give the inverse of f . We define a morphism X × k2g+2 → OW by the conditions that it is zero outside W , its restriction to wi is given by mapping Di isomorphically onto k(wi ) and the restriction to w0 is given by mapping D isomorphically onto OY . Let K be the kernel of this morphism. Since W is a Cartier divisor and X is Cohen-Macaulay, K is locally free of rank 2g + 2. Since Q | K vanishes identically on W , it induces a quadratic form q on K with values in OX (2g + 1 − 2W ) = OX (−1) which is easily seen to be everywhere nondegenerate.

50 8

USHA N. BHOSLE

Now take V ∈ R − R0 . As seen in section 3.2, we have V + Dj = k2g+2 , i = 0, 1, · · · , 2g. j=i

This implies that the composite X × V → X × k2g+2 → OW is a surjection. The kernel V of this composite is a vector subbundle of K fitting in the following commutative diagram: 0

→

K

→

X × k2g+2

↑ 0

→

V

→ OW

↑ →

→

0

→

0.

X ×V

→ OW

The orthogonal complement V of V in K is a vector bundle of rank g − 1. Since V ≈ V generically, the rank of the induced form q on V is ≤ 4 at any point x. On the other hand dim Vx ∩ Vx ≤ g − 1 at every point x, so that the rank of q | Vx ≥ 4. Thus q has constant rank 4 on V , V = V ∩ V and F = V /V has a nondegenerate quadratic form with values in OX (−1). Define E = F (g + 1); thus E is a vector bundle of rank 4 with a nondegenerate quadratic form with values in OX (2g + 1). The isomorphisms (ni ) are obtained from the fact that Vw i contains a copy of Di for all i = 0 and Vw 0 contains a copy of D. We shall show below that E is semistable. Define f (V ) = (E, n). Lemma 4.3. Let E be as in section 4.1. Then H 1 (E)− = 0. Proof. Using Serre duality and the fact that E ≈ E ∗ (2g + 1), we have h1 (E)− = h0 (E ∗ (−2g − 1) ⊗ ωX )− = h0 (F (−g) ⊗ ωX )− . Since i acts on ωX by −Id at all w, there is an isomorphism ωX →OX (g − 1) not compatible with the i-action. Consequently, h0 (F (−g) ⊗ ωX )− = h0 (F (−1))+ and it suffices to show that h0 (F (−1))+ = 0. Now, H 0 (F (−1))+ ⊂ H 0 (F )+ and we shall in fact show that h0 (F )+ = 0. From the cohomology exact sequence associated with the exact sequence 0 → V → V → F → 0, we have h0 (F )+ = 0 if h0 (V )+ = 0 = h1 (V )+ . One has H 0 (V ) ⊂ H 0 (K) = 0 as H 0 (X × k2g+2 ) ≈ H 0 (OW ) implies that H 0 (K) = 0. Then h0 (V ) = 0 = h0 (V ). As deg F = −4 and deg V = −(2g + 2), one has deg V = −2g + 2 and r(V ) = g − 1. Hence by the Riemann-Roch theorem, h1 (V ) = −χ(V ) = g 2 − 1. Now h1 (V )+ = −χ(V )+ = 0 if and only if χ(V )− = χ(V ) = 1 − g 2 . Note that Vw− = ker(Vw → (X × V )w ) has a nondegenerate quadratic form at w while Vw is isotropic for all w. Hence rw (V ) = 0 for all w. Then Proposition 2.3 gives χ(V )− = 1 − g 2 . This proves the lemma. Lemma 4.4. There is a canonical isomorphism of V onto H 0 (E)− such that the following diagram commutes: V

→

↓ H 0 (E)−

k2g+2 || n

→

(E ⊗ OW )− .

Proof. This follows as in [B1, Lemma 2.4].


51 9

Lemma 4.5. Let E be as in section 4.1. Then F = E ⊗ α−1 is a semistable orthogonal bundle with i-action. Proof. Let L0 ⊂ F be an i-invariant isotropic subsheaf with a torsionfree quotient, L = L0 ⊗ α ⊂ E. We identify (E ⊗ OW )− with k2g+2 by n. Since L is − − isotropic, L− wi ⊂ Di , i = 0, and Im (L ⊗ OY ) ⊂ D are isotropic. Hence Lwi = 0 − − for i = 0 and Im(L ⊗ OY ) ⊂ D0 . Thus Im (L ⊗ OW ) ⊂ D0 . Now, H 0 (L)− ⊂ Im (L ⊗ OW )− ∩ H 0 (E)− ⊂ D0 ∩ H 0 (E)− ⊂ D0 . Hence h0 (L)− ≤ 1. If D0 ∩ H 0 (E)− = {0}, then H 0 (L)− = 0. Since H 0 (L⊥ )− ⊂ H 0 (E)− and 1 h (E)− = 0 (Lemma 4.3), we have h0 (L⊥ )− ≤ χ(E)− . Hence h0 (L)− + h0 (L⊥ )− ≤ χ(E)− and so χ(L)− + χ(L⊥ )− ≤ χ(E)− . Using Proposition 2.3, this implies that d(L0 ) + d(L⊥ 0 ) ≤ d(F ). Suppose that D0 ∩ H 0 (E)− = {0}. If dim (L ⊗ OY )− = rw0 (L) = 0, then we still have H 0 (L)− = {0} and the previous argument goes through. Assume that rw0 (L) = 0. Since 0 → L⊥ → E → L∗ ⊗ OX (2g + 1) → 0 is exact, rw0 (L) + rw0 (L⊥ ) = rw0 (E) = 2 and, L ⊂ L⊥ being isotropic, one must have rw0 (L) = rw0 (L⊥ ) = 1, i.e. Im(L⊥ ⊗ OY )− = D0 . Then

H 0 (L⊥ )− ⊂ D0 ⊕ Di ∩ H 0 (E)− = D0 ⊕ Di ∩ V i>0

i>0

by Lemma 4.4. Since V ∈ R − R0 , we have V + D0 ⊕ = k2g+2 (section Di i>0

3.2). Hence

dim V ∩ D0 ⊕ = dim V − 1. Di i>0

We already have h (L) ≤ 1. Thus h (L)− + h0 (L⊥ )− ≤ dim V = χ(E)− , so that χ(L)− + χ(L⊥ )− ≤ χ(E)− in this case also and hence degree L0 + degree L⊥ 0 ≤ deg F . This proves that F is a semistable orthogonal bundle with i-action. 0

−

0

Corollary 4.6. The canonical map H 0 (E)− → (E ⊗ OW )− is an injection and f (E, n) = V. Thus f ◦ f = Id. Proof. This is obvious from Lemma 4.4.

Lemma 4.7. f ◦ f = Id. Proof. This follows from f ◦f = Id and the fact that two semistable quotients of the same rank and degree of the trivial (fixed rank) bundle, which are generically isomorphic, are isomorphic. This finishes the proof of Theorem 1.1.

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USHA N. BHOSLE

Remark 4.8. We can use Theorem 1.1 to construct a birational map between U2L and R exactly as in [R, § 6]. To prove an analogue of [R, Theorem 2, § 6], we need Lemma 2.2 and Remark 3.3 together with the fact that H 2 (X, O∗ ) = 0 for the nodal curve X too. Notice also that the orthogonal bundle with i-action associated with E is E ⊗ i∗ E in [R, § 6.3], the i-action being swiching factors E, i∗ E, and (E ⊗ i∗ E ⊗ OY )− ≈ (Λ2 (E) ⊗ OY )− ; hence rwo (E ⊗ i∗ E) = 2, rw (E ⊗ i∗ E) = 1 for w = w0 . Note that the tensor product of two stable vector bundles E and i∗ E on the nodal curve X may not be semistable. If the pull back E of E to the normalization ˜ is smooth). Since E ⊗ i∗ E is a vector ˜ is stable, then E ⊗ i∗ E is semistable (as X X ∗ bundle, the semistability of E ⊗ i E implies the semistablity of the generalized parabolic bundle corresponding to E ⊗ i∗ E, hence E ⊗ i∗ E is semistable. Thus there is a nonempty open subset U of the moduli space U2L such that for E ∈ U, E ⊗ i∗ E ∈ M. Hence we get only birationality rather than an isomorphism as in [R]. References Bhosle Usha N. Moduli of orthogonal and spin bundles over hyperelliptic curves, Compositio Math. 51 (1984), no.1, 15–40. [B2] Bhosle Usha N. Degenerate symplectic and orthogonal bundles on P 1 , Math. Ann. 267 (1984), no.3, 347–364. [B3] Bhosle Usha N. Pencils of quadrics and hyperelliptic curves in characteristic two, J. Reine Angew. Math. 407 (1990), 75–98. [B4] Bhosle Usha N. Vector bundles of rank 2, degree 0 on a nodal hyperelliptic curve, Algebraic geometry (Catania, 1993/Barcelona, 1994), 271–281, Lecture notes in Pure and Appl. Math., 200, Dekker, New York, 1998. [B5] Bhosle Usha N. Moduli of vector bundles in characteristic 2, Math. Nachr. 254/255 (2003), 11–26. [DR] Desale U.V. and Ramanan S. Classification of vector bundles of rank 2 on hyperelliptic curves, Invent. Math. 38 (1976/77), no.2, 161–185. [NRa] Narasimhan M.S., Ramanan S. Moduli of vector bundles on a compact Riemann surface, Ann. of Math. (2) 89 (1969), 14–51. [N] Newstead P.E. Stable bundles of rank 2 and odd degree over a curve of genus 2, Topology 7 (1968), 205–215. [R] Ramanan, S. Orthogonal and spin bundles over hyperelliptic curves, Geometry and analysis (papers dedicated to the memory of V.K. Patodi), pp. 151–166, Indian Acad. Sci., Bangalore, 1980. [B1]

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India E-mail address: [email protected]


Hilbert Schemes of Fat r-Planes and the Triviality of Chow Groups of Complete Intersections André Hirschowitz and Jaya NN Iyer Abstract. In this paper, we investigate the question of triviality of the rational Chow groups of complete intersections in projective spaces and obtain improved bounds for this triviality to hold. Along the way, we have to study the dimension and nonemptiness of some Hilbert schemes of fat r-planes contained in a complete intersection Y , generalizing well-known results on the Fano varieties of r-planes contained in Y .

1. Introduction 1.1. The triviality conjecture. The aim of this paper is to investigate the triviality of the low-dimensional rational Chow groups for certain projective varieties. If Y is a nonsingular complete intersection of multidegree (d1 , · · · , ds ) in a projective space Pn , and n is sufficiently large with respect to the degrees, it is known that, for small values of r, the rational Chow group QCHr (Y ) := CHr (Y )⊗Q is trivial, namely one-dimensional (generated by the linear sections). The precise conjectural bound on the multidegrees for the triviality follows from the study of the Hodge type of the complementary open variety Pn − Y initiated by Deligne [De] and followed by works of Deligne-Dimca [De-Di] and Esnault-Nori-Srinivas [Es], [EsNS]. A formulation of the conjectured bound was made in [Pa, Conjecture 1.9], which says: Conjecture 1.1. Suppose Y ⊂ Pn is a smooth complete intersection of multidegree (d1 , · · · , ds ), and let r be a nonnegative integer. If r(max1≤i≤s di ) +

s

di ≤ n,

i=1

then QCHr (Y ) is trivial. This conjecture includes the hypersurface case (s = 1) as well as the higherdimensional case s ≥ 2, and, as we will see, our contribution concerns mostly the latter. 2010 Mathematics Subject Classification. Primary 14C25, 14D07; Secondary 14D22,14F40. Key words and phrases. Complete intersections, Chow groups, Hilbert schemes. c Mathematical 0000 (copyright Society holder) c 2010 American

1 53

54 2

A. HIRSCHOWITZ AND J. NN IYER A.HIRSCHOWITZ N. IYER

1.2. Fat and strong planes. The central notions in our approach are those of fat and strong planes, which appear at least implicitly in [EsLV], and go back to Roitman for the 0-dimensional case. By a t-fat r-plane in a projective space, we mean the t-th infinitesimal neighborhood of an r-plane in an (r + 1)-plane. Given a subscheme Y in a projective space and a Cartier divisor Y in Y , we say that an r-plane L in Y is strong (with respect to Y ) if there exists an (r + 1)-plane L in Y containing L such that the set-theoretic intersection L ∩ Y is either L or L . The connection between the two notions is given by the following statement, proven in section 2 : Proposition 1.2. Suppose Y is in the linear system |OY (t)|. Then any strong r-plane in Y is the support of a t-fat r-plane contained in Y . Conversely, if furthermore Y is (set-theoretically) defined by equations of degree (strictly) less than t, then the support of any t-fat r-plane contained in Y is strong. 1.3. Roitman’s technique, small steps and big steps. The case of 0cycles has been handled by Roitman [Ro]. His method consists in starting from a (positive) 0-cycle Z on Y , and building a ruled cycle in Pn whose intersection with Y will be not too far from a multiple of Z. This is achieved by choosing a ruling by lines which are strong (see below), in the sense that either they cut Y in a single (multiple) point or they are inside Y . This method can be extended to the higher-dimensional case of r-cycles. Of course, the scope of this method is limited by the need for “sufficiently many strong (r + 1)-planes”. This approach has been successfully applied in [EsLV] through a single big step, showing that the restriction of (r+s)-cycles from Pn to Y is sufficiently surjective under the following numerical assumption (at least for degrees at least 3, the assumption being different when all degrees are equal to 2): s di + r n≥ . r+1 i=1 The geometric meaning of their numerical condition is the rational-connectedness of the variety of r-planes in Y . In the present work, we apply Roitman’s technique through smaller steps, typically showing that the restriction of (r+1)-cycles on a suitable complete intersection of multidegree (d1 , · · · , ds−1 ) to Y is sufficiently surjective. The analysis of small steps being somewhat simpler, we succeed in applying Roitman’s technique to small steps in essentially the whole expected range, relaxing in particular the rationalconnectedness assumption. For instance for 5-cycles on complete intersections of type (20, 30), the rational-connectedness condition requires n ≥ 1800000 while we take care of all cases with n ≥ 370000. 1.4. Our small step theorem. Our first main result reads as follows: Theorem 1.3. If, in the subvariety Y ⊂ Pn , the Cartier divisor Y ∈ |OY (t)| is covered by strong r-planes, then the restriction map QCHr+1 (Y ) → QCHr (Y ) is onto. This implies in particular that whenever QCHr+1 (Y ) is trivial, so is QCHr (Y ). Our proof of Theorem 1.3 follows the corresponding proof for the big step in [EsLV], with a single but decisive technical improvement: we introduce a different (s) filtration of the Chow group CHr , where CHr is generated by subvarieties covered

HILBERT SCHEMES OF FAT r-PLANES

55 3

by strong s-planes. We did not explore yet whether such a new filtration could also improve the bound for the big step. In order to apply the above result, we need to find the appropriate condition on the degrees for our complete intersection Y to be covered by strong r-planes. 1.5. Covering by strong planes. Thus we are led to search for the numerical condition for at least the generic complete intersection of multidegree (d1 , · · · , ds ) to be covered by strong r-planes. We recall that the strongness property is with respect to the pair (Y, Y ). We say that an s-codimensional subvariety in a projective space has type (d1 , · · · , ds ) when it is a union of irreducible components of a complete intersection of multidegree (d1 , · · · , ds ). Accordingly, we say that a pair Y ⊂ Y has type (d1 , · · · , ds ) if Y has type (d1 , · · · , ds−1 ) and Y is a divisor of degree ds in Y . In §6, we prove Proposition 1.4. Let n, r, s, d1 , · · · , ds be integers satisfying r ≥ 0, s ≥ 1, n ≥ r + s, 2 ≤ d1 ≤ · · · ≤ ds−1 < ds and the (“expected”) inequality

ρ + r ≥ n − s, s di + r + 1 ρ := (r + 2)(n − r) − r+1 i=1

where

is the dimension of the variety of ds -fat r-planes in the general complete intersection of type (d1 , · · · , ds ) (see §4). If Y ⊂ Y is any pair of type (d1 , · · · , ds ) in Pn , then Y is covered by strong r-planes. Note that the intended meaning of the “expected” inequality relates the dimension of the universal ds -fat r-plane with the dimension of our complete intersection Y. Also note the strict inequality ds−1 < ds . Apart from this restriction, our result is the expected one. The discarded case would involve a refined analysis (this is where we do not cover the whole range of Roitman’s method for small steps). Our proof of Proposition 1.4 relies on the study of the Hilbert schemes of fat rplanes contained in a general complete intersection. We show in §5 that they have the expected dimension; but we need a more accurate result saying that, when this expected dimension is nonnegative, these Hilbert schemes are nonempty. We conjecture that this is true in most cases (despite the notable exception of double lines on quadric surfaces), and prove it in the case we need for our application to Chow groups. For such a result, as illustrated in [De-Ma], two approaches are available: through intersection computations or through maximal rank problems. We follow the latter approach, using a method that can be tracked back at least to [EH, EHM]. 1.6. The main theorem. Combining the previous results, we obtain our main result: Theorem 1.5. Let n, r, s, d1 ≤ · · · ≤ ds−1 < ds be integers as above, satisfying ρ + r ≥ n − s.

If Y ⊂ Y is any pair of type (d1 , · · · , ds ) in Pn , and if QCHr+1 (Y ) is trivial, then so is QCHr (Y ).

56 4


This theorem may be applied recursively. For instance in the case of codimension two complete intersections (s = 2), our assumption for triviality reads, for d1 < d2 : d

1 +r+1

n ≥ max

r+1

+

d2 +r+1 r+1

r+1

+ r2 + r − 2

d1 +r+2 ,

r+2

+ r 2 + 3r

r+2

.

In order to compare this new bound with [EsLV]’s d2 + r d1 + r + , n≥ r+1 r+1 we fix r and d1 and let d2 vary. For sufficiently large values of d2 , the new bound dr+1 2 , while, if one uses [EsLV], the best value of n is can be estimated as (r+1)(r+1)! dr+1

2 . Hence in this context of a large d2 , we roughly divide by r + 1 estimated as (r+1)! the range where the conjecture is still open. On the other hand, it can be checked that, if d2 is not sufficiently large, [EsLV]’s bound remains the best (for instance fix d2 = d1 + 1 and let d1 go to infinity). In the higher-codimensional cases, a similar picture will occur, namely our result will provide an improved bound only for sufficiently large values of ds . Furthermore, in many cases, the best bound will be obtained by combining one or more of our small steps with a big step from [EsLV].

1.7. The case of hypersurfaces. The case of hypersurfaces (s = 1) has been considered in the first place. Concerning a general cubic hypersurface Y ⊂ Pn , C. Schoen [Sc] showed the triviality QCH1 (Y ) Q when n ≥ 7 and Paranjape [Pa] obtained the sharp bound in this case showing the triviality of 1-cycles when n ≥ 6 (in the same paper, he gave the first finite bound for general complete intersections). For hypersurfaces of the general degree d, the best known bound has been obtained “in the margin” by J. Lewis [Le] (added in proof at the very end of the paper). There, the statement concerns only the generic hypersurface, and the bound occurs as the condition for the so-called cylinder homomorphism to be surjective. This bound by Lewis is better than the bound obtained later (for the hypersurface case) in [EsLV]. It was rediscovered by A. Otwinowska [Ot]: there the statement concerns all smooth hypersurfaces, and the geometric meaning of the bound is that the hypersurfaces of degree d in Pn+1 (not Pn !) are covered by (r + 1)-planes. Surprisingly, our small step gives exactly the same bound, with a third geometric meaning for the condition, namely that the hypersurfaces are covered by d-fat rplanes. Furthermore, our statement concerns all hypersurfaces, not only smooth ones. 1.8. The base field. We work over an algebraically closed field of characteristic zero. The closedness assumption could be removed, thanks to the fact that the kernel of CHr (Yk ) → CHr (Yk¯ ) is torsion [Bl], while the characteristic zero assumption is used in the proofs of §4. Acknowledgements: It is a pleasure to dedicate this paper in honour of S. Ramanan. Both authors have experienced very fruitful mathematical interaction with him, and take the opportunity to acknowledge his deep influence. This work was initiated during the second author’s stay at MPI, Bonn in 2003 and partly done during her visits to Nice in Dec 2004 and to IAS, Princeton in 2007. The support


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and hospitality of these institutions is gratefully acknowledged. We also thank H. Esnault, J. Lewis and M. Levine for their feedback at some point during the course of this work.

2. Strong planes Throughout this section, we consider a subvariety Y in a projective space equipped with a Cartier divisor Y , and we fix an integer r. We are interested in the restriction map QCHr+1 (Y ) → QCHr (Y ), where we write QCHr (W ) for the rational Chow group of r-dimensional cycles on W . Recall that an r-plane L in Y is said to be strong (with respect to Y ) if there exists an (r + 1)-plane L in Y containing L such that the set-theoretic intersection L ∩ Y is either L or L . In this section, we prove our first main result : Theorem 2.1. If, in the subvariety Y ⊂ Pn , the Cartier divisor Y ∈ |OY (d)| is covered by strong r-planes, then the restriction map QCHr+1 (Y ) → QCHr (Y ) is onto. For the proof, we generalize our notion of strongness and define a notion of strong s-plane in Y for s ≤ r + 1. A (r + 1)-plane H in Y is said to be strong (with respect to the pair (Y, Y )) if it is contained in Y , or if its set-theoretic intersection with Y is a r-plane. Then, for s ≤ r, a s-plane in Y is said to be r-strong, or simply strong (when r is clear from the context), if it is contained in a strong (r + 1)-plane. As usual, we say that a closed subvariety W of Y is spanned or covered by strong s-planes if it is a union of strong s-planes contained in W . Now we denote by QCHr(s) (Y ) the subgroup of QCHr (Y ) which is generated by r-dimensional subvarieties of Y which are spanned by strong s-planes. This is the place where our proof differs from the corresponding proof in [EsLV]. Note that any subvariety in Y is spanned at least by strong 0-planes: since Y is covered by strong r-planes, it is also covered by strong 0-planes. Thus we have QCH(0) r (Y ) = QCHr (Y ). For s ≥ 1, if Z is spanned by strong s-planes it is spanned by strong (s − 1)planes as well. Hence one has QCHr(s) (Y ) ⊆ QCH(s−1) (Y ). For s > r one has r QCHr(s) (Y ) = {0}. We prove by descending induction on s that QCHr(s) (Y ) is in the image of (Y ) is QCHr+1 (Y ). The initial case is with s := r + 1 and follows since QCH(r+1) r reduced to 0. Before stating the induction step as a lemma, we introduce the following notation. Let Γ ⊂ Y be an (r + 1)-dimensional closed subvariety or, more generally, an (r + 1)-cycle. By [Fu, 8.1], the intersection product Γ · Y is a class in CHr (|Γ| ∩ Y ). By abuse of notation we will also write Γ · Y for its image in QCHr (Y ). Lemma 2.2. Let s be an integer with 0 ≤ s ≤ r, and W an r-dimensional irreducible subvariety of Y , spanned by strong s-planes but not by strong (s + 1)planes. Then there exist an (r + 1)-dimensional cycle Γ in Y and a positive integer α with Γ · Y ≡ αW mod QCHr(s+1) (Y ).

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Proof. We start with the case s = r which means that W is a strong r-plane. This gives us a strong (r + 1)-plane in Y which we take for Γ. Indeed, we have Γ · Y = dW. Now we suppose s < r. In order to define Γ, we start by choosing carefully an algebraic family (Hz )z∈Z of strong s-planes covering W . Note that by our assumption on s, each strong s-plane in Y is contained in a strong (s + 1)-plane also contained in Y , thus we may choose more precisely an algebraic family (Hz ⊂ Hz )z∈Z where Hz is a strong (s + 1)-plane in Y , Hz is a hyperplane in Hz and W is covered by (Hz )z∈Z . By standard arguments, we may suppose that Z is projective smooth connected of dimension r − s. We denote by HZ ⊂ HZ the two corresponding projective bundles over Z. Since W is not covered by strong (s + 1)-planes, the projection of HZ into Y is not contained in W , thus it is a positive (r + 1)-cycle. We take for Γ this Chow-theoretic projection of HZ in Y . Let us now compute Γ · Y in QCHr (Y ) (remember that we consider Γ as a cycle in Y ). We start by applying the projection formula [Fu, 8.1.7] to pr2 : Z × Y → Y : Γ · Y = pr2∗ (HZ ) · Y = pr2∗ (HZ · (Z × Y )). So now we compute HZ · (Z × Y ). This is the divisor class in HZ defined by the linear system |pr2∗ OY (Y )|. Now HZ is a projective bundle and this linear system has degree d along the fibers of this bundle. Thus we have HZ · (Z × Y ) = dHZ + ψ −1 (D), where D is a divisor in Z and ψ : HZ → Z is the bundle projection. We get Γ · Y = dpr2∗ (HZ ) + pr2∗ ψ −1 (D) in QCHr (Y ∩ pr2 (HZ )). Since HZ is generically finite over the subvariety W and since pr2∗ (ψ −1 (D)) lies in QCHr(s+1) (Y ), one obtains, for some positive multiple α of d, the relation Γ · Y ≡ αW mod QCHr(s+1) (Y ). Now we check the following statement, already mentioned in our introduction: Proposition 2.3. Suppose Y is in the linear system |OY (t)|. Then any strong r-plane in Y is the support of a t-fat r-plane contained in Y . Conversely, if furthermore Y is (set-theoretically) defined by equations of degree (strictly) less than t, then the support of any t-fat r-plane contained in Y is strong. Proof. For the first statement, our strong r-plane L is contained in a strong (r + 1)-plane L ⊂ Y . If L is contained in Y , then so is the t-th infinitesimal neighborhood of L in L . If not, then, since the set-theoretic intersection of Y and L is L, and the degree of the restriction of |OY (t)| to L is t, the scheme-theoretic intersection Y ∩ L has to be the t-th infinitesimal neighborhood of L in L . Thus in both cases, this is the desired t-fat r-plane. For the second statement, let L ⊂ Pn be a t-fat r-plane contained in the (r +1)plane L . The equations defining Y vanish on L. Since these equations can be


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chosen of degree strictly less than t, they vanish identically on L , which means that L is contained in Y , hence that L is strong. 3. Restricted flag-Hilbert schemes In this section, we collect some technical material concerning the infinitesimal theory of restricted flag-Hilbert schemes. Here by a full Hilbert scheme (for a given projective variety), we mean any open subscheme of the Hilbert scheme associated to a Hilbert polynomial, while by a Hilbert scheme, we mean any locally closed subscheme of a full Hilbert scheme. Given two full Hilbert schemes H1 and H2 of subschemes of the same ambient projective scheme P , we have the corresponding flag-Hilbert scheme D of pairs (X → Y ) in H1 × H2 . Two subschemes H1 ⊂ H1 and H2 ⊂ H2 being given, by the corresponding restricted flag-Hilbert scheme, we mean the scheme-theoretic intersection D of D with H1 × H2 . In the example we have in mind, P is a projective space, H1 is a variety of fat planes, and H2 = H2 is a full Hilbert scheme of complete intersections. We write i : X → Y for a given pair, IX and IY for the two ideal sheaves on P , NX := Hom(IX , OX ) and NY := Hom(IY , OY ) for the corresponding normal bundles. We denote by NY |X the restriction i∗ (NY ) of NY to X. We also have i∗ : NX → i∗ NY and i∗ : NY → i∗ NY |X . Note that the two codomains have the same space of sections H 0 (NY |X ). Putting this together, we have a morphism (i∗ , i∗ ) : H 0 (NX ) ⊕ H 0 (NY ) → H 0 (NY |X ). The domain of this morphism is the tangent space to the product of our two Hilbert schemes, and the tangent space to the flag-Hilbert scheme is identified as the kernel of the above map (i∗ , i∗ ) (see [Kl], [Se, Remark 4.5.4 ii]). Hence the differentials of the two projections are the restrictions to this kernel of the projections. We first state in our way the standard result in the unrestricted case: Proposition 3.1. We suppose that H1 , H2 are smooth connected and that D has codimension c at O := (X, Y ). We also suppose that i∗ : H 0 (NY ) → H 0 (NY |X ) has rank c. Then (i) D is smooth at O; (ii) the image of i∗ : H 0 (NX ) → H 0 (NY |X ) is contained in the image of i∗ : H 0 (NY ) → H 0 (NY |X ); (iii) the first projection D → H1 is smooth at (X, Y ); (iv) the second projection D → H2 is smooth at (X, Y ) if (and only if ) the rank of i∗ : H 0 (NX ) → H 0 (NY |X ) is c. Proof. (i) Since i∗ has rank c, the pair (i∗ , i∗ ) has rank at least c. It follows that, in the tangent space of H1 × H2 at O, the tangent space to D is at least c-codimensional. Since D is c-codimensional, this implies that D is smooth at O. (ii) By the previous argument, we see that the rank of the pair (i∗ , i∗ ) is exactly c, which means the stated inclusion. (iii) Using the previous item and an easy diagram chase, we see that the differential of D → H1 at O is onto. (iv) This follows by a similar diagram chase.

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Now we turn to the restricted case. Here we write K 0 (NX ) ⊂ H 0 (NX ) for the tangent space to H1 at X, and K 0 (NY ) ⊂ H 0 (NY ) for the tangent space to H2 at Y. Proposition 3.2. We suppose that H1 , H2 are smooth connected and that D has codimension c at O := (X, Y ). We also suppose that i∗ : K 0 (NY ) → H 0 (NY |X ) has rank c. Then (i) D is smooth at O; (ii) the image of i∗ : K 0 (NX ) → H 0 (NY |X ) is contained in the image of i∗ : K 0 (NY ) → H 0 (NY |X ); (iii) the first projection D → H1 is smooth at (X, Y ); (iv) the second projection D → H2 is smooth at (X, Y ) if (and only if ) the rank of i∗ : K 0 (NX ) → H 0 (NY |X ) is c. Proof. The main point is the identification of the tangent space to D : a pair (t1 , t2 ) of vectors in K 0 (NX )×K 0 (NY ) is tangent to D if the subscheme (over Spec k[]) corresponding to t1 is included in the one corresponding to t2 . This means exactly that (t1 , t2 ) is tangent to D. Hence the tangent space to D is the kernel of the restriction (i∗ , i∗ ) : K 0 (NX ) ⊕ K 0 (NY ) → H 0 (NY |X ). The rest of the proof is identical to the previous one. 4. Fat planes in complete intersections In this section, we consider • a projective space Pn , • an integer r with 0 ≤ r < n, which is the dimension of our (fat) planes, • an integer s with 1 ≤ s ≤ n − r − 1, which is the codimension of our complete intersections (or the number of their equations), • a sequence d := (d1 , · · · , ds ) of s positive integers, which is the multidegree of our complete intersections, • an integer t, with 2 ≤ t ≤ max d, which is the multiplicity of our fat r-planes. We keep the notations of the previous section for our case where H1 is the (smooth) Hilbert scheme parametrizing t-fat r-planes in Pn and H2 = H2 is the (smooth) Hilbert scheme of complete intersections of type d. We write δi for the dimension of Hi . The dimension δ1 of H1 does not depend on t (thanks to the assumption t ≥ 2), it is the dimension of the corresponding flag variety, namely (r + 2)(n − r − 1) + r + 1, in other words (r + 2)(n − r) − 1. di −t+r+1 di +r+1 s We set ρ := (r + 2)(n − r) − 1 − Σi=1 r+1 + Σdi ≥t . We will r+1 see that ρ is the expected dimension for the Hilbert scheme of t-fat r-planes in a complete intersection of type d in Pn . Recall that by a t-fat r-plane, we mean the t-th infinitesimal neighborhood of an r-plane in an (r + 1)-plane. Finally, we set c := δ1 − ρ. Hence we have ρ = δ1 − c which means that c is the (expected) number of conditions imposed on a t-fat r-plane for it to be contained in a given complete intersection of type d. The first result of this section confirms this expectation.


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Proposition 4.1. (i) The codimension of the restricted flag-Hilbert scheme D in H1 × H2 is c. (ii) For the generic complete intersection Y of type d in Pn the dimension of the Hilbert scheme of t-fat r-planes in Y is everywhere ρ. In particular this Hilbert scheme is empty if ρ is negative. Proof. (i) We consider a variety V parameterizing our complete intersections, namely the open subset of the vector space Vˆ of s-tuples of homogeneous polynomials (in n + 1 variables) of the given multidegree defining an s-codimensional subscheme in Pn . We write dV for the dimension of V . The variety V comes equipped with the tautological subscheme T ⊂ V ×Pn . The corresponding morphism V → H2 is surjective and it is easily checked to be smooth. Similarly, H1 comes equipped (thanks to t ≥ 2) with a tautological flag L ⊂ L ⊂ H1 × Pn , where L is the tautological t-fat r-plane, while L is its linear span: its fibers over H1 are (r + 1)-planes. Next, we introduce the incidence subscheme D := D ×H2 V ⊂ H1 × V . Since V → H2 is surjective and smooth, it is enough to prove that the codimension of D in H1 × V is c. The dimension dD of D is understood through the projection on H1 . Indeed the fibers of the projection D → H1 are traces on V of sub-vectorspaces in Vˆ . So we have to compute the codimension in Vˆ of tuples vanishing on a fixed t-fat r-plane L. This codimension is Σsi=1 ci , where ci is the codimension ofhomogeneous di +r+1 , polynomials of degree di vanishing on L. We easily check ci = r+1 − di −t+r+1 r+1 p where we adopt the convention that q is zero whenever p < q. Hence we end up with the desired result dD = dV + ρ. (ii) This is an immediate consequence of the first item. We need a complementary statement which is a particular case of the following conjecture: Conjecture 4.2. Apart from the exception below, for the generic complete intersection Y of type d in Pn , when ρ is nonnegative, the Hilbert scheme of t-fat r-planes in Y is nonempty. Here is the known exception : Example 4.3. For double lines on the generic quadric in P3 , we have ρ = 0 while the corresponding Hilbert scheme is empty. In the rest of this section, we reduce the above conjecture to a maximal rank problem. This maximal rank problem for the particular case we need will be handled in the next section. We want to apply the result of the previous section. So we start from a flag H ⊂ L ⊂ L ⊂ Pn where L is a t-fat r-plane with support H and linear span L . Our first task is to identify the tangent space TL H1 at L with the variety H1 of fat planes. Recall that the tangent space at L to the full Hilbert scheme is H 0 (L, NL ), where NL := Hom(IL , OL ) is the normal bundle. Hence we look for a subspace of that vector space. We choose coordinates xi where L is defined by the equations xt0 = x1 = · · · = xn−r−1 = 0 so that we may identify NL as

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the direct sum OL (1)n−r−1 ⊕ OL (t) and accordingly H 0 (NL ) as the direct sum H 0 (OL (1))n−r−1 ⊕ H 0 (OL (t)). For the following lemma, we will introduce again a notation K 0 . The reader should be aware that, in the present section, this notation is introduced in such a way that K 0 differs from H 0 only in the special case where r is zero. For each integer a, we denote by K 0 (OL (a)) the image of the restriction H 0 (OPn (a)) → H 0 (OL (a)). We also extend this notation to sequences in the natural way: by OL (d) we mean ⊕i OL (di ), and K 0 (OL (d)) stands for ⊕i K 0 (OL (di )). Finally we set p := n − r − 1. Lemma 4.4. (i) The image of the natural morphism j : T Pn → NL from the tangent sheaf of Pn to the normal sheaf NL is a subsheaf NL isomorphic to OL (1)p ⊕ OH (1) as an OL -module. (ii) More precisely, we may choose an isomorphism between NL and OL (1)p ⊕ OL (t) so that the corresponding injection OL (1)p ⊕OH (1) → OL (1)p ⊕OL (t) decomposes as ι⊕μ, where ι is the identity on the first summand and μ is the multiplication by xt−1 on the second one. 0 (iii) The tangent space TL H1 at L to the variety H1 of fat planes is the image of H 0 (T Pn ) (or H 0 (OPn (1)n+1 )) in H 0 (NL ) (or H 0 (NL )). We write K 0 (NL ) for this image. (iv) Under the identification in (i), K 0 (NL ) becomes K 0 (OL (1))p ⊕H 0 (OH (1)). Proof. Let us start with the third statement. Since H1 is the orbit in the full Hilbert scheme of L under the projective linear group, TL H1 has to be the image of the natural map H 0 (j) : H 0 (T Pn ) → H 0 (NL ). Now we turn to (i) and (ii). Using our coordinates, our morphism j, viewed as a morphism from OPn (1)n+1 to OL (1)p ⊕ OL (t), is given by the partial derivatives of our n − r equations, which gives essentially the announced matrix: just note that, thanks to the characteristic zero assumption, the image of the multiplication by the partial derivative txt−1 from OL (1) to OL (t) is the same as the image of the 0 multiplication by xt−1 0 , and this image is isomorphic to OH (1). Now we turn to (iv). We just note that by i), H 0 (NL ) is equal to H 0 (OL (1))p ⊕ 0 H (OH (1)). By definition, K 0 (NL ) is the image of H 0 (OPn (1)n+1 ) in H 0 (NL ), which can now be identified as the space K 0 (OL (1))p ⊕ H 0 (OH (1)). Now we consider a flag i : L → Y of complete intersection subschemes in Pn where Y is the general complete intersection of type d containing L. The tangent space at Y to the corresponding Hilbert scheme is H 0 (Y, NY ) which can be computed as H 0 (Y, OY (d)). It follows that H 0 (L, NY |L ) is isomorphic to H 0 (L, OL (d)). Thus we can write K 0 (L, NY |L ) or simply K 0 (NY |L ) for the image of H 0 (NY ) in H 0 (L, NY |L ). Lemma 4.5. (i) For r > 0, K 0 (NY |L ) is the whole of H 0 (L, NY |L ). (ii) In any case, the dimension of K 0 (NY |L ) is c. (iii) In any case, the natural map K 0 (NL ) → H 0 (L, NY |L ) factors through K 0 (NY |L ). (iv) If the induced map K 0 (NL ) → K 0 (NY |L ) is onto, then D → H2 is smooth at (L, Y ). Proof. (i) We know that NY is the direct sum OY (d) and a standard cohomological argument shows that all its sections come from the ambient projective


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space. Hence what we have to prove is that any section of OL (d) comes from the ambient projective space, which follows from the standard cohomological argument: the cohomology of line bundles on projective spaces of dimension at least two is trivial (here we use r ≥ 1). (ii) As we have just seen, K 0 (NY |L ) is the image of H 0 (OPn (d)) in H 0 (OL (d)), so this is just a count of monomials which we leave to the reader. (iii) and (iv) Now we apply Proposition 3.2: in our case, we have K 0 (NY ) = 0 H (NY ) and, according to Lemma 4.4, the assumption in Proposition 3.2 is precisely the previous item. The statements (iii) and (iv) here are exactly the conclusions (ii) and (iv) of Proposition 3.2. We turn to the final result of the present section where, for sake of clarity, we handle separately the case r = 0. We will write 1, 2 and t respectively for the sequence (1, · · · , 1), (2, · · · , 2) and (t, · · · , t), hence accordingly d − 1, d − 2, d − t respectively for (d1 − 1, · · · , ds − 1), (d1 − 2, · · · , ds − 2), (d1 − t, · · · , ds − t). Proposition 4.6. (i) For r ≥ 1, we consider the generic morphism m : OL (1)p ⊕ OH (1) → OL (d) of coherent OL -modules. If H 0 (m) : H 0 (OL (1)p ) ⊕ H 0 (OH (1)) → H 0 (OL (d)) is onto, then D → H2 is also onto. (ii) For r = 0, we denote by K 0 (OL (1)p ⊕ OH (1), OL (d)) the image of the natural map K 0 (OL (d − 1)) ⊕ K 0 (OL (d − t)) → Hom(OL (1)p ⊕ OH (1), OL (d)). We consider the generic morphism m in K 0 (OL (1)p ⊕ OH (1), OL (d)). If the image by H 0 (m) of K 0 (OL (1)p )⊕H 0 (OH (1)) in H 0 (OL (d)) is K 0 (OL (d)), then D → H2 is onto. Proof. (i) Since our morphism D → H2 is projective and the codomain is irreducible, it is sufficient to prove that it is dominant. We apply Lemma 4.5 (iv), hence we have to prove that the map mY : H 0 (NL ) → H 0 (L, NY |L ) is onto. This map depends upon our complete intersection Y . We express it in terms of the system of equations b := (b1 , · · · , bs ) ∈ H 0 (IL (d)) of Y , rather than in terms of Y itself. This allows us to describe the associated morphism mY : NL → NY |L or, via the identifications of Lemma 4.4, mb : OL (1)p ⊕ OH (1) → OL (d) as follows: - for the first factor, the j-th component (1 ≤ j ≤ p), from OL (1) to OL (d), is the derivative of b with respect to xj ; -for the second factor, from OH (1) to OL (d), we have the derivative of b with respect to x0 (note that indeed this derivative factors through OH (1)). What we have to prove is that, for b sufficiently general, H 0 (mb ) is onto. For this, thanks to our surjectivity assumption, it is enough to prove that b → mb is dominant (or onto). We prove that b → mb is onto. For this we take m := (m1 , · · · , mp , m0 ) : OL (1)p ⊕ OH (1) → OL (d) and search for b with m = mb . By the standard cohomological argument, we may lift m1 , · · · , mp and consider we are given (m1 , · · · , mp ) : OPn (1)p → OPn (d). Now for m0 , we see it as a section of Hom(OH , OL (d − 1)), hence as a section of OL (d − 1) annihilated by x0 , thus of the form txt−1 0 f with f a section of OL (d − t), using the characteristic zero assumption. As above, we

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may lift f as a section, still denoted by f , of OPn (d − t). At this point we may set b := x1 m1 + · · · + xp mp + xt0 f and check that it has the desired property. (ii) The proof is almost the same: we apply Lemma 4.5 (iv). This time, we have to prove that the map mY : H 0 (NL ) → H 0 (L, NY |L ) sends K 0 (NL ) onto K 0 (L, NY |L ). As above we introduce a system of equations b := (b1 , · · · , bs ) ∈ H 0 (IL (d)) of Y . Via the identifications of Lemma 4.4, we are concerned, for b sufficiently general, with the sheaf morphism mb : OL (1)p ⊕ OH (1) → OL (d) defined by the same formulas as in the previous case. Since our identifications send K 0 (NL ) to K 0 (OL (1))p ⊕ H 0 (OH (1)) and 0 K (L, NY |L ) to K 0 (OL (d)), it is enough to prove that the image of b → mb is K 0 (OL (1)p ⊕ OH (1), OL (d)). For this we take m := (m1 , · · · , mp , m0 ) ∈ K 0 (OL (1)p ⊕ OH (1), OL (d)). This means that m1 , · · · , mp come from sections still denoted by m1 , · · · , mp in t−1 H 0 (Pn , O(d − 1)), while m0 is of the form xt−1 0 f , or better of the form tx0 f , with 0 n f a section of H (P , O(d − t)). We now search for b with m = mb . Again we may set b := x1 m1 + · · · + xp mp + xt0 f and check that it has the desired property. 5. Nonemptiness In this section, we prove our conjecture 4.2 in the case we need. We restrict to the very special case where t is the greatest number in our sequence d, and we assume furthermore that t is at least 3, and that it occurs only once in d. We will prove: Proposition 5.1. Under the above restrictions, when ρ is nonnegative, for any complete intersection Y of type d in Pn , the Hilbert scheme of t-fat r-planes in Y is nonempty. We keep the notations of the previous section. Furthermore, we denote by h0 (u, e) the number of monomials of degree e in u variables, and accordingly, for any sequence e := (e1 , · · · , es ) of integers, we set h0 (u, e) := h0 (u, e1 ) + · · · + h0 (u, es ). Thanks to Proposition 4.6, it is enough to prove a maximal rank statement, which depends on whether r is zero or not. Namely, we have to prove the following two lemmas. Lemma 5.2. (the case r ≥ 1) For p satisfying (r + 2)p + r + 1 ≥ h0 (r + 2, d) − 1, and for the general morphism m : OL (1)p ⊕ OH (1) → OL (d), H 0 (m) is onto. Lemma 5.3. (the case r = 0) We suppose 2p + 1 ≥ h0 (2, d) − 1. Then, for the general morphism m in K 0 (OL (1)p ⊕ OH (1), OL (d)), the image by H 0 (m) of K 0 (OL (1)p ) ⊕ H 0 (OH (1)) in H 0 (OL (d)) is K 0 (OL (d)). Recall that K 0 (OL (1)p ⊕ OH (1), OL (d)) denotes the image of the natural map from K 0 (OL (d − 1)) ⊕ K 0 (OL (d − t)) to Hom(OL (1)p ⊕ OH (1), OL (d)). The differences between our two lemmas can be erased by switching to the point of view of graded modules. So, just for the present section, we radically change the meaning of our notations: from now on, OL denotes the graded ring k[x0 , · · · , xr+1 ]/(xt0 ) and OH denotes the quotient graded module k[x1 , · · · , xr+1 ]. For a graded OL -module G with graduation γ, we write G(a) for the module G equipped with the graduation γa := γ − a. For a graded module G, by H 0 (G) we


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mean the degree 0 component of G, while for a morphism m of graded modules, by H 0 (m) we mean the restriction of m to the degree 0 components. With these conventions, our two lemmas rephrase as the single following one: Lemma 5.4. For p satisfying (r + 2)p + r + 1 ≥ h0 (r + 2, d) − 1, and for the general morphism m : OL (1)p ⊕ OH (1) → OL (d), H 0 (m) is onto. Here we use a method which can be tracked back to [EH, EHM], where similar results were obtained in a different context. We denote by M the vector space Hom(OL (1)p ⊕ OH (1), OL (d)), by S the space of nontrivial linear forms on H 0 (OL (d)), and by Z the “incidence” subscheme in M ×S consisting of pairs (m, ) for which ◦ H 0 (m) vanishes. We denote by the second projection: : Z → S. What we want to prove is that the first projection Z → M is not dominant. This will follow if we prove the inequality dim Z ≤ dim M , since the fibers of our projection are unions of lines. We proceed by contradiction and suppose that the projection Z → M is dominant. To each λ ∈ S we attach the bilinear form λ∗ on H 0 (OL (d − 1)) × H 0 (OL (1)) defined by λ∗ (f, v) = λ(vf ). By semi-continuity, we have an open subset Zu ⊂ Z which still dominates M , and where the rank u of

∗ is constant. In the first factor H 0 (OL (d − 1)) of our product, we have a distinguished line: the line D generated (in the summand OL (t)) by xt−1 0 . Our first observation is the following: Lemma 5.5. For our general point z ∈ Zu , ∗ (z) vanishes on D × H 0 (OL (1)). Proof. In OL (d) we have the summand OL (t). And therein, we have the t−1 graded submodule xt−1 0 OL (1) consisting of multiples of x0 . This submodule may t−1 be better denoted by x0 OH (1) since the multiplication by xt−1 0 , which sends OL (1) into OL (t), factors through OH (1). This submodule xt−1 0 OH (1) is easily identified as the submodule of OL (d) which is annihilated by x0 . Hence, any morphism m ∈ M has to send the second summand OH (1) of its domain, which is annihilated by x0 , into the summand OL (t) of its codomain, and more precisely into the submodule xt−1 0 OH (1) mentioned above. Also a sufficiently general morphism m ∈ M sends OH (1) isomorphically onto that submodule. Accordingly, H 0 (m) sends H 0 (OH (1)) isomorphically onto H 0 (xt−1 0 OH (1)). So, for our general z ∈ O (1)) which implies that ∗ (z) vanishes on Zu , (z) has to vanish on H 0 (xt−1 H 0 D × H 0 (OL (1)). Our next observation stresses the role of u, which is to control the dimension of the fiber of Z → S. We denote by Su the projection of Zu in S. Lemma 5.6. The codimension of the fiber of Zu over a point λ ∈ Su is pu. Proof. Let m := (m1 , · · · , mp ; m0 ) be a point in M , where (m1 , · · · , mp ) are in H 0 (OL (d − 1)) while m0 is in H 0 (OL (d − t)). Thanks to the previous lemma, we see that λ ◦ H 0 (m) vanishes if and only ∗ if λ (m1 ) = · · · = λ∗ (mp ) = 0. Each one among these p equations imposes u independent conditions on m, since the rank of λ∗ is u. Since these equations concern different components of m, their ranks add up to the rank of m → λ◦H 0 (m) which turns out to be pu.

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Our next task consists in estimating the dimension of Su . Lemma 5.7. The dimension of Su is at most h0 (u, d) + (r + 1 − u)u. Proof. For this we have to single out the line E generated by x0 in H 0 (OL (1)) and to distinguish two cases according to whether, for our general z ∈ Zu , ∗ (z) vanishes or not on H 0 (OL (d − 1)) × E. (i) We start with the (slightly simpler) case where ∗ (z) does not vanish on 0 H (OL (d − 1)) × E. In order to bound the dimension of Su at a point λ0 , we will define, in a neighborhood U ⊂ Su of λ0 , two algebraic maps f : U → Ab and g : U → Ac so that (f, g) is injective. This will bound the dimension of Su by b + c. To this effect, we reorder our basis C := (x0 , · · · , xr+1 ) of H 0 (OL (1)) (where x0 remains an equation of H) so that, in this basis, the first u rows of the matrix of λ∗0 are linearly independent. This property will hold in a neighborhood of λ0 which we take as U . We write C for the sub-basis (x0 , · · · , xu−1 ) and C for the rest of the basis so that we have C = C C . Next, in OL (d), we have the basis consisting of monomials in each summand, which we call d-monomials. Similarly, we have the basis of (d − 1)-monomials in OL (d − 1). Associated with these bases, we have the matrix Nλ of λ∗ , which is an algebraic function of λ. Now for each element in C , we have the u coordinates of the corresponding row (in Nλ ) as a combination of the rows in C . This defines b := (r + 2 − u)u functions on U which altogether yield our map f . Now for g(λ) we take the restriction λ of λ to the subspace generated by the following set T of d-monomials: first, take the set T of d-monomials depending (at most) on variables in C , then delete those, in the summand OL (t), which are divisible by xt−1 0 . This deletion corresponds to the fact observed above that λ vanishes there. What we have to check is that λ is determined by λ and f (λ). For this, we claim that for each integer q with 0 ≤ q ≤ t the values of λ on the set Tq of those d-monomials which are of degree q with respect to variables in C are linear combinations (where coefficients are polynomials in f (λ)) of its values on T . We prove the claim by induction on q, the case q = 0 following from the vanishing mentioned above. For the general case we consider a d-monomial m := m xi where m is a (d − 1)-monomial and xi is in C . In the column corresponding to m in Nλ , the first u entries are values of λ on elements of Tq−1 , while the entry in the row corresponding to xi is λ(m), which gives us the desired linear relation. It remains to check that the number of elements in T is h0 (u, d) − u. Indeed, 1 is subtracted from h0 (u, d) because, although d contains t, xt0 is not a d-monomial, and u − 1 is subtracted due to the difference between T and T . Thus the codomain of our map g is Ac with c := h0 (u, d) − u. (ii) Now we treat the similar case where ∗ (z) vanishes on H 0 (OL (d − 1)) × E. The method is the same so we just highlight the changes. Thanks to the vanishing assumption, λ∗ is now determined by the bilinear form λ∗ induced on H 0 (OL (d − 1)) × H 0 (OH (1)). Our basis C now has the form (x1 , · · · , xr+1 ), and the subbasis C is (x1 , · · · , xu ) Accordingly, the number b is now equal to (r + 1 − u)u. On the other hand, here, there is no deletion, T is equal to T and its number of elements is h0 (u, d), which yields the desired formula.


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In order to complete the proof of 5.2, it remains to check that the estimates obtained so far make the dimension of Zu smaller than that of M , namely that the codimension (in M ) obtained for the fiber of Zu → Su is bigger than the dimension of Su . This reads: Lemma 5.8. For t ≥ 3, p satisfying (r + 2)p + r + 1 ≥ h0 (r + 2, d) − 1, and 1 ≤ u ≤ r + 2 we have h0 (u, d) + (r + 1 − u)u ≤ pu. Proof. We argue by convexity (with respect to u) and start by checking the extreme cases: (i) For u = r + 2, the desired conclusion is just the assumption. (ii) For u = 1, we contrapose and prove that p ≤ h0 (1, d) + r − 1 implies (r + 2)p + r + 3 ≤ h0 (r + 2, d). Taking the critical value s + r − 1 for p we have to prove s(r + 2) + (r + 1)2 ≤ h0 (r + 2, d). We split this inequality summand by summand, in other words we claim a) r + 2 + (r + 1)2 ≤ h0 (r + 2, t) (for the occurence of t in d) and b) r + 2 ≤ h0 (r + 2, δ) (for each other integer, δ ≥ 2, in d). For a) it is sufficient to check the first case t = 3. In this case, we have to prove 6(r + 1)2 ≤ (r + 2)[(r + 3)(r + 4) − 6] or, dividing by r + 1, 6r + 6 ≤ (r + 2)(r + 6), or 0 ≤ r 2 + 2r + 6, which is evident. Moreover, b) is clear since for each variable xi , we have the monomial xδi . It remains to check that the function f := u → h0 (u, d) + (r + 1 − u)u is convex on our interval [1, r + 2]. For this, we compute the discrete derivatives f := u → f (u + 1) − f (u) and f . We find f (u) = h0 (u + 1, d − 1) + r + 1 − 2u − 1 and f (u) = h0 (u + 2, d − 2) − 2. We see that this second derivative is nonnegative for u ≥ 1, yielding the desired convexity. 6. Spannedness This section is devoted to the proof of the desired covering statement : Proposition 6.1. Let r ≥ 0, 1 ≤ s ≤ n − − 1 and 2 ≤ d1 ≤ ... ≤ ds−1 < ds r+r+1 and assume the (necessary) be integers. We set ρ := (r + 2)(n − r) − Σsi=1 dir+1 inequality ρ + r ≥ n − s.

If Y ⊂ Y is any pair of type (d1 , ..., ds ) in Pn , then Y is covered by strong r-planes. Thanks to Proposition 2.3, this statement is an immediate consequence of the following one. Proposition 6.2. Under the same assumptions, Y is covered by ds -fat rplanes. We pose t := ds . From §4, we have the restricted flag-Hilbert scheme D ⊂ H1 × H2 . Over the first factor H1 , we have the universal t-fat r-plane, say L ⊂ H1 × Pn . Over the second factor H2 , we have the universal complete intersection of type d, say Y ⊂ H2 × Pn , and over D , we have the universal flag, say Lˆ ⊂ Yˆ ⊂ D × Pn . We have a natural projection e : Yˆ → Y, and what we have to prove is that the restriction e : Lˆ → Y is onto.

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Since e is a H2 -morphism between varieties which are projective over H2 , its image is also projective over H2 . So it is sufficient to prove that e is dominant, and, for that, to find one point in Lˆ where the fiber of e has the expected dimension ρ + r − n + s, and not more. ˆ where L is a t-fat r-plane So we compute the fiber of e at a point (L, Y, p) ∈ L, contained in the complete intersection Y and p is a point on L. This splits into two cases according to whether r is zero or not. (i) The case r = 0. This case is known since [Ro]. Hence we just give the idea of the proof, which is similar to but simpler than the other case. The variety of t-fat points at p contained in Y is identified with a subvariety in the projectivized tangent space of Y at p with equations depending on the equations of Y . The number of these equations is easily checked to be Σi (di − 1) − 1: di − 1 is the number of degrees between 2 and di , and 1 is subtracted for the degree t. Thanks to our assumption on ρ, this is at most n − s − 1 which is the dimension of this projective space. Hence this variety is nonempty. (ii) The case r ≥ 1. We consider the subscheme WY,p in the projectivized tangent space P TY,p of Y at the point p, which parametrizes lines through p contained in Y . It is defined by the homogeneous components of the Taylor expansions at p of the equations of Y . For this reason, we set d := (1, 2, · · · , d1 , · · · , 1, 2, · · · , ds ) and d := (2, · · · , d1 , · · · , 2, · · · , ds ). We immediately observe that there is a natural isomorphism between the vector space of tuples of equations of type d (in n + 1 variables) vanishing at p and tuples of equations of type d (in n variables). This yields: For (Y, p) sufficiently general in Y, WY,p is a sufficiently general complete intersection of type d in P TY,p . Next we have the following Lemma 6.3. For any (Y, p) ∈ Y, the fiber of e over (Y, p) is isomorphic with the Hilbert scheme of t-fat (r − 1)-planes in WY,p . First we check how we may complete the proof of Proposition 6.2 using this lemma. Thanks to Proposition 5.1, we just have to check that the expected dimension ρ of the Hilbert scheme of t-fat (r − 1)-planes in the generic complete intersection of type d in Pn−1 or equivalently of type d in Pn−s−1 is nonnegative. We have j+r di s ρ = (r + 1)(n − r) − 1 − Σi=1 Σj=1 r di + r + 1 s − 1) = (r + 1)(n − r) − 1 − Σi=1 ( r+1 di + r + 1 s = (r + 2)(n − r) − 1 − n + r + s − Σi=1 r+1 = ρ + r − n + s. This is nonnegative by assumption. Now we prove Lemma 6.3. First of all, we have a natural isomorphism g between the Hilbert scheme of t-fat r-planes in Pn passing through p and the Hilbert scheme of t-fat (r − 1)-planes in the projectivized tangent space P T Pnp : if we identify this projectivized tangent space with a hyperplane K ⊂ Pn not passing through p, g(L) is the scheme-theoretic intersection of L with K.


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Now we prove that g induces a bijection from the Hilbert scheme H of t-fat r-planes in Y passing through p to the Hilbert scheme H of t-fat (r − 1)-planes in WY,p . Let M be a t-fat (r − 1)-plane in WY,p and M ⊂ Pn be the unique t-fat r-plane through p corresponding to M (hence g(M ) = M ). Let (fi := Σj≤di fij )i≤s be a system of equations of Y . Note that the fi ’s vanish at p. Our claim is that the fi ’s all vanish on M if and only if the fij all vanish on M = g(M ). This follows readily from the particular case of a single equation f , with n = r + 1, which we state explicitly: Let f (x0 , x ) := x0 f1 (x ) + · · · + xδ0 fδ (x ) be a homogeneous polynomial in n + 1 variables x0 , · · · , xn where x stands for (x1 , · · · , xn ). Note that f vanishes at p := (1, 0, · · · , 0). We denote by K the hyperplane defined by x0 = 0, by M the t-fat (n − 1)-plane defined by xt1 = 0 and by N the t-fat (n − 2)-plane defined in K by the same equation. Since f is a multiple of xt1 if and only if all the fi ’s are, we have that f vanishes on M if and only if the fi ’s vanish on N . References S. Bloch, Lectures on Algebraic Cycles. Duke Univ. Math. Series IV, Durham, N.C., 1980. [De-Ma] O. Debarre, L. Manivel, Sur la vari´ et´ e des espaces lin´ eaires contenus dans une intersection compl` ete. (French) [The variety of linear spaces contained in a complete intersection] Math. Ann. 312 (1998), no. 3, 549–574. [De] P. Deligne, Groupes de monodromie en g´ eom´ etrie alg´ ebrique. II. (French) Séminaire de G´ eom´ etrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II). Dirig´ e par P. Deligne et N. Katz. Lecture Notes in Mathematics, Vol. 340. Springer-Verlag, Berlin-New York, 1973. x+438 pp. [De-Di] P. Deligne, A. Dimca, Filtrations de Hodge et par l’ordre du pˆ ole pour les hypersurfaces singuli` eres. (French) [Hodge filtration and filtration by the order of the pole for singular hypersurfaces] Ann. Sci. Ecole Norm. Sup. (4) 23 (1990), no. 4, 645–656. [EH] Ph. Ellia, A. Hirschowitz, Voie ouest I: G´ en´ eration de certains fibr´ es sur les espaces projectifs et application, J. of Alg. Geom. 1 (1992) 531–547. [EHM] Ph. Ellia, A. Hirschowitz, L. Manivel, Le probl` eme de Brill-Noether pour les fibr´ es de Steiner et application aux courbes gauches. Annales ENS Paris 32 (1999) 835–857. [Es] H. Esnault, Hodge type of subvarieties of P n of small degrees. Math. Ann. 288 (1990), no. 3, 549–551. [EsNS] H. Esnault, M.V. Nori, V. Srinivas, Hodge type of projective varieties of low degree. Math. Ann. 293 (1992), no. 1, 1–6. [EsLV] H. Esnault, M. Levine, E. Viehweg, Chow groups of projective varieties of very small degree, Duke Math. J. 87, (1997), no. 1, 29–58. [Fu] W. Fulton, Intersection theory, Ergeb.der Math.und ihrer Grenz. 3. Folge, SpringerVerlag, Berlin, 1984. [Kl] J.O. Kleppe, Nonreduced components of the Hilbert scheme of smooth space curves. Space curves (Rocca di Papa, 1985), 181–207, Lecture Notes in Math., 1266, Springer, Berlin, 1987. [Le] J.D. Lewis, Cylinder homomorphisms and Chow groups. Math. Nachr. 160 (1993), 205– 221. [Ot] A. Otwinowska, Remarques sur les groupes de Chow des hypersurfaces de petit degr´ e, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 1, 51–56. [Pa] K. Paranjape, Cohomological and cycle-theoretic connectivity, Ann. of Math. (2) 139 (1994), 641–660. [Ro] A. Roitman, Rational equivalence of zero cycles, Mat. Zametki 28 (1980), no. 1, 85–90, 169. [Sc] C. Schoen, On Hodge structures and nonrepresentability of Chow groups, Compositio Math. 88 (1993), no. 3, 285–316. [Bl]

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[Se]


E. Sernesi, Deformations of algebraic schemes, Grundlehren der mathematischen Wissenschaften, Springer, 2006, xi+339 pp.

CNRS, Laboratoire J.-A.Dieudonn´ e, Universit´ e de Nice–Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France E-mail address: [email protected] The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India Current address: Department of Mathematics and Statistics, University of Hyderabad, Gachibowli, Central University P O, Hyderabad-500046, India E-mail address: [email protected]


Vector Bundles and the Icosahedron Nigel Hitchin Dedicated to S. Ramanan on the occasion of his 70th birthday

Abstract. A plane curve C defined by a homogeneous polynomial satisfying Laplace’s equation appears canonically as the vanishing of the Pfaffian of a skew-symmetric matrix of linear forms. As a consequence there is a natural semi-stable rank two vector bundle defined on C. We consider the case of degree 3, and apply Atiyah’s classification of bundles to determine various invariant strata in the space of harmonic cubics. We encounter the MukaiUmemura threefold and link up with the classical geometry of the Clebsch diagonal cubic surface, certain distinguished rational curves on it, and the action of the icosahedral group.

1. Introduction Describing the equation of a plane curve C as the determinant of a matrix of linear forms is a classical problem. A related issue is expressing C as the Pfaffian of a skew-symmetric matrix of forms. When the curve is defined by a harmonic homogeneous polynomial (a solution to Laplace’s equation), then the action of the Lie algebra of the orthogonal group SO(3, C) provides a natural such expression, and with it, a natural semi-stable rank two vector bundle E on C. This paper concerns itself with these bundles, which have trivial determinant and H 0 (C, E) = 0. For a cubic curve, Atiyah’s classification shows that they fall into three types: a sum of line bundles L ⊕ L∗ with L2 nontrivial, a non-trivial extension of a line bundle of order two by itself, and a trivial extension. We shall show that the second and third cases are described in the space of harmonic cubics by the vanishing of a certain invariant polynomial of degree six. Our route to this result uses the Mukai-Umemura threefold and we link up with the classical geometry of the Clebsch diagonal cubic surface, certain distinguished rational curves on it, and the action of the symmetric group S5 . The original motivation for this paper comes from the author’s interest in explicit solutions to Painlevé equations, as in [10] and [11]. These were connected to the study of certain threefolds with an open orbit of SO(3, C) and finite stabilizer. 2010 Mathematics Subject Classification. Primary 14M12; Secondary 14H60. c Mathematical 0000 (copyright Society holder) c 2010 American

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The Mukai-Umemura manifold has such an action where the stabilizer is the icosahedral group. Since the first paper, where the stabilizer is the dihedral group, was written for the 60th birthdays of Narasimhan and Seshadri, it seems appropriate, sixteen years on, to discuss the icosahedron for Ramanan. 2. Representations of SO(3, C) Let V be a finite-dimensional irreducible representation space of the group SO(3, C). These occur in each odd dimension (2d + 1) and have an invariant inner product (u, v). Restricted to the circle subgroup SO(2, C) ⊂ SO(3, C), the weights are −d ≤ n ≤ d; in particular the weight zero occurs with multiplicity one so there is a unique invariant element. The usual realization of V is as S 2d – the space of homogeneous polynomials in the complex variables z1 , z2 of degree 2d under the action of SL(2, C)/ ± 1 ∼ = SO(3, C). The Lie algebra so(3) acts on V as skew-adjoint transformations and so each x ∈ so(3) defines a skew form ωx (u, v) = (x · u, v). Now fix v ∈ V and restrict ωx to the 2d-dimensional orthogonal complement W of v. The Pfaffian ωx ∧ ωx · · · ∧ ωx ∈ Λ2d W ∗ ∼ = C defines a homogeneous polynomial f (x) of degree d. The fact that we have a natural map from vectors in V to polynomials is no surprise if we recall that another realization of the (2d + 1)-dimensional irreducible representation of SO(3, C) is as spherical harmonics – homogeneous polynomials f (x1 , x2 , x3 ) of degree d which satisfy Laplace’s equation. However, v gives somewhat more than just a polynomial. The skew form ωx defines a map ωx : W → W ∗ , which is linear in x. Hence if 2 P is the projective space of the vector space so(3), we have a sequence of sheaves (2.1)

0 → OP2 (W (−2)) → OP2 (W ∗ (−1)) → E → 0 ω

where E is a rank 2 vector bundle supported on the curve C ⊂ P2 defined by f (x) = 0. The vector bundle satisfies Λ2 E ∼ = KC , the canonical bundle, and from the exact cohomology sequence, H 0 (C, E) = 0. These facts about curves defined by Pfaffians (and much more besides) can be found in [3]. In particular, the bundle is always semi-stable. To see this, note that if L ⊂ E is a subbundle, the inclusion defines a section i of L∗ E and if deg L ≥ g then by Riemann-Roch there is a non-zero section s of L. But then si is a nonzero section of E which is a contradiction. Hence we must have 1 1 deg L ≤ g − 1 = deg KC = deg E. 2 2 This natural process thus generates a curve and a rank 2 semi-stable bundle, and the question we ask is “What is this bundle?” When d = 3, the curve C is a plane cubic and then we know from Atiyah’s classification of bundles on an elliptic curve [1] that, when C is smooth, there are only three possibilities. Since KC ∼ = O, Λ2 E is trivial and H 0 (C, E) = 0 and we have the cases: • E = L ⊕ L∗ where deg L = 0 and L = O (the generic case)

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• E is a non-trivial extension 0 → L → E → L → 0 where L2 ∼ = O and L = O • E = L ⊕ L where L2 ∼ = O and L = O. These three types define a stratification of the six-dimensional projective space P(V ) where V is the seven-dimensional space of harmonic cubic polynomials f – or at least that part of it for which the curve f (x) = 0 is nonsingular. To investigate this further we have to see a link with another piece of geometry. 3. Isotropic spaces Let d = 3, and as above let f ∈ V define a smooth cubic curve C ⊂ P2 and W ⊂ V the six-dimensional orthogonal complement of f in W . Since Λ2 E ∼ = O, we have E ∗ ∼ = E and then the kernel of ω : W → W ∗ (1) on C can be identified with E(−1). Suppose U ⊂ W is a three-dimensional subspace, isotropic with respect to the skew forms ωx for all x. Then, for each x, ωx : W → W ∗ maps U to its annihilator U o and so defines a homomorphism of sheaves on P2 , ω|U : U → U o (1). Moreover the determinant of this map is the Pfaffian of ωx . The kernel of ω|U on C is then a line bundle L(−1) ⊂ U . Since this is the restriction of ω to U , L is a subbundle of E. From the treatment of determinantal loci in [3] we have deg L = 0 and H 0 (C, L) = 0 so L = O. Conversely, on C, suppose we have a line bundle L ⊂ E with deg L = 0 and L = O. From the exact cohomology sequence of (2.1) we have W ∗ ∼ = H 0 (C, E(1)). ∗ If α ∈ W annihilates L(−1) then it defines a section of (E/L)(1) = L∗ (1) which is of degree 3 and so has a three-dimensional space of sections on C. It follows that L(−1) ⊂ W sweeps out a three-dimensional subspace U . Now ω|U is a section of Λ2 U ∗ (1) on C and we have Λ2 U ∗ ∼ = U ⊗ Λ3 U ∗ so this defines a section w of U (1) which tautologically is annihilated by ω. Thus if w = 0, then ker ω|U ∼ = O(−1). But ker ω|U ∼ = L(−1) and L = O, so we deduce that w = 0 and U is isotropic for each ωx . If we look again at the trichotomy of Atiyah’s classification, we see that the three cases are equivalent to • E has two rank one subbundles • E has one rank one subbundle • E has infinitely many rank one subbundles and by what we have just seen, this condition translates into an equivalent statement about the three-dimensional isotropic subspaces of W : • W has two isotropic subspaces • W has one isotropic subspace • W has infinitely many isotropic subspaces. This provides an extension of the criterion to any f ∈ V , and not just those which define smooth cubic curves. (In fact, Atiyah’s classification has been extended in [5] to a class of singular elliptic curves; moreover the wild case of cuspidal cubics does not occur when f is harmonic). 4. The Mukai-Umemura threefold The study of three-dimensional isotropic subspaces of V is best approached via a special Fano threefold introduced by Mukai and Umemura [15]. We consider

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the Grassmannian G(3, V ) of three-dimensional subspaces of the seven-dimensional representation space V and its universal rank 3 bundle E. For x ∈ so(3) we have the skew form ωx on V , and so a section of the rank 9 vector bundle over G(3, V ): Λ2 E ∗ ⊗ so(3). Now dim G(3, V ) = 3 × (7 − 3) = 12 and then the zero set of the section is a smooth 12 − 9 = 3-dimensional manifold known as the Mukai-Umemura threefold Z. By construction it parametrizes subspaces U isotropic for all ωx and thereby has a natural action of SO(3, C). The threefold Z has the same additive integral cohomology as P3 and H 2 (Z, Z) is generated by x = c1 . However x2 ∈ H 4 (Z, Z) is 22 times a generator y. Since c1 > 0 the Todd genus is 1 so c1 c2 = 24 and we have Chern classes c1 = x

(4.1)

c2 = 24y

c3 = 4xy.

Given f ∈ V , the inner product (f, −) defines an element of V ∗ and, restricting to the universal bundle E ⊂ V , we get a section of E ∗ . Over Z, this is a rank three bundle and the section vanishes at the points which correspond to isotropic three-dimensional subspaces orthogonal to f . Proposition 4.1. On the Mukai-Umemura threefold, c3 (E ∗ ) = 2. Proof. The tangent bundle of the Grassmannian is Hom(E, V /E) and Z is the non-degenerate zero set of a section of Λ2 E ∗ ⊗ C3 . Thus, as C ∞ bundles T Z ⊕ (Λ2 E ∗ ⊗ C3 ) ∼ = Hom(E, V /E). Applying the Chern character we find 1 1 (c3 + 4c31 − 3c1 c2 ). c2 (E ∗ ) = c21 − c2 c3 (E ∗ ) = 2 10 From (4.1) we obtain 10c3 (E ∗ ) = 4 + 88 − 72 = 20 and hence the result. c1 (E ∗ ) = c1

A section with nondegenerate zero set will thus vanish at two points. We can now see the trichotomy in terms of the section s of E ∗ : • s vanishes at two points • s vanishes at one point • s vanishes on a subvariety of positive dimension. 5. The icosahedron The ten-dimensional space of all homogenous cubics in x has an SO(3, C)invariant product which can be normalized so that (f (x), (x, a)3 ) = f (a). For fixed a ∈ C3 , the polynomial (x, a)3 is not harmonic but its orthogonal projection onto V is 3 fa (x) = (x, a)3 − (a, a)(x, a)(x, x). 5 This is invariant by the group of rotations fixing a. For any cubic f we still have the inner product property (f, fa ) = f (a). The action of u ∈ so(3) on fa is 1 (u · fa )(x) = 3(x, [u, a])[(x, a)2 − (a, a)(x, x)]. 5


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Take a regular icosahedron with vertices at ±a1 , ±a2 . . . , ±a6√. The angle θ between any two axes joining opposite vertices satisfies cos θ = ±1/ 5. Now consider 1 (u · fai , faj ) = 3(aj , [u, ai ])[(aj , ai )2 − (ai , ai )(aj , aj )]. 5 If i = j then (aj , [u, ai ]) = 0 and when i = j 1 1 (aj , ai )2 − (ai , ai )(aj , aj ) = (cos2 θ − )(ai , ai )(aj , aj ) = 0 5 5 so fa1 , . . . , fa6 span an isotropic subspace U , invariant under the icosahedral group G. Its orthogonal complement in V consists of those f such that 0 = (f, fai ) = f (ai ) = 0. There are five objects which are permuted by G which realize the well-known isomorphism G ∼ = A5 . These are the five sets of three orthogonal planes in which all 12 vertices lie. Let e1 , e2 , e3 be the three unit normals to such a set then the cubic (e1 , x)(e2 , x)(e3 , x) vanishes at ai as do all its transforms by G. These span the four-dimensional permutation representation 4 of A5 . Moreover, since (e1 , x)(e2 , x)(e3 , x) satisfies Laplace’s equation, this is the orthogonal complement of U in V , hence U has dimension 3. Every icosahedron therefore defines a point in the Mukai-Umemura manifold and the three-dimensional orbit SO(3, C)/G is a dense open set in Z. Generically, a harmonic cubic f defines a section s of E ∗ which vanishes at two points in this open set, which means that the corresponding plane cubic C contains the twelve points in P2 defined by the axes of two icosahedra. Put another way, if a generic cubic curve contains one such “icosahedral set” {[a1 ], [a2 ], . . . , [a6 ]} ⊂ P2 then it contains another. 6. Degenerate icosahedra 6.1. The two types. We need to understand also the divisor of “degenerate icosahedral sets” which forms the complement of the open orbit of SO(3, C) in Z. These correspond to isotropic subspaces of two types, constituting two orbits, of dimension 2 and 1 respectively. A representative of the first type is the space with basis 3 (b, x)3 − (b, x)(b, b)(x, x), (a, x)2 (b, x), (a, x)3 5 where (a, a) = 0, (a, b) = 0 and (b, b) = 0. This subspace is invariant by the rotations about the axis b and is spanned by the weight spaces {0, 2, 3} for that action. Geometrically, the isotropic subspace is defined by [b] ∈ P2 (a point not on the null conic Q defined by (x, x) = 0) together with the point of contact [a] of a tangent to Q through [b]. The choice of tangent is a double covering of P2 \ Q, which is abstractly an affine quadric. The icosahedral set here describes five of the vectors {a1 , a2 , . . . , a6 } coalescing into a single vector a. Under this degeneration the relation (aj , ai )2 − 15 (ai , ai )(aj , aj ) = 0 implies that a is null and the sixth vector b satisfies (b, a) = 0. The second type is a subspace with basis 1 (a, x)((b, x)2 − (b, b)(x, x)), 5

(a, x)2 (b, x),

(a, x)3

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where (a, a) = 0 and (a, b) = 0. This is geometrically defined by the point [a] ∈ Q and is invariant by the Borel subgroup which fixes [a]. Decomposing with respect to a semisimple element this is the span of weight spaces {1, 2, 3}. Here all six vectors {a1 , a2 , . . . , a6 } coalesce into a single null vector a. The union of these two orbits forms an anticanonical divisor in Z, for if X1 , X2 , X3 are the vector fields on Z generated by a basis of so(3), then X1 ∧ X2 ∧ X3 vanishes on the lower-dimensional orbits and this is a section of KZ∗ = Λ3 T Z. Since c1 (Z) is a generator of H 2 (Z, Z) it must vanish with multiplicity 1. Note that this anticanonical divisor D cannot be smooth (for then it would be a K3 surface or a torus). It is instead a singular image of a map α : P1 × P1 → Z (in fact, a transverse to the diagonal in P1 × P1 has a cusp singularity y 2 = x3 (see [7]). To define α recall that we have identified the type 1 orbit as an affine quadric, the complement of the diagonal in P1 ×P1 . This is an ordered pair of points [a], [a ] on the conic Q ∼ = P1 , where the two tangents meet at [b] ∈ P2 . To extend the map α to the diagonal put c = a + tb, with (c, a) = 0 and let t → 0. Then 3 (c, x)3 − (c, x)(c, c)(x, x) = 5 1 = (a, x)3 + 3t(a, x)2 (b, x) + 3t2 [(a, x)((b, x)2 − (b, b)(x, x))] + . . . 5 and the three leading coefficients span the type 2 isotropic subspace. Consider a degenerate icosahedral set of type 1, spanned by the three cubic polynomials (b, x)3 − 3(b, x)(b, b)(x, x)/5, (a, x)2(b, x) and (a, x)3 . Then f is orthogonal to this if the curve C given by f (x) = 0 intersects the conic Q tangentially at [a] and passes through [b]. For type 2, the intersection multiplicity of C with Q at [a] must be at least 3. 6.2. The universal bundle. We should also consider the universal bundle E on the divisor D, or rather its pullback α∗ E on P1 × P1 . Note that both types of degenerate isotropic subspace contain (a, x)3 and (a, x)2 (b, x) where (b, a) = 0, or equivalently the two-dimensional subspace given by (a, x)2 (c, x) for all c orthogonal to a. We use the two-fold covering map π : P1 × P1 → P2 , the quotient by the involution interchanging the factors. Each factor is isomorphic to the diagonal which maps to the conic Q. As usual, geometrically π([a], [a ]) is the point [b] of intersection of the tangents at [a], [a ] ∈ Q. The two-dimensional vector space a⊥ ⊂ C3 for [a] ∈ Q defines a vector bundle A over Q for which the projective bundle P(A) is trivial. Indeed, it is the bundle of tangent lines to the conic and so under the map π can be identified with the family P1 × {x}, x ∈ P1 , which is a trivial bundle over the diagonal. On the other hand since (a, a) = 0 a⊥ ∼ = (C3 /OP2 (−1))∗ and so Λ2 A ∼ = OP2 (−1) = O(−2), identifying Q with P1 . Hence, since P(A) is trivial, A∼ = O(−1) ⊗ C2 . Multiplying by the factor (a, x)2 which is quadratic in a, it follows that the subbundle in α∗ E of cubics of the form (a, x)2 (c, x) with (c, a) = 0 is isomorphic to O(−5, 0) ⊗ C2 .


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The cubic (b, x)3 − 3(b, x)(b, b)(x, x)/5, together with the subspace A, spans a type 1 degenerate subspace for (b, b) = 0 and when (b, b) = 0 it lies in A. This term is homogeneous of degree 3 in b, and so defines a homomorphism from π ∗ OP2 (−3) = O(−3, −3) to α∗ E. It projects to a section of the line bundle α∗ E/A which vanishes on the diagonal Δ ⊂ P1 × P1 . Hence α∗ E/A is of the form O(k, k) and Λ3 α ∗ E ∼ = O(k − 10, k). But for Z we have the Chern number c31 = 22 and since D is an anticanonical divisor, c21 [D] = 22. However c1 (E) = −c1 (Z) and so 22 = 2k(k − 10) and k = −1. Thus α∗ E is an extension 0 → O(−5, 0) ⊗ C2 → α∗ E → O(−1, −1) → 0. This extension is classified by an element of H 1 (P1 × P1 , O(−4, 1)) ⊗ C2 . It is also by definition SO(3, C)-invariant by the diagonal action on P1 × P1 . But as representation spaces H 1 (P1 × P1 , O(−4, 1)) ⊗ C2 ∼ = H 1 (P1 , O(−4)) ⊗ H 0 (P1 , O(1)) ⊗ C2 ∼ = S2 ⊗ S ⊗ S and α∗ E is defined by a non-zero vector in the unique invariant one-dimensional subspace. The sections of α∗ E ∗ fit into the exact sequence 0 → H 0 (P1 × P1 , O(1, 1)) → H 0 (P1 × P1 , α∗ E ∗ ) → H 0 (P1 , O(5)) ⊗ C2 → 0. As a representation space we then have H 0 (P1 × P1 , α∗ E ∗ ) ∼ = S ⊗ S + S ⊗ S5 which contains with multiplicity one the seven-dimensional representation S 6 as a subspace of S ⊗ S 5 . These sections are the restriction to D of the sections (f, −) on Z that we have considered earlier. There is one important consequence of this: Proposition 6.1. A harmonic cubic f is orthogonal to a degenerate isotropic subspace if and only if Δ(f ) = 0 for a certain SO(3, C)-invariant polynomial Δ of degree 10. Proof. We want to know when a section s of E ∗ defined by (f, −) vanishes on D. If it does vanish somewhere then the map to H 0 (P1 , O(5)) ⊗ C2 gives two sections of O(5) with a common zero. These arise from S 6 ⊂ S 5 ⊗S so the condition is that we have a homogeneous polynomial p(z1 , z2 ) of degree 6 such that the two partial derivatives ∂p/∂z1 , ∂p/∂z2 , homogeneous of degree 5, have a common zero. The vanishing of the resultant is the condition. This is a degree 10 polynomial Δ in the coefficients of f , the discriminant. Its vanishing implies that there is a point [a] = [z1 , z2 ] ∈ P1 where the section of O(6) has a double zero. Then the section of O(5, 0) ⊗ C2 vanishes on {[a]} × P1 ⊂ P1 × P1 . The dual of the inclusion O(−3, −3) ⊂ α∗ E gives a homomorphism α∗ E ∗ → O(3, 3) which determines the third component of the section s. It maps H 0 (P1 × P1 , E ∗ ) to H 0 (P1 × P1 , O(3, 3)) ∼ = S3 ⊗ S3

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and our seven-dimensional representation space S 6 maps into the symmetric elements. Such a section is then the pull-back of a section of OP2 (3) - the cubic f . In particular, its divisor passes through ([a], [a]) where the other two components of s vanish. We see then that the section s vanishes at a point of D if the discriminant vanishes. Geometrically this implies that the cubic curve C meets the null conic tangentially at [a]. 7. The Clebsch cubic surface The standard permutation representation of A5 is the four-dimensional subspace 4 ⊂ C5 defined by y1 + y2 + · · · + y5 = 0. We shall investigate next the geometry of the corresponding three-dimensional projective space P3 . The ring of invariants for A5 on 4 is generated by the elementary symmetric functions σ2 , σ3 , σ4 , σ5 in (y1 , y2 , . . . , y5 ) and the degree 10 invariant (yi − yj ). i<j

y12 + y22 + · · · + y52

defines a nonsingular quadratic form on 4 and then The invariant orthogonality of subspaces defines polarity in P3 : each point has a polar plane, and each line a polar line. There is a unique invariant cubic, which we can write as either σ3 = 0 or more usually as y13 + y23 + · · · + y53 = 0. This equation defines the Clebsch cubic surface S. It is nonsingular and is invariant under the action of the full symmetric group S5 . Consider now six points [a1 ], [a2 ], . . . , [a6 ] ∈ P2 forming an icosahedral set. No three are collinear and no conic passes through them all, for then it would be invariant by the icosahedral group and so given by the null conic (x, x) = 0; but the vertices ai are not null. It follows that blowing up the six points gives a nonsingular cubic surface in P3 . More precisely, the embedding is given by the plane cubic curves that pass through the six points, which is the representation 4 of A5 . Hence, by uniqueness, the invariant cubic surface must be the Clebsch surface S. The blown up points form six disjoint lines in S, half of a double-six configuration. The other six are given by the proper transforms of conics passing through five of the six points. These we have encountered already: 1 (ai , x)2 − (ai , ai )(x, x) = 0. 5 ˜ 2 . MoreBlowing these down gives another map from S to a projective plane P over since the icosahedral group permutes these lines, it is the plane of a threedimensional representation. These two planes are the projective spaces of the two ˜ of A5 . (These can be viewed inequivalent three-dimensional representations 3 and 3 as the self-dual and anti-self-dual two-forms on 4). Since the null conic Q ⊂ P2 does not meet the points [ai ] to be blown up, it lifts to a rational curve R ⊂ S with self-intersection number 4, and hence of degree 6 in P3 . The null conic in the second projective plane similarly lifts to a degree 6 curve ˜ ⊂ S. Each of these is individually invariant by the group A5 and interchanged R by S5 .


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˜ have a rather special relationship: the polar plane The rational curves R and R ˜ and vice-versa. Furthermore the polar of a point p ∈ R is a tritangent plane to R ˜ This classical result can be line of the tangent line through p is a trisecant of R. found in [14]. It establishes a 3 : 3 correspondence between the two curves. 8. The trichotomy for P3 So far we have introduced three pieces of geometry – the vector bundle on a harmonic cubic curve, the Mukai-Umemura threefold, and the projective space P3 of the permutation representation of A5 . We seek the distinguished SO(3, C)invariant subvarieties in the six-dimensional projective space P(V ) which describe the three different types of bundles. To do this we first restrict to P3 ⊂ P(V ), where we shall see the classical geometry described in the previous section playing a role. The first step is to establish a rational correspondence between the MukaiUmemura threefold and P3 . 8.1. From Mukai-Umemura to P3 . The Mukai-Umemura threefold parametrizes three-dimensional isotropic subspaces of V . Fix a nondegenerate one, U0 , defined by the icosahedral set [a1 ], [a2 ], . . . , [a6 ] as a base point. For any other U we can consider the subspace U + U0 ⊂ V . Generically this has dimension 6 and so has a one-dimensional orthogonal complement spanned by a harmonic cubic f : f then has two orthogonal isotropic subspaces, U and U0 . This establishes a rational map from Z to P3 = P(U0⊥ ), but we must examine the indeterminacy. Proposition 8.1. If dim(U + U0 ) < 6 then it is defined by an icosahedral set with a point in common with [a1 ], [a2 ], . . . , [a6 ]. Proof. Suppose that dim(U + U0 ) < 6, then dim(U + U0 )⊥ ≥ 2 and there is a pencil of cubics passing through the two icosahedral sets. Suppose that the icosahedral set defined by U is nondegenerate: [b1 ], [b2 ], . . . , [b6 ]. If two axes of an icosahedron coincide then so do all of the axes, so if the icosahedra are distinct there must be at least 11 points for the pencil to pass through, but by Bézout’s theorem, unless there is a common component in the pencil, the maximum number of intersections is nine : we deduce that there must be a common component. If it is a line, then since no three of the points [ai ] or [bi ] are collinear there is a maximum of four – two from each set – on the line. But then we have seven remaining for the pencil of conics. By Bézout again we reach a contradiction. If the common component is a conic, then it contains at most five points from each set, and the remaining point must lie in a pencil of lines and so for some i, j, [ai ] = [bj ]. The equation is therefore of the form 1 (8.1) (c, x)((a, x)2 − (a, a)(x, x)) 5 where [a] = [ai ] = [bj ] and (c, a) = 0. If f is orthogonal to a degenerate isotropic subspace then the cubic is tangential to the conic Q at a point [a] and meets the tangent line at a point [b]. The pencil thus consists of the pencil of planes in P3 containing the tangent line to the rational curve R ∼ = Q at the corresponding point p. Unless the tangent line is contained in the Clebsch cubic S, the generic curve in the pencil is smooth at p, and we can apply the Cayley-Bacharach theorem (in the degenerate case as in [9] p.672) to deduce that any cubic which passes through the six points [b], [a2 ], [a3 ], . . . , [a6 ] and

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is also tangent to Q at [a] must pass through [a1 ]. But take the cubic which is the conic through [a2 ], [a3 ], . . . , [a6 ] together with the tangent at [a]: 1 (a, x)((a1 , x)2 − (a1 , a1 )(x, x)) = 0. 5 If a1 lies on this then (a, a1 ) = 0. Repeating for the other [ai ] we deduce that they are all collinear, which is a contradiction. ˜i , and the pencil is The tangents to R which are contained in S are the lines E the same as (8.1). For each vertex ai of an icosahedron, Proposition 8.1 has highlighted the role of the icosahedra with this as a common vertex. They form an orbit of the group of rotations fixing ai – a C∗ orbit in Z. Its closure is a rational curve with two extra points, fixed by the action, which lie in the divisor D (they are two type 1 degenerate icosahedra, determined by the two tangent lines to Q from [b] = [ai ]). These six rational curves Ci all pass through the basepoint U0 in Z. From Proposition 8.1 there is a well-defined map from the complement of these six curves to P3 : β(U ) = (U + U 0 )⊥ . We shall extend this to Z by blowing up certain subvarieties. (i) First we need to deal with the basepoint U = U0 . Blow up the basepoint in Z, replacing it by the projectivized tangent space. Since the stabilizer of U0 is finite, the tangent space is naturally so(3). For each b ∈ so(3) ∼ = C3 we orthogonally ⊥ project fb onto U0 to get pb . If this projection is zero then (f, fb ) = f (b) = 0 for all f ∈ U0 , so that any cubic which vanishes at the ai also vanishes at b. However, these cubics define a projective embedding of the blown-up plane, and so must separate points. It follows that the projection is zero if and only if [b] = [ai ] for some 1 ≤ i ≤ 6. We can therefore extend β to P2 \ {[a1 ], [a2 ], . . . , [a6 ]} by using the orthogonal projection pb . (ii) The six proper transforms of the curves Ci meet the exceptional divisor in the points [ai ] and we need to blow up these curves to extend the map. Consider pb(t) for b(t) = ai + tv as t → 0, where (v, ai ) = 0. We have 1 2 fb(t) = fai + 3t(v, x) (ai , x) − (ai , ai )(x, x) + . . . 5 Now fai lies in U0 , so

1 pb(t) = 3t(v, x) (ai , x)2 − (ai , ai )(x, x) + . . . 5

and the coefficient of t extends β to the blow-up of the projectivized tangent space at [ai ]. Its image in P3 is the Clebsch cubic. The exceptional curve Ei obtained by blowing up [ai ] maps to the pencil in Proposition 8.1. Denote by Zˆ the blown-up Mukai-Umemura threefold. Remark 8.2. Blowing up Ci picks out a cubic of the form 1 (8.2) (c, x)((ai , x)2 − (ai , ai )(x, x)) 5 with (c, ai ) = 0 which extends the map β. The normal bundle of Ci is trivial so the blow-up is Ci × P1 ∼ = P1 × P1 and β collapses the first factor. This corresponds to the fact that the cubic (8.2) contains the one-parameter family of icosahedral sets


81 11

which include [ai ]. In general the positive-dimensional fibres of the map β give us cubics containing infinitely many icosahedral sets. 8.2. The trisecant surface. The image β(D) ⊂ Zˆ in P3 consists of those polynomials f ∈ U0⊥ which are also orthogonal to a degenerate isotropic subspace. From Proposition 6.1 this is given by the vanishing of an A5 -invariant polynomial of degree 10, the restriction of the SO(3, C)-invariant polynomial Δ. On the other hand, we know that if f is orthogonal to a degenerate isotropic subspace, the cubic curve C is defined by a plane section of S which is tangent to the rational curve R. This means that [f ] ∈ P3 lies on the polar line of a tangent ˜ observed in Section line to R. From the special properties of the curves R and R ˜ 7 this is a trisecant of R. So the degree 10 polynomial vanishes on the trisecant ˜ surface T of R. ˜ if its polar line meets d tangents Now a generic line meets d trisecants of R to R. But this surface is the tangent developable surface and we can calculate the degree easily. If X is a curve of degree n and C2 ⊂ H 0 (P3 , O(1)) is the pencil of planes through a line then under the one-jet evaluation map ev : H 0 (P3 , O(1)) → J 1 (X, OP3 (1)) we have a section of

Λ2 (J 1 (X, OP3 (1))) ∼ = KX (2)

and the degree of this is 2g − 2 + 2n. It vanishes at the points of intersection of and the surface. In our case R is of degree 6 and rational so the developable has degree 0 − 2 + 12 = 10. Thus the trisecant surface, which is irreducible, is of degree 10 and given by the vanishing of the invariant polynomial Δ of Proposition 6.1. It is the locus of cubics which pass through a nondegenerate icosahedral set [a1 ], [a2 ], . . . , [a6 ] and a degenerate one. 8.3. The three cases for P3 . Consider now the trichotomy in the light of the construction of the map β : Zˆ → P3 . The trisecant surface T = β(D) consists of cubics which contain a nondegenerate icosahedral set and a degenerate one. They contain therefore two or infinitely many. The image of the six blown-up curves Ci have infinitely many. This image moreover lies in the Clebsch cubic. There only remains the rest of the Clebsch cubic, for which there is either one icosahedral set or infinitely many. The whole surface is therefore the locus of sections of E ∗ in 4 which have a degenerate zero set – the second and third cases of the trichotomy. We wish finally to determine which cubic curves contain infinitely many icosahedral sets, that is, which of the sections of E ∗ on Z vanish on a positive-dimensional variety Y ⊂ Z. Since c1 (Z) > 0, Y must intersect the anticanonical divisor D nontrivially, and so we are looking for [f ] lying in S and also in the trisecant surface T. ˜ we observed is that the polar of any point on it is Now the special property of R ˜ defines a plane section of S ⊂ P3 a tritangent plane of R. This means that [f ] ∈ R which meets the curve R tangentially at three points. In other words the associated plane cubic C passes through three and hence infinitely many icosahedral sets. A ˜ meets the cubic surface S generically in three points, which are trisecant line of R ˜ Thus for these lines T intersects S only in the three points of intersection with R. ˜ R.

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˜ where There are degenerate cases – twelve points (an A5 -invariant set) on R the tritangent plane to R meets it in two points with multiplicity three instead of ˜ at these points lies entirely in three with multiplicity two. The trisecant line to R the Clebsch cubic – these are the six exceptional divisors E1 , E2 , . . . , E6 . ˜ and the six lines. We have therefore Set-theoretically, T intersects S in R shown: Theorem 8.3. A generic point of P3 defines a cubic which contains precisely two icosahedral sets. If it contains one or infinitely many it lies on the Clebsch ˜⊂S cubic surface S and in the latter case it lies on the degree six rational curve R or on the six lines E1 , . . . , E6 . Remark 8.4. In terms of divisor classes let H be the pull-back of the hyperplane divisor on the original projective plane then a plane cubic C through [a1 ], . . . , [a6 ] lifts to a curve whose divisor is 3H − (E1 + · · · + E6 ). The embedding for a cubic surface in P3 is the anticanonical embedding so −KS ∼ 3H − (E1 + · · · + E6 ). The divisor of R, the lift of a conic in P2 , is 2H. Now consider the other six exceptional curves coming from the conics passing ˜1 ∼ 2H − (E2 + · · · + E6 ) etc. through five of the six points. These have classes E and ˜ − (E ˜1 + · · · + E ˜6 ) = 3H − (E1 + · · · + E6 ) −KS ∼ 3H ˜ is the hyperplane divisor for P ˜ 2. ˜ It follows that H ∼ 5H − 2(E1 + · · · E6 ), where H ˜ ∼ 2H ˜ ∼ 10H − 4(E1 + · · · E6 ), so the divisor class of T on S is Then R −10KS ∼ 10(3H − (E1 + · · · + E6 )) = 3(10H − 4(E1 + · · · E6 )) + 2(E1 + · · · E6 ). We may also note here that ˜ ∼ 2H + 10H − 4(E1 + · · · E6 ) = 4(3H − (E1 + · · · + E6 )) ∼ −4KS R+R so that there is an invariant quartic surface which vanishes on the pair of curves. This is 9σ22 − 20σ4 . 9. The trichotomy for P(V ) Since a generic f ∈ V defines a cubic which contains a finite number of icosahedral sets, and since the stabilizer of an icosahedron is finite, then P3 ⊂ P(V ) sweeps out an open set under the action of SO(3, C). In principle, therefore, the trichotomy for P(V ) entails looking for a hypersurface which intersects P3 in the Clebsch cubic. However, the Hilbert polynomial for the four-dimensional representation space of A5 is 1 + t10 (1 − t2 )(1 − t3 )(1 − t4 )(1 − t5 ) and for the SO(3, C) invariants in the seven-dimensional representation space (1 −

t2 )(1

1 + t15 − t4 )(1 − t6 )(1 − t10 )

so there is no degree 3 invariant for SO(3, C). However, as shown in [12], there is an invariant sextic hypersurface in P(V ) which meets P3 tangentially in the Clebsch cubic. The formula given in [12] is adapted to the context of spherical harmonics and functions on the two-sphere, but here we shall adopt the point of view in [13]


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instead, using the realization of the representation space V as S 6 – homogeneous sextic polynomials in (z1 , z2 ). In inhomogeneous form the sextic is written as u0 x6 + u1 x5 + · · · + u6 with roots x1 , x2 , . . . , x6 . Writing (ij) for xi − xj the discriminant is the degree 10 invariant Δ = u10 (ij)2 . 0 i<j

As we have seen, the vanishing of this describes the locus of cubics f which are orthogonal to a degenerate isotropic subspace. In [13], Igusa defines similar invariants, summing over permutations: A = u20 (12)2 (34)2 (56)2 fifteen

B

=

u40

=

u60

(12)2 (23)2 (31)2 (45)2 (56)2 (64)2

ten

C

(12)2 (23)2 (31)2 (45)2 (56)2 (64)2 (14)2 (25)2 (36)2

sixty

and introduces rational invariants J2 , J4 , J6 , J8 , J10 where J6 =

5AB C A3 + − 221184 13824 576

and J10 = 2−12 Δ. Theorem 9.1. A generic point of P(V ) defines a cubic which contains precisely two icosahedral sets. It contains one or infinitely many if and only if it lies on the hypersurface J6 = 0. Proof. The proof consists of using the explicit forms in Section 4 of [13] to determine a degree six invariant which vanishes on the SO(3, C) transforms of the ˜ and E1 , . . . , E6 . This can be done by evaluating on the normal rational curves R forms below: ˜ gives a sextic polynomial in z with three double zeros, and each (i) A point in R such is equivalent under the action of SO(3, C) to z 2 (z − 1)2 . (ii) The cubic polynomials in Ei are of the form 1 (c, x)((ai , x)2 − (ai , ai )(x, x)) 5 with (c, a) = 0. The line (a, x) = 0 intersects Q given by (x, x) = 0 at two points in general and c is a third point on this line. Its polar line (c, x) = 0 intersects Q in two more points, so the sextic has two double zeros and two simple ones if [c] does not lie on Q. There is a constraint however – identifying Q with P1 the four points must have cross-ratio −1. A normal form for this is z 2 (z 2 − 1). We finally need to consider the points of P(V ) whose SO(3, C) orbits do not intersect P3 . This means cubics which do not contain any nondegenerate icosahedral set. If the cubic contains two degenerate icosahedral sets, then it is tangential twice to the null conic, so in the sextic polynomial interpretation we can take it to the form z 2 (az 2 + bz + c).

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and evaluating J6 gives 1 2 2 b (b − 4ac)2 . 1024 If J6 = 0, then either b2 − 4ac = 0 which is Case (i) above or b = 0 which is Case (ii). 10. A special cubic curve We remarked above that the sextic polynomial z 2 (z − 1)2 is a normal form for an element in S 6 which represents a cubic curve containing an infinite number of icosahedral sets. There is thus a single SO(3, C) orbit for a cubic which is tangential to the null conic at three distinct points. Perhaps the simplest canonical form for this is y 2 = x3 + x2 + 4x + 4 where the null conic has equation y 2 = 4(2x − 3)(x + 1). The two meet at just three points: (4, 10), (4, −10), (−1, 0). Note that this cubic C is defined over the integers and has rational points ∞, (4, −10), (0, −2), (−1, 0), (0, 2), (4, 10) forming a cyclic group of order six. Now C contains a one-parameter family of icosahedral sets, three of which are degenerate. Moreover this family is rational. To see this, we revert to the isotropic subspace description. From Proposition 8.1 we may assume that there are two isotropic subspaces U1 , U2 of W for which U1 ∩ U2 = 0. A third subspace U3 intersects each of these trivially and hence is the graph of a linear transformation S : U1 → U2 . Thus for w1 , w2 ∈ U1 we have 0 = ωx (w1 + Sw1 , w2 + Sw2 ) = ωx (w1 , Sw2 ) + ωx (Sw1 , w2 ) since ωu (w1 , w2 ) = 0 = ωu (Sw1 , Sw2 ). This means the graph of tS for any t ∈ C is also isotropic, so we have a family of isotropic subspaces with rational parameter t. Each icosahedral set consists of six points and so we have a sixfold covering p : C → P1 with three branch points 0, 1, ∞ in P1 . At these points the icosahedron degenerates – one axis remains (the vector b in the type 1 degenerate subspace) and the other five coalesce to be a null vector (the vector a). The inverse image of each branch point consist of two points – one, [a], a ramification point of order 5 and one simple point [b]. These six points are precisely the six rational points in the canonical form above. The projection map p is given in its simplest form by the meromorphic function p2 =

(1 + x)5 (y − x − 2) . (y + 3x − 2)5

An explicit form for p is a little less simple – it involves elliptic functions for two planar embeddings (see Remark 5 in Section 12), which differ by the line bundle L of order 2 , for here we are considering a curve where the vector bundle is L ⊕ L with L2 trivial.


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Remark 10.1. The formulae above have come from [17] in the context of algebraic solutions to the hypergeometric equation. The monodromy of the covering p lies in the icosahedral group and mapping into P SL(2, C) this gives a second order differential equation with three singular points. Schwarz’s celebrated paper [16] classifying these stops short of explicit expressions and the elliptic curve above appears in an attempt to be more concrete about these. 11. Higher degree curves The picture for higher degree representations and curves is not clear – in particular there are stable bundles and not just semi-stable as in the case of the elliptic curve. One might ask for a link between semistability and isotropic subspaces, however. The argument in Section 3 generalizes to show that a d-dimensional isotropic subspace orthogonal to a harmonic polynomial f of degree d defines a line subbundle L with deg L = deg E/2, though it is not clear if the reverse is true. What is true is that isotropic subspaces are rare. We can find some, though, as we show below. Let V now be the (2d + 1)-dimensional irreducible representation space of SO(3, C) and consider a d-dimensional subspace isotropic for all skew forms (x·u, v) which is invariant by SO(2, C) ⊂ SO(3, C). Then it is a sum of d weight spaces Lm where −d ≤ m ≤ d. Let h, n+ , n− be a basis of so(3) where h generates SO(2, C) and [h, n+ ] = n+ , [h, n− ] = −n− . Proposition 11.1. Let U ⊂ V be a d-dimensional SO(2, C)-invariant isotropic subspace of V . Then U is a sum of weight spaces ±{1, 2, 3, . . . , d} or ±{0, 2, . . . , d} Proof. Because h acts as the scalar m on Lm , if a weight m > 0 occurs, then −m does not because on Lm ⊕ L−m , (h · u, v) = muv which is non-zero. Now n+ : Lm → Lm+1 , so on Lm ⊕ L−m−1 , (n+ · u, v) = uv which is non-zero. Hence if m occurs, −m − 1 does not. Similarly with n− , if m occurs −m + 1 does not. Suppose 0 is not one of the d weights and let P be the positive weights. Then the negative weights N must be the negatives of the complement of P in {1, . . . , d} by the first criterion. If m ∈ P then by the second criterion, the adjacent numbers m − 1, m + 1 must be in P . But this means that P consists of 1, 2, 3, . . . , d. If P is empty, then we get −1, −2, −3, . . . , −d If 0 is one of the weights, then by the second criterion ±1 is not a weight. Then P is a subset of 2, . . . , d. Again the d − 1 non-zero weights must be made up of P and the negative of its complement. As before adjacent positive numbers must be in P , so P is 2, 3, . . . , d so the weights are 0, 2, . . . , d or their negatives. Note that these are precisely the degenerate isotropic subspaces which we have encountered for d = 3. So if the curve f (x) = 0 has a high order contact with the conic Q, we may deduce that the bundle E is strictly semistable. 12. Further remarks 1. We have shown here that a generic cubic which passes through one icosahedral set passes through another one. According to Melliez [14] (who also introduced the use of the bundle E ∗ ), this is a result of Reye and can be found in Baker [2], page 145, Exercise 26 or Coble [6] page 236. The reader is invited to make the translation.

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2. A discussion of the real case can be found in [12]. There we consider the picture in the sphere double covering RP2 . The inverse image of the cubic curve is then the so-called nodal set of a spherical harmonic and the issue is whether the nodal ˜ are set contains the vertices of a regular icosahedron. The rational curves R and R not real and so do not appear in the story. ˜ 2 but is much more complicated when ˜ is simply the null conic in P 3. The curve R 2 viewed in P . We saw from Remark 8.4 that its divisor class is 10H − 4(E1 + · · · E6 ) so it is of degree 10. In fact it is the invariant singular curve defined by: 2(x√1 10 + x3 10 + x2 10 ) + 35(x1 8 x2 2 + x1 2 x2 8 + x2 2 x3 8 + x2 8 x3 2 + x3 8 x1 2 + x3 2 x1 8 ) + 25 5(x1 2 x2 8 + x2 2 x3 8 + x3 2 x1 8 − x1 8 x2 2 −√x2 8 x3 2 − x3 8 x1 2 ) − 30(x1 6 x2 4 + x1 4 x2 6 + x2 6 x3 4 + x2 4 x3 6 + x3 6 x1 4 + x3 4 x1 6 ) + 50 5(x1 6 x2 4 + x2 6 x3 4 + x3 6 x1 4 − x1 4 x2 6 − x2 4 x3 6 −x3 4 x1 6 )−560x21 x22 x23 (x41 + +x42 +x43 )+1060x21 x22 x23 (x21 x22 +x22 x23 +x23 x21 ) = 0. 4. The closure of the curves in the Mukai-Umemura threefold defined by icosahedra with a fixed axis are rational curves C with c1 (Z)[C] = 2, i.e degree 2. These are the minimal rational curves appearing in the classification of Fano varieties. The one˜ define degree 3 curves, parameter family of icosahedra parametrized by points of R for the plane cubic C is tangential to the null conic at three points. Whereas degree 3 rational curves were linked to icosahedral solutions to the hypergeometric equation, degree 4 curves give rise to algebraic solutions of a particular case of Painlevé’s sixth equation. This is the setting of [10],[11] where the dihedral and octahedral cases are examined. A classification of all such solutions, akin to Schwarz’s list, has been given by Boalch [4]. To find degree 4 rational curves we consider the map β : Zˆ → P3 of Section ˜ parametrizes cubics containing an infinite number of icosahedral sets, 8.1. Since R ˆ If we want to go the other way, from P3 the map β collapses a subvariety in Z. ˜ to Z, we first blow up R and then do some blowing up and down. Now take a ˜ ⊂ P3 – a line joining two points. A generic line intersects the secant to the curve R trisecant surface T in 10 points. However, the 3 : 3 correspondence between R and ˜ implies that through a generic point of R ˜ there pass three trisecants. Blowing R ˜ up R means that the proper transform of the secant is a curve which meets the blown-up trisecant surface in 10 − 3 − 3 = 4 points, and this becomes a rational curve of degree 4 in the Mukai-Umemura threefold. The corresponding solution to Painlevé VI is known and is due to Dubrovin and Mazzocco [8]. An open set in the Mukai-Umemura threefold can then be identified with the twistor space of a complex self-dual four-manifold and the secants above, after transforming by SO(3, C), give the four-parameter family of twistor lines. 5. One might ask, without reference to the vector bundle E, how a cubic curve defined by a harmonic polynomial gives rise to a line bundle L. It arises from the geometry as follows. As a varies in the cubic C consider the polynomial fa , invariant under the action of a ∈ so(3). Now ωa (fa , v) = (a · fa , v) = 0 so that fa generates a line bundle in the kernel of ω : W → W ∗ (1). We identify this bundle as follows. Consider the map γ : P2 → P(V ) defined by a → fa . Then fa spans the pull-back γ ∗ (O(−1)) and since fa is homogeneous of degree 3 in a, γ ∗ (O(−1)) ∼ = OP2 (−3). Restrict this to C to get the required line bundle. We then have an inclusion O(−3) ⊂ E ∗ (−1) over C. Projecting to ∗ E (−1)/L(−1) gives a section of L∗ (2). This vanishes when fa ∈ U which is


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at the six points [a1 ], . . . , [a6 ] defined by the icosahedral set. Thus on C we have the relation of divisor classes 6 L ∼ O(2) − [ai ]. 1

Now we saw that the cubics through the icosahedral set are given by plane sections of the Clebsch cubic surface. This means that the same cubic is embedded in two different planes – a plane in P3 and the original plane with the conic Q. Since the embedding in P3 is 6 [ai ] O(3) − 1

we see that the degree zero line bundle L is the difference of the two hyperplane divisor classes. References [1] M.F.Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. 7 (1957), 414– 452. [2] H.F.Baker, “Principles of Geometry”, Vol 3 Cambridge Univ. Press, Cambridge (1933). [3] A.Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000) 39–64. [4] P.Boalch, The fifty-two icosahedral solutions to Painlev´ e VI, J. Reine Angew. Math. 596 (2006), 183–214. [5] I. Burban and Y. Drozd, Coherent sheaves on rational curves with simple double points and transversal intersections, Duke Math. J. 121 (2004), 189–229. [6] A. B. Coble, “Algebraic geometry and theta functions,” 10 AMS Coll. Publ., 1929. [7] S.K.Donaldson, K¨ ahler geometry on toric manifolds, and some other manifolds with large symmetry, arXiv:0803.0985 [8] B. Dubrovin and M. Mazzocco, Monodromy of certain Painlev´ e-VI transcendents and reflection groups, Invent. Math. 141 (2000), 55–147. [9] P.Griffiths and J.Harris, “Principles of algebraic geometry”, John Wiley & Sons New York, 1978. [10] N.J.Hitchin, Poncelet polygons and the Painlev´ e equations, in “Geometry and analysis (Bombay, 1992)”, 151 – 85, Oxford University Press, Bombay, 1996. [11] N.J.Hitchin, A lecture on the octahedron, Bulletin of the London Math Soc., 35 (2003), 577–600. [12] N.J.Hitchin, Spherical harmonics and the icosahedron, Groups and symmetries, 215–231, CRM Proc. Lecture Notes, 47, Amer. Math. Soc., Providence, RI, 2009. [13] J-I. Igusa, Arithmetic variety of moduli for genus two, Ann. of Math. 72 (1960), 612–649. [14] F.Melliez, Duality of (1, 5) polarized abelian surfaces, Math. Nachr. 253 (2003), 55–80. [15] S.Mukai, Fano 3-folds, in “Complex projective geometry (Trieste, 1989/Bergen, 1989)” 255– 263, London Math. Soc. Lecture Note Ser. 179 Cambridge Univ. Press, Cambridge (1992). ¨ [16] H.A.Schwarz, Uber diejenigen Falle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Funktion ihres vierten Elements darstellt, J. Reine Angew. Math. 75 (1872). [17] R.Vidu˜ nas, Darboux evaluations of algebraic Gauss hypergeometric functions, arXiv:math 0504264 Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, UK E-mail address: [email protected]



Cohomology of the Toroidal Compactification of A3 Klaus Hulek and Orsola Tommasi Abstract. We prove that the cohomology groups with rational coefficients of of the moduli space of abelian threefolds the Voronoi compactification AVor 3 coincide with the Chow groups of that space, as determined by Van der Geer.

1. Introduction The moduli space Ag of principally polarized abelian varieties has several comand various toroidal compactifications, notably the Satake compactification ASat g pactifications. Among the toroidal compactifications the so called Voronoi compactification AVor is distinguished by the fact that it represents a geometrically g meaningful functor, as was shown by Alexeev [A] and Olsson [O]. Toroidal compactifications are defined by suitable fans in the cone of semi-positive symmetric real (g × g) matrices and in the case of AVor the fan is given by the second Voronoi g decomposition. For a definition of the second Voronoi fan we refer the reader to [V] or for a more modern reference to [AN]. A general discussion of toroidal compactifications of Ag can be found in the survey article [HS]. In genus 3 all known toroidal compactifications of the moduli space A3 of principally polarized abelian varieties coincide with the Voronoi compactification AVor 3 . We recall the explicit description of the second Voronoi decomposition in the case g = 3 in Section 5. A detailed description of the geometry of the space AVor 3 can be found in [Ts], and the Chow ring of this space has been computed by Van der Geer [vdG]. In this note we compute the cohomology groups with rational coefficients and prove that they coincide with the Chow groups of this space. Theorem 1.1. The Betti numbers of AVor are b0 = b12 = 1, b2 = b10 = 2, 3 b4 = b8 = 4 and b6 = 6. Our approach is similar to that of [vdG], and is based on a study of the stratification of AVor defined by the torus rank, which we introduce in Section 2.1. 3 2010 Mathematics Subject Classification. Primary 14K10; Secondary 14C15, 14F25, 14D22. Key words and phrases. Abelian varieties, Voronoi compactification, Chow ring, cohomology ring. Partial support from DFG under grant Hu/6-1 is gratefully acknowledged. We are also grateful to the referee for careful reading and valuable suggestions concerning the presentation. c Mathematical 0000 (copyright Society holder) c 2010 American

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We shall give the proof of the main result in Section 2.4 modulo the computation of the cohomology of the various strata, which will be done in the subsequent sections. As a corollary we obtain Corollary 1.2. The cycle map defines an isomorphism CH • (AVor ) ⊗ Q ∼ = H • (AVor ; Q) 3

3

between the Chow ring and the cohomology ring of AVor with rational coefficients. 3 Proof. The Betti numbers coincide with the rank of the Chow groups as determined by Van der Geer ([vdG]). Since the intersection pairing is non-degenerate, the cycle map gives an isomorphism. Although this result is not particularly surprising, we could not find a reference to it in the literature, so we decided to fill the gap with this note. We would like to remark that there are other possible approaches that yield the same result. For instance, one can consider the Torelli map M3 → AVor from 3 the moduli space of Deligne–Mumford stable curves of genus 3 to the toroidal compactification of A3 . The moduli space M3 has a stratification by topological type. Since the Torelli map for genus 3 is surjective, we can stratify AVor by taking 3 obtained the images of the strata of M3 . It is easy to show that all strata of AVor 3 in this way are isomorphic to finite quotients of products of moduli spaces Mg,n with g ≤ 3 and 0 ≤ n ≤ 2(g − 3). Then one can use the known results about the cohomology of these spaces Mg,n to calculate the cohomology of AVor 3 . Vor In this note, we will work with the stack A3 rather than the associated coarse moduli space. We recall that AVor is a smooth Deligne–Mumford stack. Hence the 3 rational cohomology of the stack and the associated coarse moduli space coincide. Finally we remark that the same techniques also apply to the (easier) case of genus 2. Remark 1.3. There is an isomorphism CH • (AVor ) ⊗ Q ∼ = H • (AVor ; Q). 2

2

Notation. Ag Xg ASat g AVor g XgVor Mg,n

moduli stack of principally polarized abelian varieties of genus g universal family over Ag Satake compactification of Ag Voronoi compactification of Ag universal family over AVor g moduli stack of non-singular curves of genus g with n marked points Mg := Mg,0 Sd symmetric group in d letters

→ ASat the natural map from the Voronoi For every g, we denote by ϕg : AVor g g be the universal family, to the Satake compactification. Let πg : XgVor → AVor g

COHOMOLOGY OF THE TOROIDAL COMPACTIFICATION OF A3

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qg : XgVor → XgVor / ± 1 the quotient map from the universal family to the universal the universal Kummer morphism. Kummer family and kg : XgVor / ± 1 → AVor g Throughout the paper, we work over the field C of complex numbers. 2. Stratification and outline of the proof 2.1. A stratification. The object of this note is the rational cohomology of the toroidal compactification AVor of the moduli space of abelian varieties of 3 dimension 3. We shall make use of a natural stratification of AVor which was also 3 used by Van der Geer [vdG], whose notation we adopt. Recall that there is a natural map ϕ3 : AVor → ASat to the Satake compactifi3 3 Sat cation. The moduli space ASat admits a stratification A = A3 A2 A1 A0 . 3 3 This defines a filtration {βt }0≤t≤3 on AVor , by setting 3 ⎛ ⎞ ⎝ Aj ⎠ . βt := ϕ−1 3 0≤j≤g−t

is the locus of semi-abelian varieties with torus rank at In other words, βt ⊂ AVor 3 least t. 2.2. Cohomology of the strata. We shall now state the results about the cohomology with compact support of the various strata. Proofs will be given in the subsequent Sections 3 – 5. The stratum β0 \ β1 of the filtration {βt } is A3 . Its cohomology was computed by Hain in [H]. Theorem 2.1. The rational cohomology groups with compact support of A3 are given by ⎧ Q(−6) k = 12, ⎪ ⎪ ⎪ ⎪ ⎨ Q(−5) k = 10, Q(−4) k = 8, Hck (A3 ; Q) = ⎪ ⎪ F k = 6, ⎪ ⎪ ⎩ 0 otherwise, where F is a two-dimensional mixed Hodge structure which is an extension 0 → Q → F → Q(−3) → 0. Proof. This is a rephrasing of [H, Thm 1], by using the isomorphism Hck (A3 ; Q) ∼ = H 12−k (A3 ; Q)∗ ⊗ Q(−6) given by Poincaré duality on the 6-dimensional space A3 .

The cohomology with compact support of the other strata is as follows. Proposition 2.2. The cohomology with compact support of β1 \ β2 is given by Hc10 (β1 \ β2 ; Q) = Q(−5) Hc8 (β1 \ β2 ; Q) = Q(−4)2 Hc6 (β1 \ β2 ; Q) = Q(−3)2 For torus rank 2 we obtain

Hc5 (β1 \ β2 ; Q) = Q Hc4 (β1 \ β2 ; Q) = Q(−2) Hck (β1 \ β2 ; Q) = 0 for k ∈ / {4, 5, 6, 8, 10}.

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Proposition 2.3. The cohomology with compact support of the stratum β2 \ β3 is given by Hc8 (β2 \ β3 ; Q) = Q(−4) Hc6 (β2 \ β3 ; Q) = Q(−3)2 Hc4 (β2 \ β3 ; Q) = Q(−2)

Hc2 (β2 \ β3 ; Q) = Q(−1) Hck (β2 \ β3 ; Q) = 0 for k ∈ / {2, 4, 6, 8}.

In the proof of the two propositions above, we make use of the following fact factors through AVor (see [Ts]): the natural map β1 → ASat 2 2 , giving rise to the commutative diagram k2 / AVor β1 B 2 BB BBπ2 BB ϕ2 B! ASat 2

∼ (X Vor / ± 1) → AVor is the universal Kummer variety over AVor . where k2 : β1 = 2 2 2 Finally, we use the toroidal description of AVor to compute the cohomology of 3 the stratum with torus rank 3. The corresponding result is Proposition 2.4. The cohomology groups Hck (β3 ; Q) are trivial for degree k ∈ / {0, 2, 4, 6}, and are given in the other cases by Hc6 (β3 ; Q) = Q(−3) Hc4 (β3 ; Q) = Q(−2)2

Hc2 (β3 ; Q) = Q(−1) Hc0 (β3 ; Q) = Q.

Moreover, the generators of these cohomology groups with compact support can be identified with the fundamental classes of the strata of AVor corresponding to the 3 (3) (4) (4) cones σlocal , σI , σII , σ (5) and σ (6) (to be defined in Section 5). 2.3. Spectral sequences in cohomology. Our proofs of results on the cohomology of AVor and its strata are based on an intensive use of long exact sequences 3 and spectral sequences in cohomology with compact support. We shall recall the definition of the sequences we use most often in the proofs. Since the cohomology with rational coefficients of a Deligne–Mumford stack coincides with that of its coarse moduli space, in this section we will work with quasi-projective varieties. A more stack-theoretical approach can be obtained by recalling that AVor is the finite quotient of the fine moduli scheme AVor g g (n) of abelian varieties with level-n structure for n ≥ 3. Then the same constructions can be obtained by working on AVor g (n) equivariantly. Recall that if X is a quasi-projective variety and Y a closed subvariety of X, then the inclusion Y → X induces a Gysin long exact sequence in cohomology with compact support: · · · → Hck−1 (Y ; Q) → Hck (X \ Y ; Q) → Hck (X; Q) → Hck (Y ; Q) → · · · By functoriality of mixed Hodge structures ([PS, Prop. 5.54], this exact sequence respects mixed Hodge structures. Next, assume we have a filtration ∅ = Y0 ⊂ Y1 ⊂ Y2 ⊂ · · · ⊂ YN = X by closed subvarieties of X. In this case, there is a spectral sequence Erp,q ⇒ Hcp+q (X; Q) associated to the filtration {Yi }. The E1 term is given by E1p,q = Hcp+q (Yp \ Yp−1 ; Q). This spectral sequence can be constructed by taking a compactification X of X


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with border S := X \ X. Let us denote by Yi the closure of Yi in X, and consider the filtration {Yi := Yi ∪ S}0≤i≤N of the pair (X, S). In particular, one has H • (Yj , Yj−1 ; Q) = Hc• (Yj \ Yj−1 ; Q) for all j ≥ 1. One can describe the spectral sequence associated to {Yi } as the spectral sequence associated to the bigraded ex ; Q) and E α,β = H α+β (Yα , Yα−1 ; Q), act couple (D, E) with Dα,β = H α+β (X, Yα−1 α+β α+β (X; Q). Arguing as in [Ar, Lemma 3.8], which converges to Hc (X, S; Q) = H this ensures the compatibility with mixed Hodge structures by functoriality. For the definition of exact couples, see [PS, §A.3.2]. Note that the d1 differentials of the spectral sequence in cohomology with compact support associated to {Yi } coincide with the differentials of the Gysin long exact sequences associated to the closed inclusions Yi \ Yi−1 → Yi+1 \ Yi−1 . Leray spectral sequences play an intensive role in our computation of the cohomology of the strata β1 and β2 \ β1 . Typically, we will be in the following situation: let X and Y be quasi-projective varieties, and f : X → Y a fibration with fibres which are homotopy equivalent under proper maps to a fixed quasi-projective variety B. Let us denote by H(p) the local system on X induced by the pth cohomology group with compact support of the fibre of f . In this situation, one can consider the Leray spectral sequence of cohomology with compact support associated to f . This is the spectral sequence Erp,q ⇒ Hcp+q (X; Q) with E2p,q ∼ = H p (Y ; H(q) ). Note that the Leray spectral sequence associated to f respects Hodge mixed structures (e.g. see [PS, Cor. 6.7]). 2.4. Proof of the main theorem. The results on the cohomology with compact support stated in Section 2.2 enable us to compute the cohomology of AVor 3 p,q using the spectral sequence E•p,q ⇒ Hcp+q (AVor = Hcp+q (β3−p \ β4−p ; Q) 3 ; Q), E1 associated to the filtration β3 ⊂ β2 ⊂ β1 ⊂ β0 = AVor 3 . Lemma 2.5. The E1 term of the spectral sequence in cohomology with compact support associated to the filtration β3 ⊂ β2 ⊂ β1 ⊂ β0 = AVor is as given in Table 2. 3 2,3 3,3 The only non-trivial differential of this spectral sequence is d2,3 1 : E1 → E1 , which is injective. In particular, the spectral sequence degenerates at E2 . Proof. The description of the E2 term of the spectral sequence follows from the description of the compactly supported cohomology of the strata given in Section 2.2 and from the definition of the spectral sequence in Section 2.3. An inspection of the spectral sequence in Table 2 yields that E1p,q (and hence Erp,q ) is always trivial if p + q is odd, with the exception of E12,3 (hence possibly also Er2,3 for r ≥ 2). Therefore, all differentials not involving Er2,3 terms are necessarily trivial, since they are maps either from or to 0. This leaves us with only three possibly non-trivial differentials to investigate. The first two are the differentials d2−r,2+r : Er2−r,2+r → Er2,3 for r = 1, 2. Note r that in both cases, the Hodge structure on Er2,3 is pure of weight 0, whereas the Hodge structure on Er2−r,2+r is pure of weight 4. Since the weights are different, the differential d2−r,2+r can only be the 0 morphism. r 2,3 → E13,3 , which can have rank Next, we investigate the differential d2,3 1 : E1 either 0 or 1. Assume for the moment that d2,3 1 is the 0 morphism. Then the spectral 2,3 ∼ 5 Vor sequence degenerates at E1 , so that H 5 (AVor 3 ; Q) = Hc (A3 ; Q) = E1 = Q holds. Vor This means that the cohomology of A3 in degree 5 is pure of Hodge weight 0. But AVor is a smooth proper stack, being the quotient by a finite group of the stack 3

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Table 2. E1 term of the spectral sequence converging to Hc• (AVor ; Q) = H • (AVor 3 3 ; Q)

q 9 8 7 6 5 4 3 2 1 0 ( torus rank )

0 0 0 Q(−3) 0 Q(−2)2 0 Q(−1) 0 Q 0 (3)

0 0 Q(−4) 0 Q(−3)2 0 Q(−2) 0 Q(−1) 0 1 (2)

0 Q(−5) 0 Q(−4)2 0 Q(−3)2 Q Q(−2) 0 0 2 (1)

Q(−6) 0 Q(−5) 0 Q(−4) 0 F 0 0 0 3 (0)

p

AVor 3 (n) of principally polarized abelian varieties with a level-n structure, which is represented by a smooth projective scheme for n ≥ 3. In particular, the Hodge 2,3 structure on H k (AVor must be 1. 3 ; Q) is pure of weight k. Hence, the rank of d1 Therefore, this differential is injective with cokernel isomorphic to Q(−3). This ensures E22,3 = 0 and E23,3 = Q(−3). Note that Lemma 2.5 directly implies that the cohomology of AVor is all alge3 braic, with Betti numbers as stated in Theorem 1.1. In the remainder of this paper we will discuss the various strata defined by the torus rank and compute their cohomology. 3. Torus rank 1 To compute the cohomology with compact support of β1 \β2 we will use the map k2 : β1 \ β2 → A2 realizing β1 \ β2 as the universal Kummer variety over A2 . The fibre of β1 \ β2 over a point parametrizing an abelian surface S is K := S/ ± 1. The cohomology of K is one-dimensional 2 in 1degree 0 and 4. The only other non-trivial cohomology group is H 2 (K; Q) ∼ H (S; Q). = To compute Hc• (β1 \ β2 ; Q), we consider the Leray spectral sequence associated to k2 . Note that the 0th and the fourth cohomology group of the fibre induce trivial local systems on A2 . Moreover, the second cohomology group of the fibre induces the rank 6 local system V(1,1) ⊕ Q(−1) on A2 . Here we denote by V(1,1) the symplectic local system on A2 determined by the irreducible representation of Sp(4, Q) associated to the partition (1, 1). We start by determining the cohomology with compact support of A2 with values in the local system V(1,1) . Lemma 3.1. The rational cohomology groups with compact support of the moduli spaces M2 and A2 with coefficients in V(1,1) vanish in degree k = 3. In degree 3, one has Hc3 (A2 ; V(1,1) ) = Hc3 (M2 ; V(1,1) ) = Q.


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Proof. We prove the claim about the cohomology of M2 first. Following the approach of [G2], we use the forgetful map p2 : M2,2 → M2 to obtain information. Note that the fibre of p2 is the configuration space of 2 distinct points on a genus 2 curve. The cohomology of M2,2 , with the action of the symmetric group, was computed in [T, Cor. III.2.2]. This result allows us to conclude Hc3 (M2 ; V(1,1) ) = Q, Hck (M2 ; V(1,1) ) = 0 for k = 3. (Note that this is in agreement with the Hodge Euler characteristic of M2 in the local system V(1,1) computed in [G2, §8.2].) Next, we determine Hc• (A2 ; V(1,1) ). To this end, we write A2 as the disjoint union of the locus A1,1 of decomposable abelian surfaces, and the image of the Torelli map t : M2 → A2 . Since the Torelli map is injective on the associated coarse moduli spaces, it induces an isomorphism between the cohomology of M2 and that of t(M2 ) in every system of coefficients that is locally isomorphic to a Q-vector bundle. Therefore, the Gysin long exact sequence with V(1,1) -coefficients associated to A1,1 → A2 yields Hck−1 (A1,1 ; V(1,1) ) → Hck (M2 ; V(1,1) ) → Hck (A2 ; V(1,1) ) → Hck (A1,1 ; V(1,1) ). In Lemma 3.3 below, we will show that Hc• (A1,1 ; V(1,1) ) is trivial. In view of the Gysin exact sequence above, this implies that Hck (A2 ; V(1,1) ) is isomorphic to Hck (M2 ; V(1,1) ). This implies the claim. Remark 3.2. Getzler’s result would have been sufficient for the purposes of this note. This follows again from the fact that AVor is a finite quotient of AVor 3 3 (n), so in particular its Hodge Euler characteristic determines the cohomology of the space as graded vector space with Q-Hodge structures. Lemma 3.3. The cohomology with compact support of A1,1 in the local system of coefficients given by the restriction of V(1,1) is trivial. Proof. We consider the restriction of k2 to A1,1 . Let S = E1 × E2 be an element of A1,1 , and let K := k2−1 (S). Recall that V(1,1) ⊕ Q(−1) is the local system H(2) on A1,1 induced by H 2 (K; Q). Therefore, the cohomology of A1,1 with values in V(1,1) ⊕ Q(−1) coincides with the cohomology of A1,1 with values in the local system induced by the part of 2 H 1 (S; Q) which is invariant under the symmetries of E1 × E2 and under the interchange of the two factors E1 , E2 (which can be done topologically albeit not algebraically). Using the K¨ unneth formula one sees that the latter local system is one-dimensional and induces the local system Q(−1). From this one obtains Hc• (A1,1 ; V(1,1) ) = 0. This allows us to show the following result, which directly implies that the cohomology with compact support of β1 \ β2 is as stated in Proposition 2.2. Proof of Proposition 2.2. We compute the cohomology with compact support of β1 \ β2 by using the Leray spectral sequence associated to the Kummer fibration k2 : β1 \ β2 → A2 . By the description of the fibre of k2 given at the beginning of this section, the local systems H(0) and H(4) are the constant one, whereas H(2) is the direct sum of the constant local system Q and V(1,1) . The cohomology with compact support of A2 is well known: it is one-dimensional in degree 4 and 6, and trivial elsewhere. This can be easily deduced from the results in [M2] on the Chow ring of M2 . The cohomology of A2 in the local system

96 8


Table 3. E2 term of the Leray spectral sequence converging to the cohomology with compact support of β1 \ β2

q 4 3 2 1 0

Q(−4) 0 Q(−3) 0 Q(−2) 4

0 0 Q 0 0 3

0 Q(−5) 0 0 0 Q(−4) 0 0 0 Q(−3) 5 6

p

V(1,1) was computed in Lemma 3.1. From this, one obtains that the E2 term of the Leray spectral sequence in cohomology with compact support associated to k2 is as in Table 3. From an inspection of the spectral sequence, one finds that all E2p,q have pure Hodge structures, which have the same Hodge weight if and only if the sums p + q coincide. Therefore, all differentials dr (r ≥ 2) of the spectral sequence are morphisms between Hodge structures of different weight. Hence all differentials are trivial for this reason. This means that the spectral sequence degenerates at E2 , thus implying Proposition 2.2. 4. Torus rank 2 Recall that k2 : β1 → AVor is the universal family of Kummer varieties over 2 Under this map, the elements of AVor with torus rank 2 are mapped to 3 elements of AVor of torus rank 1. If we denote by βt the stratum of AVor of semi2 2 abelian varieties of torus rank ≥ t, we get a commutative diagram

AVor 2 .

AVor 3O ? β 2 \ β3 O

AVor AVor 2O 1O ? k2 / β1 \ β2 k1 / A1? {= t9 O tt {{ t { t q1 q2 { t tt π2 {{ π1 tt {{ X1 π2−1 (β1 \ β2 ) The map π2 is the restriction of the universal family over AVor 2 . In particular, the fibres of π2 over points of β1 \ β2 are rank 1 degenerations of abelian surfaces, i.e. compactified C∗ -bundles over elliptic curves. A geometric description of these C∗ -bundles is given in [M1]. We want to describe this situation in more detail. For this consider the universal Poincaré bundle P → X1 ×A1 Xˆ1 and let U = P(P ⊕ OX1 ×A1 X1 ) be the associated P1 -bundle. Using the principal polarization we can naturally identify Xˆ1 and X1 , which we will do from now on. We denote by Δ the union of the 0-section and the ∞-section of this bundle. Set U = U \ Δ, which is simply the C∗ -bundle given by the universal Poincaré bundle P with the 0-section removed and denote the bundle map by f : U → X1 ×A1 X1 . Then there is a map ρ : U → β2 \ β3 with finite fibres. Note that the two components of Δ are identified under the map ρ. The restriction


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of ρ to both U and to Δ is given by a finite group action, although the group is not the same in the two cases (see the discussion below). We now consider the situation over a fixed point [E] ∈ A1 . For a fixed degree 0 line bundle L0 on E the preimage f −1 (E × {L0 }) is a semi-abelian surface, namely ˆ This semi-abelian the C∗ -bundle given by the extension corresponding to L0 ∈ E. surface admits a Kummer involution ι which acts as x → −x on the base E and by t → 1/t on the fibre over the origin. The Kummer involution ι is defined universally on U . Consider the two involutions i1 , i2 on X1 ×A1 X1 defined by i1 (E, p, q) = (E, −p, −q) and i2 (E, p, q) = (E, q, p) for every elliptic curve E and every p, q ∈ E. These two involutions lift to involutions j1 and j2 on U that act trivially on the fibre of f : U → X1 ×A1 X1 over the origin. Lemma 4.1. The diagram (4.1)

/ X1 ×A1 X1

U ρ|U

(β2 \ β3 ) \ ρ(Δ)

ρ

/ Sym2 (X1 / ± 1), A1

where ρ : X1 ×A1 X1 → Sym2A1 (X1 / ± 1) is the natural map, is commutative. Moreover ρ|U : U → ρ(U ) ⊂ β2 \ β3 is the quotient of U by the subgroup of the automorphism group of U generated by ι, j1 and j2 . Proof. Since the map ρ in the diagram (4.1) has degree 8 and ι, j1 , j2 generate a subgroup of order 8 of the automorphism group of U , it suffices to show that the map ρ|U factors through each of the involutions ι and j1 , j2 . Recall that the elements of β2 \β3 correspond to rank 2 degenerations of abelian threefolds. More precisely, every point of ρ(U ) corresponds to a degenerate abelian threefold X whose normalization is a P1 × P1 -bundle, namely the compactification of a product of two C∗ -bundles on the elliptic curve E given by k1 ◦ k2 ([X]). The degenerate abelian threefold itself is given by identifying the 0-sections and the ∞-sections of the P1 × P1 -bundle. This identification is determined by a complex parameter, namely the point on a fibre of U → X1 ×A1 X1 . Since a degree 0 line bundle L0 and its inverse define isomorphic semi-abelian surfaces and since the role of the two line bundles is symmetric, the map ρ|U factors through ι andj2 . Since j1 is the commutator of ι and j2 the map ρ|U also factors through j1 . A consequence of the lemma above is that the cohomology with compact support of ρ(U ) can be computed by taking the invariant part of the cohomology of the total space of the C∗ -bundle f : U → X1 ×A1 X1 . Hence, the invariant part of the Leray spectral sequence associated to f gives a Leray spectral sequence converging to Hc• (ρ(U ); Q). Thus, we have to consider the part of E2p,q (f ) = Hcq (C∗ ; Q) ⊗ Hcp (X1 ×A1 X1 ; Q) that is invariant under the action of ι, j1 and j2 . Since j1 and j2 both fix the fibre of f over the origin, they act trivially on the cohomology of C∗ . Instead, the Kummer involution ι acts as the identity on Hc2 (C∗ ; Q) and as the alternating representation on Hc1 (C∗ ; Q). The action of ι, j1 and j2 can be determined by considering the induced actions on X1 ×A1 X1 . Here one uses that all three involutions respect the map X1 ×A1 X1 →

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Table 4. E2 term of the spectral sequence converging to the cohomology with compact support of ρ(U )

q 2 1

Q(−2) 0 2

0 Q(−3) 0 Q(−4) 0 Q(−2) 0 0 3 4 5 6

p

A1 , whose fibre over [E] ∈ A1 is isomorphic to E × E. Note in particular that the involution (E, p, q) ↔ (E, −p, q) induced by ι acts as the alternating representation 2 1 on the linear subspace Hc (E; Q) ⊂ Hc2 (E × E; Q), on which i1 and i2 both act trivially. This discussion yields that the invariant part of the spectral sequence E2 term is as shown in Table 4. Lemma 4.2. The cohomology groups with compact support of ρ(U ) are 1-dimensional in degree 6 and 8 and trivial otherwise. 2,2 Proof. It suffices to show that the differential d2,2 → E24,1 in Table 4 2 : E2 is an isomorphism. To describe the differential d2,2 geometrically, it is useful to consider the re2 to the preimage of ρ(U ). Moreover, one striction of the Torelli map t : M3 → AVor 3 can use the stratification of M3 by topological type to describe β2 and ρ(U ). In particular, this allows one to find a geometric generator for Hc4 (ρ(U ); Q). Consider stable curves C1 ∪ C2 ∪ C3 , where the component C1 is smooth of genus 1, the component C2 is a smooth rational curve and the component C3 is a rational curve with exactly one node, satisfying #(C1 ∩ C2 ) = 1, #(C1 ∩ C3 ) = 0 and #(C2 ∩ C3 ) = 2. Denote by G the closure in t−1 (ρ(U )) of the locus of such curves, and denote by t∗ [G] the push-forward to ρ(U ) of the cycle class of G. Then the fundamental class of t∗ [G] generates Hc4 (ρ(U ); Q). Recall that the locus in M3 of irreducible curves with two nodes maps surjectively to β2 under the Torelli map. Moreover, all curves in M3 that have two nodes and map to β2 can be constructed by taking a stable curve of genus 1 with 4 marked points and identifying the marked points pairwise. There is a well known relation between cycle classes of dimension 2 in M1,4 , called Getzler’s relation (see [G1]). This relation is S4 -invariant and it induces a relation between dimension 2 cycles in t−1 (β2 ), which if pushed forward under t induces a relation in Hc4 (β2 ; Q). The latter relation involves non-trivially the push-forward of the fundamental class of G ⊂ t−1 (β2 ). In particular, restricting to ρ(U ) ⊂ β2 yields that t∗ [G] vanishes in Hc4 (ρ(U ); Q). Hence, the differential d2,2 2 must be an isomorphism. 2,2 Remark 4.3. There is also another way to see that the differential d2,2 2 : E2 → in Table 4 is an isomorphism. Namely, one can compactify the C∗ -bundle U to the P1 -bundle U = P(P ⊕ OX1 ×A1 X1 ) and compute the invariant part of the exact sequence in rational cohomology of the pair (U , Δ). This then shows that the invariant part of Hc4 (U ; Q) vanishes as claimed. We decided to include the

E24,1


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above proof involving Getzler’s relation since the relation to M3 is of independent interest. Proof of Proposition 2.3. We compute the cohomology with compact support of β2 \β3 by exploiting the Gysin long exact sequence associated to the inclusion ρ(Δ) → (β2 \ β3 ): (4.2) · · · → Hck−1 (ρ(Δ); Q) → Hck (ρ(U ); Q) → Hck (β2 \ β3 ; Q) → Hck (ρ(Δ); Q) → · · · The map ρ identifies the two components of Δ, each of which is isomorphic to X1 ×A1 X1 . Moreover, it factors through the finite group G generated by the following three involutions: the involution which interchanges the two factors of X1 ×A1 X1 , the involution which acts by (x, y) → (−x, −y) on each fibre E × E and finally the involution which acts by (x, y) → (x + y, −y). This can be read off from the construction of the toroidal compactification (see [HS, Section I] for an outline of this construction. Also note that the stratum Δ corresponds to the stratum in the partial compactification in the direction of the 1-dimensional cusp associated to a maximal-dimensional cone in the second Voronoi decomposition for g = 2. A detailed description can be found in [HKW, Part I, Chapter 3]). Hence Hc• (ρ(Δ); Q) ∼ = Hc• (E × E/G; Q) ⊗ Hc• (A1C ; Q). A straightforward calculation shows that the G-invariant cohomology of E × E has rank 1 in even dimension and vanishes otherwise. In particular this quotient behaves cohomologically like P2 . Since Hck (ρ(U ); Q) and Hck (ρ(Δ); Q) both vanish if k is odd, the exact sequence (4.2) splits into short exact sequences 0 → Hck (ρ(U ); Q) → Hck (β2 \ β3 ; Q) → Hck (ρ(Δ); Q) → 0.

This implies the claim.

Remark 4.4. We would like to take this opportunity to correct a slight error in [vdG, 3.8] where it was claimed that the map ρ factors through Sym2A1 (X1 / ± 1) rather than through the quotient by G. This, however, does not effect the results of [vdG]. 5. Torus rank 3 The stratum β3 ⊂ AVor lying over A0 ⊂ ASat is entirely determined by the fan 3 3 of the toroidal compactification. For this we first have to describe the Voronoi fan Σ in genus 3. Consider the free abelian group L3 ∼ = Z3 with generators x1 , x2 , x3 and let M3 = Sym2 (L3 ). Then M3 is isomorphic to the space of 3 × 3 integer symmetric matrices with respect to the basis xi via the map which assigns to a matrix A ∗ the quadratic form t xAx. We shall use the basis of M3 given by the forms Ui,j , 1 ≤ i ≤ j ≤ 3 given by ∗ Ui,j = 2δi,j xi xj . Let Sym≥0 2 (L3 ⊗R) be the cone of positive semidefinite forms in M3 ⊗Q. The group GL(3, Z) acts on Sym≥0 2 (L3 ⊗ R) by GL(3, Z) g : M −→ t g −1 M g −1 .

100 12


Let σ (6) := R≥0 α1 + R≥0 α2 + R≥0 α3 + R≥0 γ1 + R≥0 γ2 + R≥0 γ3 , where αi = x2i for all i = 1, 2, 3 and γi = (xj −xk )2 for {i, j, k} = {1, 2, 3}. Since the forms αj , γi form a basis of M3 , this is a basic 6-dimensional cone in Sym≥0 2 (L3 ⊗R). The Voronoi fan in genus 3 is the fan Σ in Sym≥0 (L ⊗ R) given by σ (6) and 3 2 all its faces, together with their GL(3, Z)-translates. We use the notation σ (6) = α1 ∗ α2 ∗ α3 ∗ γ1 ∗ γ2 ∗ γ3 , and similarly for the faces of σ (6) . (6) To describe AVor and its 3 , we have to know all possible GL(3, Z)-orbits of σ faces. An i-dimensional cone corresponds to a (6 − i)-dimensional stratum in AVor 3 . Since strata of dimension at least 4 necessarily lie over Al with l ≥ 1, we only need to know the orbits of cones of dimension ≤ 3. The following lemma can be proved using the methods of [Ts] (see [E, Chapter 3]). Lemma 5.1. There are two GL(3, Z)-orbits of 3-dimensional cones, represented by the cones (3) (3) σlocal = α1 ∗ α2 ∗ α3 , σglobal = α1 ∗ α2 ∗ γ3 . (3)

(3)

The stratum associated to σlocal lies over A0 , that associated to σglobal lies over A1 . There are two GL(3, Z)-orbits of 4-dimensional cones, given by (4)

σI

= α1 ∗ α2 ∗ α3 ∗ γ1 ,

(4)

σII = α1 ∗ α2 ∗ γ1 ∗ γ2 .

In dimension 5 and 6 there is only one GL(3, Z)-orbit. The strata of all cones of dimension at least 4 lie over A0 . Let H3 = {τ = (τi,j )1≤i,j≤3 : τ = t τ, Im τ > 0} be the Siegel upper half plane of genus 3. We consider the rank 6 torus T = T 6 with coordinates √ ti,j = e2π −1τi,j (1 ≤ i, j ≤ 3). ∗ . If σ (l) is an These coordinates correspond to the dual basis of the basis Ui,j l-dimensional cone in Σ then, since the fan Σ is basic, it follows that the associated affine variety Tσ(l) ∼ is then a = Cl × (C∗ )6−l . The corresponding stratum in AVor 3 quotient of {(0, 0, 0)} × (C∗ )6−l by a finite group. We consider the torus embedding T → Tσ(6) ∼ = C6 , where the latter isomorphism holds since σ (6) is a basic cone of dimension 6. Let T1 , . . . , T6 be the coordinates of C6 corresponding to the basis α1 , . . . , γ3 . If one computes the dual basis of α1 , . . . , γ3 in terms of the dual basis ∗ of Ui,j , one obtains that the torus embedding T → C6 is given by

T1 = t1,1 t1,3 t1,2 , T4 = t−1 2,3 ,

T2 = t2,2 t2,3 t1,2 , T5 = t−1 1,3 ,

T3 = t3,3 t1,3 t2,3 , T6 = t−1 1,2 .

Let us start by considering the stratum associated to (3)

σlocal = α1 ∗ α2 ∗ α3 . Let S1 , S2 and S3 be coordinates corresponding to α1 , α2 and α3 , and let t2,3 , t1,3 , t1,2 be as above. Then T (3) ∼ = C3 × (C∗ )3 ⊂ C6 = Tσ(6) σlocal


101 13

−1 −1 with coordinates S1 , S2 , S3 , t−1 2,3 , t1,3 , t1,2 , where the inclusion is defined by consid(3)

ering σlocal as a face of σ (6) . The stratum which we add is {(0, 0, 0)} × (C∗ )3 modulo a finite group G = (3) Gσ(3) , namely the stabilizer of the cone σlocal in GL(3, Z). In order to understand local the action of the group G explicitly, we recall that it is naturally a subgroup of the parabolic subgroup which belongs to the standard 0-dimensional cusp ⎧⎛ ⎫ ⎞ g 0 ⎨ ⎬ ⎠ : g ∈ GL(3, Z) ∼ P = ⎝ = GL(3, Z) ⊂ Sp(6, Z). ⎩ ⎭ 0 t g −1 (3)

(3)

Lemma 5.2. The stratum associated to σlocal is an affine variety Ylocal = (C ) /G whose only non-trivial cohomology with compact support is in degree 6. ∗ 3

(3)

Proof. Since the stratum associated to σlocal is the quotient of the smooth variety (C∗ )3 by a finite group, its cohomology and cohomology with compact support are related by Poincaré duality. Hence, it suffices to show that the rational cohomology of the stratum is concentrated in degree 0. −1 −1 Denote by T 3 the rank 3 torus with coordinates (v1 , v2 , v3 ) = (t−1 2,3 , t1,3 , t1,2 ). (3)

(3)

The stratum which we add for σlocal is then isomorphic to T 3 /G. Since σlocal = α1 ∗ α2 ∗ α3 with αi = x2i , we see that the group G is the group generated by the permutations of the xi and the involutions (x1 , x2 , x3 ) → ( 1 x1 , 2 x2 , 3 x3 ) with i = ±1. Note that the element −id acts trivially both on H3 and on M3 . Hence the group G is an extension 1 → (Z/2Z)2 → G → S3 → 1, where S3 denotes the symmetric group in 3 letters. Next, we have to analyze how this group acts on H3 and on the torus T 3 . The permutation of xi and xj interchanges τi,k and τj,k but fixes τi,j . Hence S3 also acts as group of permutations on the coordinates of T 3 . The action of the involutions generating (Z/2Z)2 can be seen for example from ⎞⎛ ⎞ ⎛ ⎛ ⎞⎛ ⎞ τ1,1 τ1,2 τ1,3 τ1,1 −τ1,2 −τ1,3 −1 0 0 −1 0 0 ⎝ 0 1 0⎠ ⎝τ1,2 τ2,2 τ2,3 ⎠ ⎝ 0 1 0⎠ = ⎝−τ1,2 τ2,2 τ2,3 ⎠ . 0 0 1 0 0 1 τ1,3 τ2,3 τ3,3 −τ1,3 τ2,3 τ3,3 Hence, the involution (x1 , x2 , x3 ) ↔ (x1 , x2 , −x3 ) induces the involution given by (v1 , v2 , v3 ) ↔ (v1−1 , v2−1 , v3 ) and similarly for the other involutions. This allows us to describe the quotient T 3 /G explicitly, as given by the image of the map T3 ∼ = (C∗ )3 −→ C4 (v1 , v2 , v3 ) −→ (u1 + u2 + u3 , u1 u2 + u1 u3 + u2 u3 , u1 u2 u3 , u4 ) = (s1 , s2 , s3 , t), where 1 1 1 u1 = v1 + , u2 = v2 + , u3 = v3 + , u4 = v1 v2 v3

1 1 1 v2 − v3 − . v1 − v1 v2 v3

Then the image is the hypersurface W ⊂ C4 given by s3 t2 − ( + 2s1 )2 + (s2 + 4)2 = 0. 4 2 Note that W is a cone with vertex the line t = s23 + 2s1 = s2 + 4 = 0 in C4 over a plane projective conic. Then the claim follows from the contractibility of W .

102 14


Alternatively, one can also show that the cohomology H • (T 3 /G; Q) is concentrated in degree 0, by proving that the only cohomology in H • (T 3 ; Q) which is fixed under the group G is in degree 0. The situation with the lower-dimensional strata is similar: Lemma 5.3. Let σ (l) be an l-dimensional subcone of α1 ∗ α2 ∗ α3 ∗ γ1 ∗ γ2 ∗ γ3 , with l ≥ 4. Then the stratum of γ3 associated to σ (l) has non-trivial cohomology with compact support only in the maximal degree 2(6 − l). Proof. Recall that all GL(3, Z)-orbits of σ (l) were described in Lemma 5.1. Hence it suffices to consider the cases in which σ (l) is one of the following cones: (4) (4) σI , σII , σ (5) := α1 ∗ α2 ∗ α3 ∗ γ1 ∗ γ2 and σ (6) . As mentioned above, if σ (l) is an l-dimensional cone in Σ then we have Tσ(l) = l C × (C∗ )6−l , because the fan Σ is basic. The corresponding stratum in AVor is 3 then a quotient of {(0, 0, 0)} × (C∗ )6−l ∼ = (C∗ )6−l by a finite group Gσ(l) . To prove the claim, it suffices to show that the part of the cohomology of (C∗ )6−l which is invariant for the action of G coincides with H 0 ((C∗ )6−l ; Q). Since (C∗ )6−l is smooth, the result about cohomology with compact support will follow from Poincaré duality. (4) For instance, consider the case of σII . Using toric coordinates, one finds that the corresponding stratum is given by a quotient of (C∗ )2 by the action of the finite group Z/2Z × S3 . The factor S3 acts on γ1 ∗ α2 ∗ α3 by permuting γ1 , α2 and α3 , whereas the action of the factor Z/2Z is generated by the involution x1 ↔ −x1 . One can compute explicitly the action of Z/2Z×S3 and prove (H • ((C∗ )2 ; Q))Z/2Z×S3 = H 0 ((C∗ )2 ; Q). Analogous considerations yield the claim in the case of the other strata. Concluding, the proof of Proposition 2.4 now follows from Lemmas 5.2 and 5.3. References V. Alexeev, Complete moduli in the presence of semiabelian group action. Ann. of Math. (2) 155 (2002), 611–708. [AN] V. Alexeev, I. Nakamura, On Mumford’s construction of degenerating abelian varieties, Tohoku Math. J. (2) 51 (1999), 399–420. [Ar] D. Arapura, The Leray spectral sequence is motivic, Invent. Math. 160 (2005), no. 3, 567–589. [E] C. Erdenberger, A finiteness result for Siegel modular threefolds, Ph. D. Thesis, Leibniz Universit¨ at Hannover (2007). Available at http://www.iag.uni-hannover.de/∼ag-iag/ data/phdthesis erdenberger.pdf [vdG] G. van der Geer, The Chow ring of the moduli space of abelian threefolds, J. Algebraic Geom. 7 (1998), 753–770. [G1] E. Getzler, Intersection theory on M1,4 and elliptic Gromov–Witten invariants, J. Amer. Math. Soc. 10 (1997), no. 4, 973–998. [G2] E. Getzler, Topological recursion relations in genus 2, in: Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), World Sci. Publ., River Edge, NJ, 1998, 73–106. [H] R. Hain, The rational cohomology ring of the moduli space of abelian 3-folds, Math. Res. Lett. 9 (2002), no. 4, 473–491. [HKW] K. Hulek, C. Kahn, S. H. Weintraub, Moduli spaces of abelian surfaces: compactification, degenerations, and theta functions, de Gruyter Expositions in Mathematics 12. Walter de Gruyter & Co., Berlin, 1993. [HS] K. Hulek, G. K. Sankaran, The geometry of Siegel modular varieties, Higher dimensional birational geometry (Kyoto, 1997), Adv. Stud. Pure Math., 35, 89–156, Math. Soc. Japan, Tokyo, 2002. [A]


[M1]

[M2]

[O] [PS] [T]

[Ts] [V]

103 15

D. Mumford, On the Kodaira dimension of the Siegel modular variety. In: Algebraic geometry – open problems, proceedings, Ravello 1982, eds. C. Ciliberto, F. Ghione and F. Orecchia. Lecture Notes in Mathematics 997, Springer-Verlag, Berlin-New York, 1983, 348–375. D. Mumford, Towards an enumerative geometry of the moduli space of curves. In: Arithmetic and geometry, Vol. II. Progr. Math., 36, Birkh¨ auser Boston, Boston, MA, 1983, 271–328. M. Olsson, Compactifying moduli spaces for abelian varieties, Lecture Notes in Mathematics, 1958. Springer-Verlag, Berlin, 2008. C. A. M. Peters, J. H. M. Steenbrink, Mixed Hodge Structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 52, Springer-Verlag, Berlin, 2008. O. Tommasi, Geometry of discriminants and cohomology of moduli spaces, Ph.D. thesis, Radboud University Nijmegen (2005). Available at http://webdoc.ubn.ru.nl/mono/t/tommasi o/geomofdia.pdf R. Tsushima, A formula for the dimension of spaces of Siegel cusp forms of degree three, Amer. J. Math. 102 (1980), no. 5, 937–977. G. F. Voronoi, Nouvelles applications des param` etres continus ` a la th´ eorie des formes quadratiques. Deuxi` eme m´ emoire. Recherches sur les parall´ elo` edres primitifs. Seconde partie. Domaines de formes quadratiques correspondant aux diff´ erents types de parall´ elo` edres primitifs, J. Reine Angew. Math. 136 (1909), 67–178.

¨t Hannover, Institut fu ¨ r Algebraische Geometrie, WelfengarLeibniz Universita ten 1, D-30167 Hannover, Germany E-mail address: [email protected] ¨t Hannover, Institut fu ¨ r Algebraische Geometrie, WelfengarLeibniz Universita ten 1, D-30167 Hannover, Germany E-mail address: [email protected]



Quasi-Complete Homogeneous Contact Manifold Associated to a Cubic Form Jun-Muk Hwang and Laurent Manivel Dedicated to S. Ramanan

Abstract. Starting from a cubic form, we give a general construction of a quasi-complete homogeneous manifold endowed with a natural contact structure. We show that it can be compactified into a projective contact manifold if and only if the cubic form is the determinant of a simple cubic Jordan algebra.

1. Introduction This note is at the crossroad of two different lines of study. On the one hand, we propose a general construction of a homogeneous quasiprojective manifold Xc associated to a cubic form with a mild genericity property. These manifolds are rationally chain connected (Proposition 2.2), a property which relates our study to that of certain types of homogeneous spaces considered in [2, 3, 4]. On the other hand, we show that our manifolds Xc are endowed with natural contact structures (Proposition 3.1). Our construction thus appears as part of the general study of contact projective and quasi-projective manifolds. Of course the projective case is the most interesting one, the main open problem in this area being the Lebrun-Salamon conjecture: the only Fano contact manifolds should be the projectivizations of the minimal nilpotent orbits in the simple Lie algebras. As explained in section 4, our construction is in fact modeled on these homogeneous contact manifolds, which are known to be associated to very special cubic forms: the determinants of the simple cubic Jordan algebras. Under both points of view, one of the most interesting questions one may ask about the quasi-projective contact manifolds Xc is about their compactifications. Even the existence of a small compactification (that is, with a boundary of codimension at least two) is not clear. Also, it is extremely tempting to try to construct new projective contact manifolds by compactifying some Xc in such a way that the contact structure extends. We show that this is possible if and only if the cubic c 2010 Mathematics Subject Classification. 14M99, 14M20, 14M17. The first author was supported by the Korea Research Foundation Grant funded by the Korean Government (MEST)(KRF-2006-341-C00004). c 2010 American c Mathematical 0000 (copyright Society holder)

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JUN-MUK HWANG AND LAURENT MANIVEL

is the determinant of a simple cubic Jordan algebra (Proposition 5.2). This can be interpreted as an evidence for the Lebrun-Salamon conjecture. 2. Homogeneous spaces defined from cubics Let V be a complex vector space of dimension p. Let c ∈ S 3 V ∗ be a cubic form on V . Let B : S 2 V → V ∗ be the system of quadrics defined by B(v1 , v2 ) = c(v1 , v2 , ·). In all the sequel we make the following Assumption on c. The homomorphism B is surjective. Let W be a complex vector space of dimension 2. Fix a choice of a non-zero 2-form ω ∈ ∧2 W ∗ . Let n := n1 ⊕ n2 ⊕ n3 where n1 := V ⊗ W,

n2 := V ∗ ,

n3 := W.

Define a graded Lie algebra structure on n, by [v1 ⊗ w1 , v2 ⊗ w2 ] = ω(w1 , w2 )B(v1 , v2 ), [v1∗ , v2 ⊗ w2 ] = v1∗ (v2 ) w2 . The Jacobi identity holds because dim W = 2. Let N be the nilpotent Lie group with Lie algebra n. For a point ∈ PW , denote by ˆ ⊂ W the corresponding 1-dimensional subspace. Let a := V ⊗ ˆ ⊂ V ⊗ W = n1 be the abelian subalgebra of n and A ⊂ N be the corresponding additive abelian subgroup. We have the smooth subvariety A ⊂ N × PW defined by A := {(g, ), g ∈ A }. This variety A can be viewed as a family of abelian subgroups parametrized by PW . Let ψ : Xc → PW be the family of relative quotients Xc := {N/A , ∈ PW } with the quotient map ξ : N × PW → Xc . Then dim Xc = dim N + 1 − p = 2p + 3. Observe that ψ is a locally trivial fibration whose fibers are isomorphic to affine spaces. But it is not a vector nor an affine bundle. In fact the transition functions are quadratic, because the nilpotence index of N is three. The variety Xc is homogeneous under the action of the group G := N SL(W ) (semi-direct product). Let o ∈ N be the identity and ∈ PW be a fixed base point. Then x := ξ(o × ) will be our base point for Xc . Its stabilizer is H = A B , if B denotes the stabilizer of in SL(W ). Moreover a Borel subgroup of G is B = N B , and we have a sequence of quotients ξ

ψ

G→G/B → G/H = Xc → G/B = PW. Proposition 2.1. (1) Xc is simply connected. (2) Let L := ψ ∗ OPW (1). Then Pic(Xc ) = ZL.

HOMOGENEOUS CONTACT MANIFOLD

107 3

Proof. As a variety, each A is nothing but an affine space. So the variety Xc being fibered in simply connected manifolds over the projective line, is simply connected. This proves (1). The character group X(G) of G being trivial, the forgetful map α : P icG (Xc )→P ic(Xc ) is injective ([9] Proposition 1.4). Moreover the Picard group of G is trivial, so α is in fact an isomorphism (see the proof of Proposition 1.5 in [9]). But P icG (Xc ) X(H) and an easy computation shows that X(H) = X(B ). This implies (2). Note that G is generated by H and SL(W ) such that H ∩ SL(W ) is a Borel subgroup of SL(W ). Thus we can apply Proposition 4.1 in [4] to deduce: Proposition 2.2. The variety Xc is rationally chain connected. In particular, Xc is quasi-complete, i.e., there is no non-constant regular function on Xc . In fact, it is easy to show that for any n ∈ N , the image of {n} × PW under ξ is a smooth rational curve on Xc with normal bundle of the form O(1)p ⊕ Op+2 . 3. Contact structures Consider the tangent spaces To× (N × PW ) = n1 ⊕ n2 ⊕ n3 ⊕ T (PW ) Tx (Xc ) = n1 /a ⊕ n2 ⊕ n3 ⊕ T (PW ). Using the subspace ˆ ⊂ W = n3 , we define the hyperplane Dx := n1 /a ⊕ n2 ⊕ ˆ ⊕ T (PW )

inside Tx (Xc ). This hyperplane is invariant under the action of the stabilizer H of x in G = N SL(W ), so we get a well-defined hyperplane distribution D ⊂ T (Xc ) with T (Xc )/D ∼ = L. Proposition 3.1. The distribution D ⊂ T (Xc ) defines a contact structure. Remark. Observe that since ˆ ⊕ V ∗ ⊕ ˆ ⊕ Hom(, ˆ W/), ˆ Dx = (V ⊗ W/) ˆ there is a natural W/-valued symplectic pairing on Dx . Note that W/ˆ is the fiber of L at x . This shows that the bundle D has an L-valued symplectic form, and indeed this symplectic form comes from the contact structure 0 → D → T (Xc ) → L → 0. Proof. Let Yc be the variety defined as the complement L× of the zero section in the total space of the line bundle dual to L. Let θ be the L-valued 1-form on Xc defining D. To check that θ is a contact form, it suffices to show that the 2-form dθ˜ where θ˜ is the pull-back of θ to Yc , is symplectic (see [1], Lemma 1.4). To check this we make a local computation. Let m ∈ PW be some point distinct from and m ˆ ⊂ W be the corresponding line. Then nm := (V ⊗m) ˆ ⊕V ∗ ⊗W ⊂ n defines a complement to a ⊂ n. We can define a local analytic chart on Xc around x by x(X, p) = ξ(exp(X) × p),

108 4


where X ∈ nm and p ∈ PW − m. Let us write down θ in that local chart. Since the chart preserves the fibration over PW we just need to compute over . The differential eX of the exponential map at X, seen as an endomorphism of nm , is defined by the relation exp(X + teX (Y ) + O(t2 ))x = exp(X)exp(tY )x . Now we can use the fact that N being 3-nilpotent, the Campbell-Hausdorff formula in N is quite simple: we have exp(X)exp(Y ) = exp(H(X, Y )) for X, Y ∈ n, with 1 1 1 H(X, Y ) = X + Y + [X, Y ] + [X, [X, Y ]] + [Y, [Y, X]]. 2 12 12 1 We easily deduce that eX (Y ) = Y + 12 [X, Y ]+ 12 [X, [X, Y ]]. Now we can decompose this formula with respect to the three-step grading of n. If Z = eX (Y ) = Z1 + Z2 + Z3 , we find that Z1

=

Y1 ,

1 Y2 + [X1 , Y1 ], 2 1 1 1 Z3 = Y3 + [X1 , Y2 ] + [X2 , Y1 ] + [X1 , [X1 , Y1 ]], 2 2 12 which can be inverted as Z2

=

Y1

= Z1 ,

Y2 Y3

1 = Z2 − [X1 , Z1 ], 2 1 1 5 = Z3 − [X1 , Z2 ] − [X2 , Z1 ] + [X1 , [X1 , Z1 ]]. 2 2 12

ˆ we deduce Since the hyperplane Dx is defined by the condition that Y3 belongs to , that the contact form is given at x(X, ), in our specific chart, by the formula 1 1 5 ˆ θx(X,) (Z) = Z3 − [X1 , Z2 ] − [X2 , Z1 ] + [X1 , [X1 , Z1 ]] mod . 2 2 12 Even more explicitly, if we write Z1 = z1 ⊗m and X1 = x1 ⊗m, we have [X1 , Z1 ] = 0, [X1 , Z2 ] = Z2 (x1 )m and [X2 , Z1 ] = −X2 (z1 )m, so 1 θx(X,) (Z) = Z3 + (X2 (z1 ) − Z2 (x1 ))m. 2 Now we pull-back θ to Yc = L× . A local section of L× around is given by m∗ −z∗ over the point p = + zm of PW . Over φ = y(m∗ − z∗ ), we get the 1-form on L× given in our local chart by y θ˜x(X,p),φ (Z, Y ) = y(m∗ − z∗ )(Z3 ) + (X2 (z1 ) − Z2 (x1 )). 2 If Z3 = Z31 m + Z32 , this can also be written as: y θ˜x(X,p),φ = y(dX31 − zdX32 ) + (X2 dX1 − X1 dX2 ). 2 We can easily differentiate this expression and evaluate it at x . We obtain dθ˜x ,ym∗ = dy ∧ dX 1 − ydz ∧ dX 2 + ydX2 ∧ dX1 .

3

3

Since y is a non zero scalar this 2-form is everywhere non-degenerate. By homogeneity this remains true over the whole of L× , and the proof is complete.


109 5

4. Projective homogeneous contact varieties Consider a complex simple Lie algebra g and the adjoint variety Yg = POmin ⊂ Pg, the projectivization of the minimal nilpotent orbit Omin . Then Yg is homogeneous under the action of the adjoint group G = Aut(g). Suppose that P ic(Yg ) Z (this is the case if and only if g is not if type A). Then the variety F of lines on Yg is G-homogeneous and we can describe a line as follows. Choose T ⊂ B ⊂ G a maximal torus and a Borel sugbroup. Let gψ denote the root space in g associated to the highest root ψ. Then Yg = Ggψ and the stabilizer of gψ is the maximal parabolic subgroup Pα of G defined by the unique simple root α such that ψ − α is a root. Moreover the line = gψ , gψ−α is contained in the adjoint variety Yg , and F = G.. There is a five-step grading on g defined by the highest root ψ, as follows. Define Hψ ∈ [gψ , g−ψ ] by the condition that ψ(Hψ ) = 2. Then the eigenvalues of ad(Hψ ) are 0, ±1, ±2 and the eigenspace decomposition yields the five-step grading g = g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 . We have g2 = gψ , while gα and gψ−α are respectively lines of lowest and highest weights in g1 . Since gψ−α defines, exactly as gψ , a point of the adjoint variety, we can use the root ψ − α to define another five-step grading. Since ad(Hψ ) and ad(Hψ−α ) commute, we get a double grading on g. Moreover, the stabilizer s ⊂ g of the line ⊂ g decomposes as follows (where the grading defined by ad(Hψ ) can be read horizontally): g−α g00 g10 gα g11 g21 gψ−α gψ Let W = gψ−α ⊕ gψ C2 and V = g∗21 . The map g10 ⊗ g21 → g31 = gψ−α defined by the Lie bracket is a perfect pairing, as well as g11 ⊗ g21 → g32 = gψ , giving a natural identification g10 ⊕ g11 V ⊗W and isomorphisms φ g10 −→ g11 ∼ = V. The positive part of the vertical grading of s thus reads (V ⊗W ) ⊕V ∗ ⊕W = n. Note that the degree zero part of this grading reads sl(W ) × h00 , where g00 = [g−α , gα ]⊕h00 is an orthogonal decomposition with respect to the Killing form. The cubic form c on V is defined (up to scalar) once we identify g10 with g11 , through the map φ. We also need to choose a generator Xψ of gψ . Then we can define c by the formula [φ(X), [φ(X), X]] = c(X)Xψ

∀X ∈ g10 V.

Remark. This construction is closely related to the ternary models for simple Lie algebras considered in [7], section 2. These models are of the form g = h × sl(U ) ⊕ (U ⊗V ) ⊕ (U ∗ ⊗V ∗ ),

110 6


where U is three dimensional, and V is an h-module. To define a Lie bracket on g, one needs a cubic form c on V , a cubic form c∗ on V ∗ , and a map θ : V ⊗V ∗ →h. Then the Jacobi identity implies a series of conditions on these data, including that h ⊂ Aut(c) ∩ Aut(c∗ ). These conditions should ultimately lead to a cubic Jordan algebra structure on V . If we choose a maximal torus in sl(U ) and use the associated grading on U, U ∗ , we get an hexagonal model as in Figure 2 of [8] : g−α g−ψ

g−1−1 g−2−1

gα−ψ

g10 g00

g−10

gψ−α g21

g11

gψ

gα The subalgebra we denoted n is the sum of the factors in the last three columns. and we can add the factor sl(W ) from the middle column in order to get s. Once we have defined the cubic c associated to the simple Lie algebra g, we have the associated homogeneous space Xc with its natural contact structure. A direct verification gives: Proposition 4.1. The homogeneous space Xc is an open subset of Yg , with a codimension two boundary. Its contact structure is the restriction of the natural contact structure on Yg . It is tempting, but illusory, as we shall see, to try to construct new projective contact manifolds as suitable compactifications of our homogeneous spaces Xc for other types of cubics. 5. Compactifications Since all regular functions on Xc are constant, we can expect that Xc admits ¯ c containing Xc as an open a small compactification, that is, a projective variety X subset in which the boundary of Xc has codimension two or more. By Theorem 1 in [3] and Proposition 2.1, it is enough to check that the algebra of sections ∞ R(Xc , L) = Γ(Xc , Lk ) k=0

is of finite type, as well as all the R(Xc , Lm ), for m ≥ 1. We have not been able to prove this but we can make the following observations. Since the Lie algebra g of G = N SL(W ) preserves the contact structure we have defined on Xc , there must be a morphism ϕ from Xc into Pg∗ (see [1], Section 1). In fact, any contact vector field on a contact manifold defines a holomorphic section of the contact line bundle L. Thus g defines a linear subsystem in |L|. The morphism is always etale over its image, and since our Xc is simply connected we conclude that ϕ embeds Xc as a coadjoint orbit in g∗ . We thus have a natural projective compactification of Xc in Pg∗ .


111 7

Note that the inclusion of Xc in Pg∗ is just the projectivization of the moment map of the symplectic variety Yc = L× . We have a commutative diagram Yc ↓ Xc

μ

g∗ ↓ ν −→ Pg∗ −→

Here μ denotes the (G-equivariant) moment map and ν is its quotient by the C∗ action. We have g∗ = n∗ ⊕ sl(W )∗ and the component μ of μ on n∗ is not injective, since the N action on Xc preserves the P1 -fibration. Consider μ (Yc ) ⊂ n∗ . Proposition 5.1. Suppose that the cubic hypersurface Zc ⊂ PV be smooth. Then the boundary of μ (Yc ) has codimension at least two. Proof. We can describe explicitly the closure of μ (Yc ) as the set of triples (φ1 , φ2 , φ3 ) ∈ n∗ such that ω(φ1 , φ3 ) = c(φ2 , φ2 , .), ∗

∗

where ω : (V ⊗W ) × W → V is the natural bilinear map. If φ3 = 0, we are in μ (Yc ). Thus on the boundary, we must have φ3 = 0, and then c(φ2 , φ2 , .) = 0. But under our smoothness assumption on Zc , this implies that φ2 = 0. So the boundary of μ (Yc ) has dimension at most 2p = dim μ (Yc ) − 2, the number of parameters for φ1 . This seems to be a first step towards proving that Xc has a small compactification. But we have not been able even to find conditions on c that would ensure ¯ c ⊂ Pg∗ is small. that the compactification X What is rather surprising is that the cubics whose associated variety Xc has a smooth contact compactification can be completely classified. By this, we mean a ¯ c compactifying Xc , with a contact structure extending smooth projective variety X that of Xc . ¯ c of Xc if Proposition 5.2. There exists a smooth contact compactification X and only if c is the cubic norm of a semi-simple Jordan algebra. Proof. We will deduce this statement from a study the variety of minimal rational tangents Cx ⊂ PDx . Note that the space of lines on Xc through x is just ξ −1 (x ) ∼ = A ∼ =V. We claim that the tangent map sending a line through x to its tangent direction in PDx is equal, up to scalars, to the rational map τ : V → PDx = P(V ⊕ V ∗ ⊕ C ⊕ C) τ (v) := [v : B(v, v) : c(v, v, v) : 1]. Indeed, a line through x in Xc is of the form g = ξ(g × PW ) for g ∈ A . To write this line in the local chart we used in the proof of Proposition 3.1 (we use the same notations), we must write ξ(g × p) = exp(Z)ξ(o × p). If g = exp(X) with X ∈ V ⊗, this amounts to solving the equation exp(X) = exp(Z)exp(W ), with Z ∈ nm and W ∈ V ⊗p. So X = H(Z, W ), and if we write

112 8


X = v⊗ for some v ∈ V , we must have W = v⊗p and then we get, up to term of order at least two in z, Z1

=

Z2

=

Z3

=

−zv⊗m, 1 z − [Z1 , W ] = B(v, v), 2 2 1 z − [Z1 , [Z1 , W ]] = − c(v). 6 6

This proves the claim. We can now conclude the proof as follows. By the results of [5], the closure of the image of this map must be smooth if there exists a smooth contact compactification of Xc . So the closure of the image of the map τ must be smooth. But then we can apply Corollary 26 in [6]. References [1] Beauville A., Fano contact manifolds and nilpotent orbits, Comment. Math. Helv. 73 (1998), no. 4, 566–583. [2] Bien F., Borel A., Sous-groupes ´ epimorphiques de groupes lin´ eaires alg´ ebriques I, C. R. Acad. Sci. Paris S´ er. I Math. 315 (1992), no. 6, 649–653. [3] Bien F., Borel A., Sous-groupes ´ epimorphiques de groupes lin´ eaires alg´ ebriques II, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 13, 1341–1346. [4] Bien F., Borel A., Koll´ ar J., Rationally connected homogeneous spaces, Invent. Math. 124 (1996), no. 1-3, 103–127. [5] Kebekus S., Lines on complex contact manifolds II, Compos. Math. 141 (2005), no. 1, 227– 252. [6] Landsberg J.M., Manivel L., Legendrian varieties, Asian J. Math. 11 (2007), no. 3, 341–359. [7] Manivel L., Configurations of lines and models of Lie algebras, J. Algebra 304 (2006), no. 1, 457–486. [8] Mukai S., Simple Lie algebra and Legendre variety, preprint 1998. [9] Mumford D., Fogarty J., Kirwan F., Geometric invariant theory, Third edition, Springer 1994. Jun-Muk Hwang: Korea Institute for Advanced Study, Hoegiro 87, Seoul, 130-722, Korea E-mail address: [email protected] Laurent Manivel: Institut Fourier, UMR 5582 CNRS/UJF, Universit´ e Joseph Fourier, 38402 Saint Martin d’H` eres, France E-mail address: [email protected]


Maximal Weights in K¨ ahler Geometry: Flag Manifolds and Tits Distance (with an Appendix by A. H . W. Schmitt) I. Mundet i Riera Dedicated to Professor Ramanan, on the occasion of his 70th birthday

Abstract. We review the definition of maximal weights and polystability in Kaehler geometry given in [M]. We compute the maximal weights in flag varieties, and we relate them to the Tits distance between points of the boundary at infinity of the corresponding symmetric space. In the appendix by A.H.W. Schmitt a proof is given of an algebraic analogue of the characterization of polystability in terms of maximal weights.

1. Introduction The purpose of this paper is twofold. First, we survey the main constructions and results in [M], notably • the maximal weight functions on the boundary at infinity ∂∞ (K\K C ) associated to a Hamiltonian action of a compact Lie group K on a Kähler manifold X extending to a holomorphic action of K C , and • the characterization of points in X whose K C -orbit intersects the zero locus of the moment map in terms of maximal weights. Second, we compute the maximal weights in some particular examples. The only new result in this paper is a formula relating the maximal weights for the standard actions on flag manifolds to the Tits distance between points in the boundary at infinity of symmetric spaces of noncompact type. This formula is analogous to formula (10) in [KLM], but here we adopt a different point of view, putting more emphasis on the symplectic aspects of the formula. Compared to [KLM], our perspective is wider in the sense that we consider actions on arbitrary Kähler manifolds (the results in [KLM] are related to actions on flag manifolds), but it is more restrictive in the sense that we only consider symmetric spaces of the form K\K C ([KLM] considers arbitrary symmetric spaces of noncompact type). However, it is very likely that the construction given in [M] extends to the setting considered in [HS], and such an extension should make use of arbitrary symmetric spaces of noncompact type. 2010 Mathematics Subject Classification. Primary 53D20; Secondary 32M05. c Mathematical 0000 (copyright Society holder) c 2010 American

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I. MUNDET I RIERA (APPENDIX BY A.H.W. SCHMITT)

1.1. Let us recall the setting of [M]. Denote by X a connected Kähler manifold endowed with an action by holomorphic isometries of a compact Lie group K. Assume that this action extends to a holomorphic action of the complexification of K, which we denote by G, and that the action of K on X is Hamiltonian with respect to the symplectic form ω defined as the imaginary part of the Hermitian pairing in X. Denote by k the Lie algebra of K. The action of K on X being Hamiltonian, there exists a K-equivariant map (the so-called moment map) μ : X → k∗ such that, for any s ∈ k, we have dμ, s = ιXs ω, where Xs is the vector field on X generated by the infinitesimal action of s and ιXs ω is the contraction of ω by Xs . We assume that the length of the vector Xs (x) grows at most linearly as a function of the distance of x to some base point x0 ∈ X. More precisely, we make the following Assumption 1.1. There exists a constant C > 0 such that for any x ∈ X and any s ∈ k we have |Xs (x)| ≤ C |s| (1 + dX (x0 , x)), where dX denotes the geodesic distance between points of X. Two natural situations in which this assumption holds are the case of compact X and that in which X is a Hermitian vector space with an action of K by unitary transformations. Let ∂∞ (K\G) denote the boundary at infinity of the symmetric space K\G (see Section 2.1 for a reminder of its definition). In [M] we defined, for each point x ∈ X, a map λx = λ(X,ω) : ∂∞ (K\G) → R ∪ {∞}, x and we defined x to be analytically polystable if and only if: (1) for any y ∈ ∂∞ (K\G) we have λx (y) ≥ 0, (2) if y ∈ ∂∞ (K\G) satisfies λx (y) = 0, then there exists another point y ∈ ∂∞ (K\G) satisfying λx (y ) = 0 and there is a geodesic in K\G which converges on one side to y and on the other side to y . The main result in [M] is: Theorem 1.2. x is analytically polystable if and only if μ−1 (0) ∩ G · x is nonempty. Furthermore, if μ−1 (0) ∩ G · x is nonempty then it consists of a unique K-orbit. The interest of such a result stems from the fact that the quotient μ−1 (0)/K is in a natural way a singular symplectic manifold (usually called the symplectic or Marsden–Weinstein quotient), which thanks to the theorem can be identified with the quotient of the set X pol = {x ∈ X | x is analytically polystable} by G. It clearly follows from Theorem 1.2 that x is polystable if and only if any point in its G-orbit is polystable. Such a statement is also a consequence of the following equivariance property of maximal weights, which was proved in [M] (see Lemma 2.1 below for a sketch of the proof). Lemma 1.3. For any x ∈ X, g ∈ G and y ∈ ∂∞ (K\G) we have λg·x (y) = λx (y · g). In the proof of this lemma we crucially use Assumption 1.1.

¨ MAXIMAL WEIGHTS IN KAHLER GEOMETRY AND TITS DISTANCE

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1.2. Suppose that X is a projective variety, that the action of G on X lifts to a linear action on a very ample line bundle L on X, and that the Kähler form on X is the restriction of the Fubini–Study form of the projective embedding given by L. In this situation a classical theorem of Kempf and Ness states that for any x ∈ X (1.1)

G · x ∩ μ−1 (0) = ∅

⇐⇒

G · lx is closed ,

where lx is any nonzero element in the fiber of L → X over x (see Theorem 4.11 in [T]). If the condition on the right hand side in (1.1) holds, then x is said to be polystable. So putting together Theorem 1.2 and (1.1) we deduce that in the projective situation analytic polystability is equivalent to polystability. Furthermore, in the projective situation the maximal weight function can be described in purely algebraic terms: up to some constant factors, it coincides with the maximal weights defined in Geometric Invariant Theory (GIT) [MFK] (see Section 2.5). It follows that there is an algebraic characterization, in terms of GIT maximal weights, of polystable points in X. This characterization, which extends the classical Hilbert– Mumford criterion, seems to be new in GIT, at least as an explicit statement. This is not so surprising: from the point of view of algebraic geometry it is not as natural to consider the set of polystable points X pol ⊂ X as it is to consider the sets of stable and semistable points, since unlike the latter X pol is not Zariski open. But of course the condition of a point being polystable appears recurrently in problems of GIT, as for example in the construction of moduli spaces of principal sheaves (see for example [S]). Shortly after the appearance of [M], A. Schmitt provided a completely algebraic proof of the Hilbert–Mumford criterion for polystability, which is included as an appendix of the present paper. 1.3. We now describe our formula for the maximal weights of flag manifolds. Recall that a flag manifold of G is a homogeneous space of the form X = G/P , where P is a parabolic subgroup of G. Fix one such X. It is clear that X carries a G-invariant complex structure. In order to apply the previous constructions to X we need to endow it with a K-invariant Kähler structure admitting a moment map μ : X → k∗ . This is exactly the same thing as choosing a K-equivariant diffeomorphism of X with a coadjoint orbit O∗ ⊂ k∗ (see Section 3.2 for details). Assume that this choice has been made, and denote by ω the resulting invariant symplectic form on X. Since X is compact, Assumption 1.1 holds, so we can indeed apply to X the previous definitions, and maximal weights satisfy the G-equivariance property stated in Lemma 1.3. Fixing an invariant Euclidean norm on k we obtain: • a G-invariant Riemannian metric on K\G, which can be used to define the Tits distance on ∂∞ (K\G) (see Section 3.1), and • a K-equivariant isomorphism k∗ k, which identifies the coadjoint orbit O∗ X with an adjoint orbit O ⊂ k. Assume that the orbit O is contained in the sphere S(k) ⊂ k of vectors of norm 1 (this can always be assumed if we conveniently rescale the symplectic form on X; on the other hand, rescaling the symplectic form has the effect of multiplying the maximal weight function by the same factor, so it is easy to modify the formula in Theorem 1.4 so that the hypothesis that O ⊂ S(k) is unnecessary). Let e : O → ∂∞ (K\G)

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be the map which sends s ∈ O to the class of the geodesic ray t → [eits ]. Let j : X → ∂∞ (K\G) be the composition of the isomorphism X → O with the map s. In Section 3.3 we prove: Theorem 1.4. For any x ∈ X and any y ∈ ∂∞ (K\G) we have λ(X,ω) (y) = − cos dT (j(x), y), x where dT denotes the Tits distance. The idea that Tits distance is related to stability notions on homogeneous spaces appears in the work of M. Kapovich, B. Leeb and J. Millson [KLM]. In particular, Theorem 1.4 can be seen, adopting a suitable point of view, as a restatement of formula (10) in [KLM]. We discuss in some detail the relation between Theorem 1.4 and the results in [KLM] in Section 3.4 below. We remark that in the proof of Theorem 1.4 we only use the definition of the Tits distance combined with the equivariance property of the maximal weight function stated in Lemma 1.3. As a consequence, we obtain a new proof of the following well known fact (see Section 2.1 for some explanations of the notions appearing in the statement). Corollary 1.5. Let y, y ∈ ∂∞ (K\G) be two arbitrary points. Let ξ : [0, ∞) → K\G be a geodesic ray representing y. For any t ∈ [0, ∞) there exists a unique geodesic ray ηt : [0, ∞) satisfying ηt (0) = ξ(t) and representing y . We then have dT (y, y ) = lim Angle(ξ (t), ηt (0)), t→∞

where the angle takes values in [0, π]. Another consequence of Theorem 1.4 is a certain symmetry property of maximal weights of flag manifolds. To state this property we introduce some notation. Given any y ∈ ∂∞ (K\G) we denote by Fy the orbit G · y and by ωy the natural symplectic form on Fy . Combining Theorem 1.4 with the symmetry of Tits distance, we deduce: Corollary 1.6. For any y, y ∈ ∂∞ (K\G) we have (F

y ,ωy ) λ(F (y ) = λy y y

,ωy )

(y).

Acknowledgements. We thank the referee for a very careful reading of the paper, and for pointing out a mistake and a number of typos in an earlier version of this paper. 2. Maximal weights and polystability In this section we recall the main constructions in [M] and we compute some examples. For the reader’s convenience we have expanded some of the explanations in [M], especially those recalling the (well known among experts) construction of the boundary at infinity of K\G and the action of G on it; however, the proofs of the new results in [M] are only sketched, so the reader might wish to consult [M] for more details.


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2.1. The boundary ∂∞ (K\G). Let K be a compact Lie group, and let G be its complexification. Before defining the boundary ∂∞ (K\G) we first recall the definition of the symmetric space K\G. The coset space K\G has a natural structure of smooth manifold with respect to which the action of G on the right is smooth. The stabilizer of the class of the identity [1] ∈ K\G is K, so there is a natural linear action of K on T[1] (K\G). Furthermore we can identify T[1] K\G k\g. Using the splitting g = k ⊕ ik we obtain an isomorphism T[1] (K\G) ik, which is K-equivariant when we consider on ik k the adjoint action of K. Consequently, any Euclidean norm on k invariant under the adjoint action induces a G-invariant Riemannian metric on K\G. Assume from now on that such a structure has been chosen. It follows that the action of G on K\G is by isometries. Although there are different possible choices of G-invariant Riemannian metrics on K\G, the following properties always hold, regardless of the choice: (1) any geodesic in K\G can be parameterized by a map t → [eits g] ∈ K\G, where s ∈ k and g ∈ G, (2) the curvature of K\G is everywhere nonpositive (see [E] for a computation), (3) K\G is a symmetric space: this means that for any x ∈ K\G there is an isometry which fixes x and which acts on the tangent space at x as multiplication by −1. It is a very natural and classical problem to compactify the symmetric space K\G (which, as a manifold, is diffeomorphic to an Euclidean space) in such a way that the action of G on K\G extends to the compactification. One can compactify symmetric spaces in many different ways, and which compactification is most adequate depends on one’s particular needs (a description of many of the possible compactifications can be found for example in [BJ]). The set of points which one adds to compactify is called the boundary. The compactification which we will describe here is homeomorphic to a closed ball and the boundary, which we denote by ∂∞ (K\G), is usually called the visual boundary and is homeomorphic to a sphere. A geodesic ray is a map γ : [0, ∞) → K\G giving a parametrization by arc length of a portion of geodesic. Let d denote the distance function between points in K\G. Two geodesic rays γ0 , γ1 are declared to be equivalent, and denoted γ0 ∼ γ1 , if the distance d(γ0 (t), γ1 (t)) is bounded independently of t. This defines an equivalence relation. As a set, the boundary at infinity ∂∞ (K\G) is: ∂∞ (K\G) = { geodesic rays }/ ∼ . Since G acts on K\G by isometries, it acts on the set of geodesic rays preserving the equivalence relation ∼, so there is a natural action of G on ∂∞ (K\G). Now we define a topology on ∂∞ (K\G). Let S(k) ⊂ k denote the unit sphere. For any s ∈ S(k) we define es ∈ ∂∞ (K\G) to be the class of the geodesic ray ηs : (0, ∞) → K\G defined as ηs (t) = [eits ]. Then the map e : S(k) s → [es ] ∈ ∂∞ (K\G) is a bijection (see Section II.2 in [B]) and we take on ∂∞ (K\G) the unique topology with respect to which e is a homeomorphism. Then the action of G on ∂∞ (K\G) is by homeomorphisms. For each s ∈ S(k) and any g ∈ G define s · g ∈ S(k) by the property that es · g = es·g .

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This action of G on S(k) extends the antiadjoint action of K on S(k), defined as s · h = Ad(h−1 )(s) for any s ∈ k and h ∈ K. However, unlike the action of K, the action of G is in general not smooth. We now describe the compactification obtained by adding to K\G the boundary ∂∞ (K\G). Define K\G := (K\G)∪∂∞ (K\G). There is a unique topology on K\G which satisfies these two properties: (1) the inclusions of K\G and ∂∞ (K\G) in K\G give homeomorphisms with their image, (2) if {xi } ⊂ K\G is a diverging sequence then one can write xi = ηsi (ti ), where si ∈ S(k) and ti are real numbers converging to ∞; then xi → es if and only if si → s. It is straightforward to prove that this topology on K\G is compact. Since the parameterized geodesics do not depend on the choice of invariant metric on k and the distance functions on K\G induced by two choices of invariant metric on k are uniformly comparable, the boundary ∂∞ (K\G) and the compactification K\G are completely canonical as topological spaces endowed with a G action. 2.2. Example: compactifying U(n)\ GL(n, C). It is instructive to study in concrete terms the previous definitions in the case of the symmetric space U(n)\ GL(n, C), which parameterizes Hermitian metrics on Cn . Let s ∈ S(u(n)). The matrix is is Hermitian symmetric, so it diagonalizes and has real eigenvalues, say λ1 < · · · < λr . Let Vj = Ker(λj − is) be the eigenspace corresponding to λj and define V k = V1 ⊕ · · · ⊕ Vk for any integer k ≥ 1. Now let g ∈ GL(n, C) be any element, and define Vj∞ = (g −1 (V j−1 ))⊥ ∩ g −1 (V j ), where V ⊥ denotes the orthogonal of V . Then we have a direct sum decomposition n C = Vj∞ . Define ρg (s) ∈ u(n) by the conditions that ρg (s) preserves each Vj∞ and that the restriction of ρg (s) to Vj∞ is given by multiplication by −iλj . We claim that (2.2)

ρg (s) = s · g.

We remark that if we prove the previous formula for any s and g then we obtain as a consequence the fact that the map e : S(u(n)) → ∂∞ (U(n)\ GL(n, C)) is a bijection, since, as we have observed earlier, any geodesic ray in U(n)\ GL(n, C) is of the form t → eits g for some s ∈ u(n) and g ∈ GL(n, C). We sketch two proofs of (2.2). First proof. We use the fact that the curvature of U(n)\ GL(n, C) is nonpositive. This implies the following convexity property: if γj : [0, t] → U(n)\ GL(n, C), j = 1, 2, are two geodesic segments parameterized at uniform speed, then for any λ ∈ [0, t] we have d(γ1 (λ), γ2 (λ)) ≤ max{d(γ1 (0), γ2 (0)), d(γ1 (t), γ2 (t))}, where d is the distance function between points in U(n)\ GL(n, C). Now define the map log : GL(n, C) → u(n)


119 7

by the condition that log(g) = u if g = keiu is the Cartan decomposition of g, so that k ∈ U (n) and u ∈ u(n). Assume that the following limit exists: 1 log(eiτ s g). τ We claim that then s · g = u. In fact, we have the following uniform upper bound: u = lim

τ →∞

(2.3)

d([eiτ u ], [eiτ s g]) ≤ d(1, [g])

To prove (2.3) define, for any t > 0, ut = t some t > 0 and define the maps γ1t , γ2 : R → U(n)\ GL(n, C),

for any τ ≥ 0. −1

its

log(e g), so that eitut = eits g. Fix

γ1t (η) = [eiηut ],

γ2 (η) = [eiηs g].

Both γ1 and γ2 parameterize geodesics at uniform speed, so the convexity property implies, for any τ ≥ 0, d(γ1t (τ ), γ2 (τ )) ≤ max{d(γ1t (0), γ2 (0)), d(γ1t (t), γ2 (t))} = max{d([1], [g]), d([eitut ], [eits g])} = max{d([1], [g]), 0} = d([1], [g]). Since by assumption ut → u, the continuity of the exponential map implies γ1t (τ ) → eiτ u as t → ∞, which combined with the continuity of the distance function implies (2.3). The conclusion is that (2.2) follows from this formula: 1 log(eiτ s g). τ To prove (2.4) one can argue as follows. Take any > 0 smaller than inf{λj − λj−1 }/3. Using the variational description of eigenvalues and eigenspaces of log(h), one proves that for big enough t the eigenvalues of ut are contained in [λj − , λj + ], and the number of eigenvalues in [λj − , λj + ] is equal to dim Vj . Furthermore, if we let Vjt be the direct sum of the eigenspaces of ut with eigenvalue contained in [λj − , λj + ], then Vjt converges to Vj∞ in the Grassmannian variety.

(2.4)

ρg (s) = lim

τ →∞

Second proof. The second proof is more direct and elementary, since it does not use any curvature properties, but it is perhaps not as enlightening as the first one. Define for convenience u := ρg (s) and Wi := Vi∞ . We first observe that (2.5)

d([eits g], [eitu ]) = d([eits ge−itu ], [1]),

since GL(n, C) acts on the symmetric space by isometries. To give a uniform upper bound for the term on the right hand side of (2.5) is equivalent to giving some constant C > 0 such that for any t ≥ 0 and any x ∈ Cn we have (2.6)

C −1 |x|2 ≤ |eits ge−itu x|2 ≤ C|x|2 .

Using induction on j one proves that for any j we have g(W1 ⊕ · · · ⊕ Wj ) = V j . Consequently, we can write g = j≥k gjk where gjk is a map from Wj to Vk . In because g is invertible. Now let x ∈ Cn be particular each map gjj is invertible, any element, and write x = j xj with xj ∈ Wj . We compute: eits ge−itu x = gjj (xj ) + gjk (xj )et(λk −λj ) . j

j>k

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Since the vectors {gjj (xj )} are pairwise orthogonal and the matrices gjj are invertible, there is some constant C1 independent of x such that gjj (xj )|2 ≤ C1 |x|2 . C1−1 |x|2 ≤ | j

On the other hand, setting η = inf{λj − λj−1 } there is a constant C2 > 0, independent of x, such that | gjk (xj )et(λk −λj ) |2 ≤ C2 |x|2 e−tη . j>k

Since η > 0, combining this estimate with the previous one we obtain, for a sufficiently big C > 0 and any t, the inequality (2.6). 2.3. Maximal weights. We now come back to the general setting presented in the introduction, so X is a connected Kähler manifold endowed with a Hamiltonian action of K with moment map μ : X → k∗ , and there is a holomorphic action of G on X extending the action of K. Let x ∈ X be any point. The maximal weight function λx : ∂∞ (K\G) → R ∪ {∞} is defined as λx (es ) := lim μ(eits · x), s ∈ R ∪ {∞}. t→∞

This limit exists because μ(eits · x), s is nondecreasing as a function of t, which also implies that (2.7)

λx (es ) ≥ μ(x), s.

The most important property of the maximal weight function proved in [M] is the following equivariance formula, which was stated in the introduction: Lemma 2.1. For any x ∈ X, g ∈ G and y ∈ ∂∞ (K\G) we have (2.8)

λg·x (y) = λx (y · g).

Proof. (Sketch.) Define a function Ψx : K\G → R as: 1 iu μ(eitu · x), u dt. Ψx ([e ]) = 0

We call Ψx the integral of the moment map. For any x ∈ X and g ∈ G we have dΨg·x = ρ∗g dΨx , where ρg : K\G → K\G is the map given by multiplication by g on the right (to prove this formula one uses the integrability of the complex structure and of the symplectic form on X, so the condition that X is Kähler seems to be crucial for this result to hold). Integrating we deduce the following cocycle property (2.9)

Ψx ([gh]) = Ψx ([g]) + Ψg·x ([h]).

Now the maximal weight function is related to the integral of the moment map as follows: 1 λx (es ) = lim Ψx (eits · x). t→∞ t In fancy words, λx is a sort of renormalized limit of Ψx as one approaches the boundary of K\G. Combining this formula with the cocycle property (2.9) one deduces (2.8). This deduction, however, is not completely immediate, and some control is needed on the growth of Ψx ; in [M] we obtain such control using Assumption 1.1.


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2.4. Example: linear actions on vector spaces. Suppose that X = Cn , endowed with the standard K¨ ahler structure, and that K acts on X in a linear way and respecting the standard Hermitian form. Let ξ : k → End Cn be the infinitesimal action. Then a moment map is given, for any v ∈ X, by i μ(v) = − ξ ∗ (v ⊗ v ∗ ). 2 So if s ∈ k then i i μ(v), s = − v ⊗ v ∗ , ξ(s) = − v, ξ(s)v ∈ R. 2 2 Hence, if X = Vj is the decomposition of X defined as Vj = Ker(ξ(s) − λj Id), where {λj } are the eigenvalues of ξ(s) then setting Vj X(s)− = λj ≤0

we have: λx (es ) =

0 if x ∈ X(s)− , ∞ if x ∈ / X(s)− .

2.5. Example: projective spaces. Suppose that X = CP n = P(Cn+1 ) is endowed with the Fubini–Study symplectic form, and that K acts on X through a linear representation on Cn+1 by unitary automorphisms. Let ξ : k → End Cn+1 be the infinitesimal action. A moment map μ : X → k∗ can be described as follows. Let x ∈ X be any point and let v ∈ Cn+1 \ {0} be any vector representing x; then

i ∗ v ⊗ v∗ μ(x) = − ξ . 2 v2 So for any s ∈ k we have μ(v), s = − 2i v, ξ(s)v/v2 ∈ R. Let V = Vη be the decomposition in eigenspaces of ξ(s), so that ξ(s) acts on Vη as multiplication by η. Since ξ(s) is skew Hermitian, Vη can only be nonzero if η is purely imaginary. Let v = vη , with vη ∈ Vη , and define τ ∈ iR by the property that iτ = sup{iη | vη = 0}. Assume that |s| = 1. A simple computation gives: λx (es ) = −iτ /2 = iτ /2. Now assume that there exists a real number σ > 0 such that all eigenvalues of σξ(s) belong to iZ. Then one can define a one parameter subgroup α : C∗ → GL(n + 1, C) by the condition that α(eit ) = exp(tσξ(s)), and the set of characters of α is equal to {−iση | Vη = 0}. It follows that x0 := limz→0 α(z) · x is the point in CP n represented by vτ . The character of C∗ acting on the fiber of OPn (−1) over x0 is equal to −iστ . The latter integer, which we denote by μO(1) (x; α), is what is usually called the maximal weight in GIT (see for example the paragraph before Theorem 3.9 in [T]). It follows that (2.10)

λx (es ) = −σμO(1) (x; α)/2.

Every one parameter subgroup α can be obtained applying the previous construction to a suitable s. Conversely, it is easy to check that for any s ∈ k satisfying |s| = 1 there is some one parameter subgroup α such that (2.10) holds (although in general its relation with s might not be as simple as in the case considered earlier). The general case of projective actions, as described in Section 1.2, can be

122 10


reduced to the previous one by taking a linearization of the action. Then the previous computations suggest that there should be a purely algebraic Hilbert–Mumford criterion for polystability (what is missing in the previous discussion is the algebraic analogue of the condition on two points in the boundary at infinity of K\G being geodesically connected) . This statement, due to Schmitt, is given and proved in the appendix to this paper. 2.6. Example: actions on Grassmannians. We now generalize the previous computation. Assume that X is equal to Gr(k, n), the Grassmannian of k-dimensional subspaces of Cn . As Kähler structure we take on X the restriction n of the Fubini–Study Kähler structure via the Pl¨ ucker embedding X → CP ( k )−1 . Consider the action of U (n) on X induced by the fundamental representation of U (n) on Cn . We identify the elements of X with orthogonal projections π ∈ End Cn whose image is k-dimensional. Then the moment map is given, for any s ∈ u(n), by i μ(π)(s) = − Tr(π ◦ s). 2 r Consider the decomposition Cn = j=1 Vj in eigenspaces of s, where s acts on Vj as multiplication by λj . A simple (but instructive) computation gives the following formula for the maximal weight: if V ∈ Gr(k, n), then ⎞ ⎛ r−1 1 λV (es ) = ⎝λr dim V + (λj − λj+1 ) dim(V ∩ Vj )⎠ . 2 j=1 Note that Grassmannians are particular cases of flag manifolds, so the formula for maximal weights given by Theorem 1.4 also includes them. We hence have two different ways of describing maximal weights of Grassmannians: the previous one, which is explicit and concrete, and the one given by Theorem 1.4, which is more synthetic and geometric but also more abstract. 3. Flag manifolds: maximal weights and Tits distance In this section we prove Theorem 1.4. To begin with, let us recall the definition of the Tits distance and of the relevant Kähler structures on flag manifolds. As always in this paper, K denotes a compact Lie group and G its complexification. Let k be the Lie algebra of K. Denote by B : k × k → R a negative definite symmetric pairing, invariant under the adjoint action of K on k. If K is semisimple then B can be for example the Killing pairing: B(u, v) = Tr(ad u ad v). 3.1. Tits distance. Define, for any u, v ∈ k, the positive definite pairing u, v = −B(u, v). This gives rise to a G-invariant Riemannian metric on K\G, which allows one to talk about angles between tangent vectors. Given any point x ∈ K\G and any y ∈ ∂∞ (K\G) we define the tangent vector v(x, y) ∈ Tx (K\G) by the property that the geodesic ray γ : [0, ∞) → K\G satisfying γ(0) = x and γ (0) = v(x, y) represents y.


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The Tits distance between two points y, y ∈ ∂∞ (K\G) is defined as dT (y, y ) = sup Angle(v(x, y), v(x, y )), x∈K\G

where the angle takes values in [0, π]. For example, if there exists a geodesic converging on one side to y and on the other side to y , then dT (y, y ) = π. The converse is also true, but this is not at all obvious, since the Tits distance is defined by taking the supremum over a noncompact space. A proof can be given combining the computation in Lemma 3.1 and the results in Section 5.2 of [M]. 3.2. Symplectic geometry of flag manifolds. There are different possible ways of looking at flag manifolds of G. From the point of view of symplectic geometry it is natural to identify them with coadjoint orbits in k∗ . To be precise, if X = G/P is a flag manifold then to identify X with a coadjoint orbit is exactly the same thing as choosing a K-invariant symplectic form on X and a moment map μ : X → k∗ : indeed, μ(X) ⊂ k∗ is a coadjoint orbit because K acts transitively on X. The symplectic structure on the coadjoint orbits O∗ = μ(X) coincides with the Kostant–Souriau symplectic form ΩO∗ , which is defined as follows: if σ ∈ O∗ ⊂ k∗ and u, v ∈ Tσ O∗ are tangent vectors then we can write u = ad∗ (a)(σ), v = ad∗ (b)(σ) for some elements a, b ∈ k, and we have ΩO∗ (u, v) = σ([a, b]). In our discussion, however, we prefer to identify flag manifolds with adjoint orbits in k instead of coadjoint orbits in k∗ . Using the invariant pairing B we get a K

equivariant isomorphism of vector spaces δ : k −→ k∗ , which puts in correspondence adjoint orbits with coadjoint orbits. If O ⊂ k is an adjoint orbit, the Kostant– Souriau symplectic form translates into a symplectic form ωO = δ ∗ ΩO∗ on O (here O∗ = δ(O)), which can be described as follows. Let y ∈ O and let α, β ∈ Ty O. We can write α = [a, y] and β = [b, y] for some a, b ∈ k, and we have ωO (α, β) = B([a, b], y). A moment map μ : O → k∗ of the action of K on O can be computed as follows: if y ∈ O and a ∈ k then (3.11)

μ(y), a = B(y, a).

To understand the converse construction identifying adjoint orbits O ⊂ k with flag manifolds, it is useful to use as a bridge the G-orbits in ∂∞ (K\G), which can be identified both with adjoint orbits and with flag manifolds. For these identifications to be G-equivariant, however, we need to invert the action of G on ∂∞ (K\G), which was previously defined as an action on the right. So for any y ∈ ∂∞ (K\G) and any g ∈ G we define g · y := y · g −1 . If O is an adjoint orbit, then picking any u ∈ O and setting s = u/|u| we have the following two (well known) facts: (1) the G-orbit and the K-orbit through es in ∂∞ (K\G) are the same: G·es = K · es (see for example Lemma 5.2 in [M]); (2) the stabilizer of es in G is a parabolic subgroup P ⊂ G (see the beginning of Section 5.1 in [M]).

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It follows that the orbit G · es ⊂ ∂∞ (K\G) is G-equivariantly isomorphic to G/P . Denote by φ : G · es → G/P a G-equivariant isomorphism. Then the map f : O → G/P defined as f (k · u) = φ−1 (k · es ) is a K-invariant isomorphism, so the action of G on G/P can be transported to an action on O which extends the adjoint action. Furthermore, G/P has a natural complex structure which combines with ωO to give a K¨ ahler structure. The previous discussion implies that if we take any x ∈ X and we denote by u ∈ k the point corresponding to x via the identification of X with an adjoint orbit, then the maximal weight function is given by ωO ) λ(X, (ev ) = lim B(u · e−itv , v) x

(3.12)

t→∞

for any v ∈ S(k). 3.3. Comparing maximal weights and Tits distance. Taking (3.12) into account, Theorem 1.4 is equivalent to the following lemma: Lemma 3.1. Assume that y, y ∈ ∂∞ (K\G), and that y = eu , y = ev with u, v ∈ S(k). Then − cos dT (y, y ) = lim B(u · e−itv , v). t→∞

Proof. An immediate consequence of the definition of the action of G on ∂∞ (K\G) is the following formula: for any x ∈ K\G, y ∈ ∂∞ (K\G) and g ∈ G, v(x · g, y) = Dρg (v(x, y · g −1 )),

(3.13)

where ρg is as in the proof of Lemma 2.1. Using (3.13) we have − cos dT (y, y ) = − inf u · g, v · g g∈G

= sup −u · g, v · g g∈G

= sup B(u · g, v · g). g∈G

−itv

Since v · e

= v for any t, we have

sup B(u · g, v · g) ≥ lim B(u · e−itv , v · e−itv ) = lim B(u · e−itv , v). t→∞

g∈G

t→∞

We now prove the converse inequality. Let X denote the orbit G · y ⊂ ∂∞ (K\G), endowed with the Kähler structure ωO described in Section 3.2 (so O is the adjoint orbit through u), and denote by μ : X → k∗ the moment map of the action of K on X. Let g ∈ G be any element. We have: O) lim B(u · e−itv , v) = λ(X,ω (ev ) y

t→∞

by formula (3.12)

(X,ω )

= λg−1 ·yO (ev · g) ≥ μ(g

−1

by Lemma 2.1

· y), v · g

= B(u · g, v · g)

by formula (2.7)

by formula (3.11).

We deduce that sup B(u · g, v · g) = lim B(u · e−itv , v),

g∈G

so the lemma is proved.

t→∞


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Corollary 1.5 follows immediately from Lemma 3.1. Indeed, we may assume, as in the statement of Lemma 3.1, that y = eu , y = ev , and (using the action of some element in G if necessary) we can also assume that ξ(t) = [eitu ]. But then − cos Angle(ξ (t), ηt (0)) = B(u · e−itv , v), so by Lemma 3.1 we have − cos dT (y, y ) = lim − cos Angle(ξ (t), ηt (0)). t→∞

Since both the Tits distance and the angle on the right hand side take values in [0, π], we can apply arccos to both sides of the equality and obtain Corollary 1.5. 3.4. Relation to the work of Kapovich, Leeb and Millson. We have already mentioned that some of the notions and results of this paper also appear, applied to the case of flag manifolds, in the paper [KLM] by Kapovich, Leeb and Millson, although often the same concepts receive a different name and the proofs follow a different scheme. We now briefly explain these analogies and differences. (1) The integral of the moment map in the case of flag manifolds is a Buseman function; this follows from the formula for the derivative of Buseman functions (see formula (4) in [KLM]). (2) It follows that the asymptotic slope, defined in (7) of [KLM], coincides with the maximal weight function for flag manifolds. (3) In view of the preceding, it is clear that formula (10) in [KLM] is completely parallel to Theorem 1.4. But the logic of the proof of (10) in [KLM] is different from ours, since Kapovich et al. rely on the property of Tits distance stated in Corollary 1.5, whereas our arguments are based on the general equivariance property of the maximal weight function (and we do not use any property of Tits distance apart from its definition). (4) As we mentioned in Section 1.2, in the case of projective manifolds the maximal weight function is essentially the same thing as the maximal weights in GIT. So Theorem 1.4 establishes a relation between Tits distance and GIT on flag manifolds. In Section 4 of [KLM] such a relation is proved for the case X = CP n . (5) Finally, the notion of nice semistability introduced in Definition 3.13 of [KLM] is very likely equivalent to the notion of analytic polystability introduced in [M]. Appendix: An algebraic proof for Mundet i Riera’s polystability criterion in GIT A.H.W. Schmitt Dedicated to Professor S. Ramanan on the occasion of his 70th birthday A.1. Statement of the result. Let G be a reductive linear algebraic group over the algebraically closed field k and ρ : G −→ GL(V ) a representation on the finite dimensional k-vector space V . A point v ∈ V is called semistable, if the closure of its G-orbit in V does not contain the origin, and polystable, if v = 0 and its orbit G · v is closed in V . Note that a polystable point is semistable. We remind the reader of the following.

126 14


Theorem A.1 (Hilbert–Mumford criterion). Let v ∈ V and v ∈ G · v be a point with closed G-orbit. Then, there exists a one parameter subgroup λ : Gm (k) −→ G with lim λ(z) · v ∈ G · v. z→∞

Given a one parameter subgroup λ : Gm (k) −→ GL(V ) and γ ∈ Z, we define the eigenspace to the weight γ as V (γ) := v ∈ V | λ(z) · v = z γ · v ∀z ∈ Gm (k) . Let γ1 < · · · < γs+1 be the weights with non-trivial eigenspaces. The vector space V then decomposes as V = V (γ1 ) ⊕ · · · ⊕ V (γs+1 ). For a point v ∈ V \ {0}, we define μ(v, λ) := max γi | v has a non-trivial component in V (γi ) . We can restate the Hilbert–Mumford criterion as follows. Corollary A.2. i) A point v ∈ V is semistable, if and only if μ(v, λ) ≥ 0 holds for every one parameter subgroup λ : Gm (k) −→ G. ii) A point v ∈ V is polystable, if and only if it is semistable and for every one parameter subgroup λ : Gm (k) −→ G with μ(v, λ) = 0 there is a group element g ∈ G with lim λ(z) · v = g · v. z→∞

Remark A.3. Let v ∈ V be a semistable point and λ : Gm (k) −→ G a one parameter subgroup with μ(v, λ) = 0. Then, v∞ := lim λ(z) · v z→∞

is the component of v in the weight space V (0) and a semistable point. For a parabolic subgroup Q ⊂ G, there is a Levi subgroup L ⊂ Q, i.e., a reductive subgroup, such that Q = Ru (Q) L, where Ru (Q) is the unipotent radical of Q. Any two Levi subgroups L and L of Q are conjugate by an element r ∈ Ru (Q). Example A.4. i) For a one parameter subgroup λ : Gm (k) −→ G, QG (λ) := g ∈ G | lim λ(z) · g · λ(z)−1 exists in G z→∞

is a parabolic subgroup of G and every parabolic subgroup of G is obtained in this way. The unipotent radical of QG (λ) is Ru QG (λ) = g ∈ G | lim λ(z) · g · λ(z)−1 = e , z→∞

and

LG (λ) := g ∈ G | λ(z) · g = g · λ(z) ∀z ∈ Gm (k) is a Levi subgroup of QG (λ).


127 15

For a one parameter subgroup λ : Gm (k) −→ GL(V ) with weights γ1 < · · · < γs+1 , we define the (partial) flag V• (λ) : {0} V1 := V (γ1 ) V2 := := V (γ1 ) ⊕ V (γ2 ) · · · Vs := V (γ1 ) ⊕ · · · ⊕ V (γs ) V. Then, QGL(V ) (λ) is the GL(V )-stabilizer of the flag V• (λ), LGL(V ) (λ) identifies with GL(V (γ1 ))×· · ·×GL(V (γs+1 )), and Ru (QGL(V ) (λ)) consists of those matrices whose diagonal blocks are unit matrices. The latter description implies the following: If v ∈ V is a point with μ(v, λ) = γi and g ∈ Ru (QGL(V ) (λ)), then v and g · v have the same component in V (γi ). ii) By the conjugacy of Levi subgroups, we can obtain every pair (Q, L), consisting of a parabolic subgroup Q of G and a Levi subgroup L of Q, as (QG (λ), LG (λ)) for an appropriate one parameter subgroup λ : Gm (k) −→ G. iii) For every point v ∈ V , every one parameter subgroup λ : Gm (k) −→ G, and every g ∈ QG (λ), one has, by i), μ(v, λ) = μ(v, g · λ · g −1 ). Two parabolic subgroups Q and Q are called opposite, if Q ∩ Q is a Levi subgroup of both Q and Q . Likewise, two one parameter subgroups λ and λ are said to be opposite1, if QG (λ) and QG (λ ) are opposite parabolic subgroups of G. In [M], Mundet i Riera proves a characterization of polystable points in a symplectic manifold with respect to a Hamiltonian action by a compact real group. Specialized to the GIT setting, his result reads as follows. Theorem A.5 (Mundet i Riera). A point v ∈ V is polystable, if and only if it is semistable and for every one parameter subgroup λ : Gm (k) −→ G with μ(v, λ) = 0 there is an opposite one parameter subgroup λ with μ(v, λ ) = 0. A.2. A Strengthening of the Hilbert–Mumford Criterion. Proposition A.6. A point v ∈ V is polystable, if and only if it is semistable and for every one parameter subgroup λ : Gm (k) −→ G with μ(v, λ) = 0 there is a group element g ∈ Ru (QG (λ)) with limz→∞ λ(z) · v = g · v. The proof rests on the following. Proposition A.7 (Kraft/Kuttler). Let G be a reductive affine algebraic group and H ⊂ G a reductive closed subgroup. Let v ∈ G/H a point and λ : Gm (k) −→ G a one parameter subgroup, such that v∞ := limz→∞ λ(z) · v exists. Then, there exists an element g ∈ Ru (QG (λ)) with v∞ = g · v. Proof. We refer to [GLSS], Proposition 2.1.2, p. 1188, or [S], Proposition 2.4.2.5, p. 181. For a different proof and a generalization to perfect fields, the reader may consult Theorem 3.4 in [BMRT]. 1Mundet i Riera does, as Mumford, require that λ−1 is conjugate to λ . The proof we present here works also for that definition.

128 16


Proof of Proposition A.6. The stated condition is clearly sufficient. To see that it is also necessary, let v ∈ V be a polystable point. Its stabilizer Gv is reductive. If Char(k) = 0, then the orbit map ov : G −→ V , g −→ g · v induces an isomorphism G/Gv −→ G · v. So, the result follows directly from Proposition A.7 with H = Gv . In positive characteristic, the orbit map ov induces a bijective and proper morphism G/Gv −→ G · v, so that the same reasoning applies. A.3. Proof of Theorem A.5. Let v ∈ V be a polystable point and λ : Gm (k) −→ G a one parameter subgroup with μ(v, λ) = 0. Then, v∞ := limz→∞ λ(z) · v exists, by Remark A.3. Proposition A.6 grants the existence of an element g ∈ Ru (QG (λ)) := g −1 · λ · g stabilizes v. We with v∞ = g · v. Note that λ stabilizes v∞ . Thus, λ −1 . Obviously, set λ := λ and LG (λ ) = QG (λ ) ∩ QG (λ) = LG (λ). μ(v, λ ) = 0, QG (λ) = QG (λ), Let us assume, to the converse, that the point v ∈ V satisfies Mundet i Riera’s criterion. We will verify that v satisfies the property stated in Proposition A.6, too. So, let λ : Gm (k) −→ G be a one parameter subgroup with μ(v, λ) = 0 and set v∞ := limz→∞ λ(z) · v. By assumption, there is a one parameter subgroup λ : Gm (k) −→ G with μ(v, λ ) = 0 which is opposite to λ. Taking into account Example A.4, iii), and the conjugacy of Levi subgroups, we may assume QG (λ ), LG (λ ) = QG (λ ), QG (λ ) ∩ QG (λ) . Likewise, we find an element g ∈ Ru (QG (λ)), such that QG (g · λ · g −1 ), LG (g · λ · g −1 ) = QG (λ), QG (λ) ∩ QG (λ ) . Now, λ and g · λ · g −1 lie in the connected component of the identity of the center of L := LG (λ ) = LG (g · λ · g −1 ). In particular, there is a maximal torus T of G which contains both λ and g · λ · g −1 . It follows that (A.14) sign g · λ · g −1 , χ = −sign λ , χ , ∀χ ∈ X(T ). On the other hand, μ(v, λ ) = 0 = μ(v, λ)

Example A.4, iii)

=

μ(v, g · λ · g −1 ).

The latter equality is compatible with (A.14), if and only if v is contained in the eigenspace to the weight zero of both λ and g · λ · g −1 . In particular, g · λ · g −1 fixes v or, equivalently, λ fixes g −1 · v. We will be done, if we can check that v∞ = g −1 · v. Recall from Remark A.3 that v∞ is the component of v in the eigenspace of λ for the weight zero. Noting that ρ maps QG (λ) into QGL(V ) (λ), Example A.4, i), shows that v, v∞ , and g −1 · v have the same component in the weight zero eigenspace of λ. Since both v∞ and g −1 · v are elements of that eigenspace, they do agree. Remark A.8. Theorem A.5 implies Proposition A.7. Indeed, if λ is a one parameter subgroup with μ(v, λ ) = 0 which is opposite to λ, then we find elements −1 g ∈ Ru (QG (λ)) and g ∈ Ru (QG (λ )), such that LG (g · λ · g ) = LG (g · λ · g −1 ). Moreover, μ(v, g · λ · g

−1

) = μ(v, λ ) = 0 = μ(v, λ) = μ(v, g · λ · g −1 ).


129 17

Again, this is only possible, if v lies in the weight space to weight zero of g · λ · g −1 . Then, g −1 · v is fixed by λ and as before we infer g −1 · v = lim λ(z) · v. z→∞

References W. Ballmann, Lectures on spaces of nonpositive curvature, With an appendix by Misha Brin DMV Seminar 25, Birkh¨ auser Verlag, Basel, 1995. [BMRT] M. Bate, B. Martin, G. Roehrle, R. Tange, Closed orbits and uniform S-instability in Invariant Theory, preprint arXiv:0904.4853. [BJ] A. Borel, L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications, Birkh¨ auser, Boston (2006). [E] P. Eberlein, Structure of manifolds of nonpositive curvature, Global differential geometry and global analysis 1984 (Berlin, 1984), 86–153, Lecture Notes in Math. 1156 Springer, Berlin, 1985. [GLSS] T.L. G´ omez, A. Langer, A.H.W. Schmitt, I. Sols, Moduli spaces for principal bundles in arbitrary characteristic, Adv. Math. 219 (2008), 1177–1245. [HS] P. Heinzner, G.W. Schwartz, Cartan decomposition of the moment map, Math. Ann. 337 (2007), 197–232. [KLM] M. Kapovich, B. Leeb, J. Millson, Convex functions on symmetric spaces, side lengths of polygons and stability inequalities for weighted configurations, J. Differential Geom. 81 (2009), no. 2, 297–354. [MFK] D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, 3rd edition, Erg. Math., Springer Verlag (1994). [M] I. Mundet i Riera, A Hilbert–Mumford criterion for polystability in K¨ ahler geometry, Trans. Amer. Math. Soc., to appear. [S] A.H.W. Schmitt, Geometric Invariant Theory and Decorated Principal Bundles, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), vii+389 pp. [T] R.P. Thomas, Notes on GIT and symplectic reduction for bundles and varieties, Surveys in differential geometry X, 221–273, Surv. Differ. Geom., 10, International Press (2006). [B]

` `tiques, Universitat de Departament d’Algebra i Geometria, Facultat de Matema Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain E-mail address: [email protected] ¨t Berlin, Arnimallee 3, 14195 Berlin, Germany Freie Universita E-mail address: [email protected]



Orthogonal Bundles Over Curves in Characteristic Two Christian Pauly Dedicated to S. Ramanan on his 70th birthday Abstract. Let X be a smooth projective curve of genus g ≥ 2 defined over a field of characteristic two. We give examples of stable orthogonal bundles with unstable underlying vector bundles and use them to give counterexamples to Behrend’s conjecture on the canonical reduction of principal G-bundles for G = SO(n) with n ≥ 7.

Let X be a smooth projective curve of genus g ≥ 2 and let G be a connected reductive linear algebraic group defined over a field k of arbitrary characteristic. One associates to any principal G-bundle EG over X a reduction EP of EG to a parabolic subgroup P ⊂ G, the so-called canonical reduction — see e.g. [Ra], [Be], [BH] or [H] for its definition. We only mention here that in the case G = GL(n) the canonical reduction coincides with the Harder-Narasimhan filtration of the rank-n vector bundle associated to EG . In [Be] (Conjecture 7.6) K. Behrend conjectured that for any principal Gbundle EG over X the canonical reduction EP has no infinitesimal deformations, or equivalently, that the vector space H 0 (X, EP ×P g/p) is zero. Here p and g are the Lie algebras of P and G respectively. Behrend’s conjecture implies that the canonical reduction EP is defined over the same base field as EG . We note that this conjecture holds for the structure groups GL(n) and Sp(2n) in any characteristic, and also for SO(n) in any characteristic different from two — see [H] section 2. On the other hand, a counterexample to Behrend’s conjecture for the exceptional group G2 in characteristic two has been constructed recently by J. Heinloth in [H] section 5. In this note we focus on SO(n)-bundles in characteristic two. As a starting point we consider the rank-2 vector bundle F∗ L given by the direct image under the Frobenius map F of a line bundle L over the curve X and observe (Proposition 4.4) 2010 Mathematics Subject Classification. 14H60, 14H25. The author was partially supported by the Ministerio de Educaci´ on y Ciencia (Spain) through the grant SAB2006-0022. c Mathematical 0000 (copyright Society holder) c 2010 American

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CHRISTIAN PAULY

that the SO(3)-bundle A := End0 (F∗ L) is stable, but that its underlying vector bundle is unstable and, in particular, destabilized by the rank-2 vector bundle F∗ OX . We use this observation to show that the SO(7)-bundle F∗ OX ⊕ (F∗ OX )∗ ⊕ A equipped with the natural quadratic form gives a counterexample to Behrend’s conjecture. Replacing A by Aˆ = End(F∗ L) we obtain in the same way a counterexample for SO(8), and more generally for SO(n) with n ≥ 7 after adding direct summands of hyperbolic planes. Note that Behrend’s conjecture holds for SO(n) with n ≤ 6 because of the exceptional isomorphisms with other classical groups. The first three sections are quite elementary and recall well-known facts on quadratic forms, orthogonal groups and their Lie algebras, as well as orthogonal bundles in characteristic two. In the last section we give an example (Proposition 6.1) of an unstable SO(7)-bundle having its canonical reduction only defined after an inseparable extension of the base field. This note can be considered as an appendix to the recent paper [H] by J. Heinloth, whom I would like to thank for helpful discussions. I also thank the referee for useful suggestions and, in particular, for an improvement of the proof of Proposition 4.4. 1. Quadratic forms in characteristic two The purpose of this section is to recall definitions and basic properties of quadratic forms in characteristic two. Let k be a field of characteristic two and let V be a vector space of dimension n over k. We denote by Sym2 V and Λ2 V the symmetric and exterior square of V , i.e., Sym2 V = V ⊗ V /I, and Λ2 V = V ⊗ V /J , where I and J are the subspaces of V ⊗ V linearly generated by the tensors v ⊗ w + w ⊗ v and v ⊗ v respectively, with v, w ∈ V . Note the inclusion I ⊂ J . We denote by F (V ) := V ⊗k k the Frobenius-twist of V . We observe that there exists a well-defined k-linear injective map F (V ) → Sym2 V,

v ⊗ λ → λ(v · v),

where v · v denotes the symmetric square of v. More precisely, we have the following exact sequence of k-vector spaces 0 −→ F (V ) −→ Sym2 V −→ Λ2 V −→ 0. Let σ denote the involution on V ⊗ V defined by σ(v ⊗ w) = w ⊗ v and let S2 (V ) ⊂ V ⊗ V denote the σ-invariant subspace. Then one has the exact sequence 0 −→ S2 (V ) −→ V ⊗ V −→ Λ2 V −→ 0, where the last arrow denotes the canonical projection of V ⊗ V onto Λ2 V . By choosing a basis of V , it can be checked that the image of S2 (V ) under the canonical projection V ⊗ V → Sym2 V equals the subspace F (V ).

ORTHOGONAL BUNDLES OVER CURVES IN CHARACTERISTIC TWO

133 3

By definition, see e.g. [Bo] section 23.5, a quadratic form on the vector space V is a k-valued function on V such that Q(av + bw) = a2 Q(v) + b2 Q(w) + abβ(v, w)

a, b ∈ k; v, w ∈ V,

where β is a bilinear form on V . The bilinear form β associated to a quadratic form Q is determined by the formula β(v, w) = Q(v + w) + Q(v) + Q(w). Is is clear that β is symmetric, hence β can be regarded as an element in S2 (V ∗ ). We denote the vector space of quadratic forms on V by Quad(V ) and introduce the k-linear map Φ : Sym2 (V ∗ ) −→ Quad(V ),

ϕ · ψ → Q, with Q(v) = ϕ(v)ψ(v).

Here ϕ · ψ ∈ Sym2 (V ∗ ) denotes the image of ϕ ⊗ ψ ∈ V ∗ ⊗ V ∗ under the canonical projection V ∗ ⊗ V ∗ → Sym2 (V ∗ ). By choosing a basis of V and using formula (5) of [Bo] section 23.5, it is easily seen that Φ is an isomorphism. We will therefore identify Quad(V ) with the symmetric square Sym2 (V ∗ ). The assignment Q → β gives rise to the k-linear polarisation map P : Sym2 (V ∗ ) −→ S2 (V ∗ ),

Q → β.

Over a field k of characteristic different from 2 we recall that the polarisation map P is an isomorphism. In our situation one can easily work out that ker P = F (V ∗ )

and

coker P = F (V ∗ ).

Quadratic forms Q ∈ ker P correspond to squares of linear forms on V and the last equality asserts that the bilinear form β is alternating, i.e., β(v, v) = 0 for all v ∈V. We say that Q is non-degenerate if β induces an isomorphism β˜ : V → V ∗ for n even, and if dim ker β˜ = 1 and Q| ker β˜ = 0 for n odd. We say that a linear subspace W ⊂ V is isotropic for Q if Q|W = 0. 2. Orthogonal groups and their Lie algebras in characteristic two In this section we work out in detail the structure of the Lie algebras of the orthogonal groups SO(7) and SO(8) over a field k of characteristic two, as well as of some of their parabolic subgroups P ⊂ SO(n). Following the principle of [H], we describe the representation so(n)/p of a Levi subgroup L ⊂ P , which we will use later in section 5. The main reference is [Bo] section 23.6. By definition the orthogonal group O(n) is the subgroup of GL(n) = GL(V ) stabilizing a non-degenerate quadratic form Q ∈ Sym2 (V ∗ ). Note that in characteristic two O(n) ⊂ SL(n). We also recall that, if n is odd, the group O(n) is connected. If n is even, O(n) has two connected components distinguished by the Dickson invariant ([D] page 301). In order to keep the same notation as in characteristic = 2, we denote by SO(n) the connected component of O(n) containing the identity.

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2.1. The group SO(8). We consider V = k8 endowed with the non-degenerate quadratic form Q = x1 x7 + x2 x8 + x3 x5 + x4 x6 ∈ Sym2 (V ∗ ). We choose this non-standard quadratic form in order to obtain a simple description of the Lie algebra p of the parabolic subgroup P fixing an isotropic 2-plane, which will appear in section 5. The bilinear form β associated to Q is given by the matrix ⎞ ⎛ O O O I ⎜ O O I O ⎟ ⎟ ⎜ ⎝ O I O O ⎠, I O O O

where I denotes the identity matrix

1 0

0 1

and O the zero 2 × 2 matrix. The

computation carried out in [Bo] section 23.6 shows, after a renumbering of the lines and columns, that the Lie algebra so(8) ⊂ End(V ) of SO(8) consists of the matrices of the form ⎛ ⎞ X1 X2 X3 D1 ⎜ X4 X5 D2 t X3 ⎟ ⎜ ⎟ (2.1) ⎝ X6 D3 t X5 t X2 ⎠ , D4 t X6 t X4 t X1 t where the Xi are 2 × 2 matrices, the Xi denote their transpose, and the Di denote 0 λ matrices of the form λ 0 .

We consider the maximal parabolic subgroup P ⊂ SO(8) preserving the isotropic 2-plane W ⊂ V = k 8 defined by the equations xi = 0 for i = 3, . . . , 8. Its orthogonal space W ⊥ is then defined by the equations x7 = x8 = 0. The Levi subgroup L ⊂ P is isomorphic to GL(2) × SO(4). The Lie algebra p of P consists of the matrices (2.1) that satisfy the conditions X4 = X6 = 0

and

D4 = 0.

The vector space so(8)/p sits naturally in the exact sequence of L-modules (2.2)

0 −→ Hom(W, W ⊥ /W ) −→ so(8)/p −→ D −→ 0,

where D is the one-dimensional subspace generated by the element e ⊗ f + f ⊗ e ∈ W ∗ ⊗ W ∗ = Hom(W, W ∗ ) ∼ = Hom(W, V /W ⊥ ), where e and f are linearly independent. 2.2. The group SO(7). We consider V = k7 endowed with the non-degenerate quadratic form Q = x1 x6 + x2 x7 + x3 x4 + x25 ∈ Sym2 (V ∗ ). The bilinear form β associated to Q is given ⎛ O O 0 ⎜ O I 0 ⎜ ⎝ 0 0 0 I O 0

by the 7 × 7 matrix ⎞ I O ⎟ ⎟. 0 ⎠ O

135 5


The radical ker β˜ of Q is generated by e5 . A straightforward computation shows that the Lie algebra so(7) ⊂ End(V ) of SO(7) consists of the matrices of the form ⎛ ⎞ X1 X2 0 D1 ⎜ X3 Δ 0 t X2 ⎟ ⎜ ⎟ (2.3) ⎝ A1 A2 0 A3 ⎠ , D2 t X3 0 t X1 where the Ai are 2 × 1 matrices and Δ is a diagonal matrix of the form

λ 0

0 λ

. We consider the maximal parabolic subgroup P ⊂ SO(7) preserving the isotropic 2-plane W ⊂ V = k7 defined by the equations xi = 0 for i = 3, . . . , 7. Its orthogonal space W ⊥ is then defined by the equations x6 = x7 = 0. The Levi subgroup L ⊂ P is isomorphic to GL(2) × SO(3). The Lie algebra p of P consists of the matrices (2.3) that satisfy the conditions X3 = 0,

A1 = 0,

and

D2 = 0.

As above we have an exact sequence of L-modules (2.4)

0 −→ Hom(W, W ⊥ /W ) −→ so(7)/p −→ D −→ 0. 3. Orthogonal bundles

By definition an orthogonal bundle over a smooth projective curve X defined over a field k is a principal O(n)-bundle over X, which we denote by EO(n) . Equivalently a O(n)-bundle corresponds to a pair (E, Q), where E is a rank-n vector bundle and Q is a non-degenerate OX -valued quadratic form on E, i.e., an element Q ∈ H 0 (X, Sym2 (E ∗ )), such that at every point x ∈ X the quadratic form Qx ∈ Sym2 (Ex∗ ) is non-degenerate. By abuse of notation we will write EO(n) = (E, Q). Since the parabolic subgroups of O(n) are the stabilizers of isotropic flags in kn , one observes that the notion of (semi-)stability of the O(n)-bundle EO(n) translates into the following condition on the pair (E, Q) (see e.g. [Ram] Definition 4.1): we say that (E, Q) is semi-stable (resp. stable) as an orthogonal bundle if and only if deg F ≤0 rkF for all isotropic subbundles F ⊂ E. μ(F) :=

(resp. < 0)

A noteworthy result is the following Proposition 3.1 ([Ram] Proposition 4.2). Assume that the characteristic of k is different from two. The pair (E, Q) is semi-stable as an orthogonal bundle if and only if the underlying vector bundle E is semi-stable. In the next section we will show that the assumption on the characteristic can not be removed. 4. Stable orthogonal bundles with unstable underlying bundle We denote by F : X → X the absolute Frobenius of the curve X of genus g ≥ 2 defined over an algebraically closed field k of characteristic two. We denote by KX the canonical line bundle of X.

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4.1. An orthogonal rank-3 bundle. Let E be a rank-2 vector bundle of arbitrary degree over X. The determinant endows the bundle End0 (E) of traceless endomorphisms of E with an OX -valued non-degenerate quadratic form det : End0 (E) −→ OX . Its associated bilinear form β is given by β(v, w) = Tr(v ◦ w) for u, v local sections of End0 (E) and the radical of det equals the line subbundle OX → End0 (E) of homotheties. Lemma 4.1. There is an exact sequence (4.1)

0 −→ OX −→ End0 (E) −→ F ∗ (E) ⊗ (det E)−1 −→ 0,

where the first homomorphism is the inclusion of homotheties into End0 (E). Proof. Consider V = k2 and W = End0 (V ) equipped with the determinant. Note that for any g ∈ GL(V ) the conjugation Cg : W → W is orthogonal and leaves the identity Id invariant. Thus we obtain group homomorphisms GL(V ) → SO(W ) → GL(W/ Id), g → Cg → C g . A straightforward computation shows that the composite map is given by g → det1 g F (g), where F (g) is the Frobenius-twist of g. Proposition 4.2. If E is stable, then (End0 (E), det) is stable as an orthogonal bundle. Proof. It suffices to observe that an isotropic line subbundle M of End0 (E) corresponds to a nonzero homomorphism φ : E → E ⊗ M −1 of rank 1. If L denotes the line bundle imφ ⊂ E ⊗ M −1 , we obtain the inequalities μ(E) < deg L < μ(E ⊗ M −1 ), which implies deg M < 0. Let B denote the sheaf of locally exact differentials [Ray], which can be defined as the cokernel (4.2)

0 −→ OX −→ F∗ OX −→ B −→ 0,

of the inclusion OX → F∗ OX given by the Frobenius map. Note that in characteristic 2 the sheaf B is a theta-characteristic, i.e., B 2 = KX . We denote by e ∈ Ext1 (B, OX ) = H 1 (X, B −1 ) the non-zero extension class determined by the exact sequence (4.2). By tensoring this exact sequence with B −1 and taking the long exact sequence of cohomology spaces, we obtain the following Lemma 4.3. The extension class e generates the one-dimensional kernel of the Frobenius map, i.e., −1

e = ker F : H 1 (X, B −1 ) −→ H 1 (X, KX ) . In particular, we have a direct sum F ∗ (F∗ OX ) = OX ⊕ KX . Let L be a line bundle of arbitrary degree. Then the rank-2 vector bundle F∗ L has determinant equal to B ⊗ L, is stable and is destabilized by pull-back by the Frobenius map F ([LP]). More precisely, the bundle F ∗ (F∗ L) ⊗ B −1 ⊗ L−1 is the unique non-split extension (see e.g. [LS]) (4.3)

0 −→ B −→ F ∗ (F∗ L) ⊗ B −1 ⊗ L−1 −→ B −1 −→ 0. ι

π


137 7

Proposition 4.4. (i) The vector bundle A := End0 (F∗ L) does not depend on L. (ii) The vector bundle A is the unique non-split extension φ

ψ

0 −→ F∗ OX −→ A −→ B −1 −→ 0. We denote by φ and ψ generators of the one-dimensional spaces Hom(F∗ OX , A) and Hom(A, B −1 ). (iii) The restriction of the quadratic form det to the subbundle F∗ OX equals the evaluation morphism F ∗ F∗ OX −→ OX . In particular the restriction of β to F∗ OX is identically zero. (iv) Let x and y be local sections of F∗ OX and A. Then the two bilinear forms β(φ(x), y) and x, ψ(y) on the product F∗ OX × A differ by a nonzero multiplicative scalar. Here ., . denotes the standard pairing between F∗ OX and its dual (F∗ OX )∗ . Proof. First we consider the exact sequence (4.1) for the bundle F∗ L 0 −→ OX −→ End0 (F∗ L) −→ F ∗ (F∗ L) ⊗ B −1 ⊗ L−1 −→ 0. We then observe that there exists a natural inclusion F∗ End(L) = F∗ OX → End0 (F∗ L), which extends the inclusion of homotheties OX → End0 (F∗ L). Note that the subbundle F∗ OX corresponds to endomorphisms of F∗ L which are F∗ OX -linear. A local computation shows that these are trace-free. Since det F∗ OX = B, we obtain an exact sequence 0 −→ F∗ OX −→ End0 (F∗ L) −→ B −1 −→ 0. We denote by a its extension class in Ext1 (B −1 , F∗ OX ) = H 1 (X, (F∗ OX ) ⊗ B). We have (F∗ OX ) ⊗ B = (F∗ OX )∗ ⊗ KX and by Serre duality H 1 (X, (F∗ OX ) ⊗ B) = H 0 (X, F∗ OX )∗ = H 0 (X, OX )∗ , which implies that dim Ext1 (B −1 , F∗ OX ) = 1. Hence there exists (up to isomorphism) a unique non-split extension of B −1 by F∗ OX . In order to show assertions (i) and (ii) it will be enough to check that a = 0. But the push-out of a under the map F∗ OX → B gives the exact sequence (4.3), which is a non-split extension. This proves the claim. The proof of the equalities dim Hom(F∗ OX , A) = dim Hom(A, B −1 ) = 1 is standard. The restriction of the bilinear form β to F∗ OX is identically zero, since β factorizes through the line bundle quotient B and β is alternating (see section 1). Hence the restriction of the quadratic form det lies in F ∗ (F∗ OX )∗ ⊂ Sym2 (F∗ OX )∗ , i.e., corresponds to an OX -linear map α : F ∗ F∗ OX = OX ⊕ KX → OX . We note that α|OX = id and α|KX = 0. This proves (iii). Finally we observe that the bilinear form (x, y) → β(φ(x), y) factorizes through the quotient B × B −1 , since β is identically zero on F∗ OX by part (iii) and since the radical of β equals OX . On the other hand the bilinear form (x, y) → x, ψ(y) also factorizes through B × B −1 . This proves (iv), since both non-zero bilinear forms factorize through the unique (up to a multiplicative scalar) perfect pairing on B × B −1 .

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CHRISTIAN PAULY

Since F∗ OX is stable and μ(F∗ OX ) = Narasimhan filtration of A is

g−1 2

> 0, we deduce that the Harder-

0 ⊂ F∗ OX ⊂ A. However, since F∗ L is stable, Proposition 4.2 implies that the orthogonal bundle (A, det) is stable. Remark. Similarly, it can be shown that the rank-4 vector bundle Aˆ := End(F∗ L) does not depend on the line bundle L, that it fits into the exact sequence ∗

0 −→ F∗ OX −→ End(F∗ L) −→ (F∗ OX ) −→ 0, ˆ det) is a stable orthogonal giving its Harder-Narasimhan filtration, and that (A, bundle. 4.2. An orthogonal rank-4 bundle. Besides the example given in the last remark of section 3.1, we can consider the unstable decomposable rank-4 vector bundle B = F∗ OX ⊕ (F∗ OX )∗ and endow it with the OX -valued quadratic form Q(x + x∗ ) = q(x) + q ∗ (x∗ ) + x, x∗ , where x, x∗ are local sections of the rank-2 bundles F∗ OX and (F∗ OX )∗ respectively. The bracket x, x∗ denotes the standard pairing between F∗ OX and its dual (F∗ OX )∗ and q, q ∗ are projections onto OX of the direct sums (see Lemma 4.3) (4.4)

F ∗ (F∗ OX ) = OX ⊕ KX ,

−1 F ∗ (F∗ OX )∗ = OX ⊕ KX .

It is clear that Q is non-degenerate and that Q restricts to q and q ∗ on the direct summands F∗ OX and (F∗ OX )∗ respectively. Proposition 4.5. The orthogonal rank-4 bundle (B, Q) is stable. Proof. Let M be an isotropic line subbundle of F∗ OX ⊕ (F∗ OX )∗ and assume that deg M ≥ 0. By stability of (F∗ OX )∗ the line bundle M is contained in F∗ OX , which gives by adjunction a nonzero map F ∗ M → F ∗ F∗ OX → OX , contradicting isotropy of M . Next, let S be an isotropic rank-2 subbundle of F∗ OX ⊕ (F∗ OX )∗ and assume μ(S) ≥ 0. By the previous considerations S is stable, hence there is no non-zero map S → (F∗ OX )∗ . Therefore S is a rank-2 subsheaf of F∗ OX , but this contradicts isotropy of S. Remark. Clearly the two bundles Aˆ and B are non-isomorphic. 5. Counterexamples to Behrend’s conjecture We consider the rank-7 vector bundle E = F∗ OX ⊕ A ⊕ (F∗ OX )∗ equipped with the non-degenerate quadratic form Q defined by Q(x + y + x∗ ) = x, x∗ + det y, where x, x∗ and y are local sections of F∗ OX , (F∗ OX )∗ and A respectively. Note that the SO(7)-bundle ESO(7) = (E, Q) has a reduction to the Levi subgroup L = GL(2) × SO(3) ⊂ P ⊂ SO(7)


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and that its L-bundle EL given by the pair (F∗ OX , A) is stable, since F∗ OX and A are stable GL(2)− and SO(3)-bundles respectively. Since μ(F∗ OX ) > μ(A) = 0, we obtain that the canonical reduction of the unstable SO(7)-bundle (E, Q) is given by the P -bundle EP := EL ×L P . Using the exact sequence of L-modules (2.4) we obtain the equality Hom(F∗ OX , A) = H 0 (X, EP ×P so(7)/p). Note that H 0 (X, EP ×P D) ⊂ Hom(F∗ OX , (F∗ OX )∗ ) = 0 by stability of F∗ OX . But the former space is nonzero by Proposition 4.4 (ii). Hence this provides a counterexample to Behrend’s conjecture. The following unstable SO(8)-bundles also provide counterexamples to Behrend’s conjecture F∗ OX ⊕ Aˆ ⊕ (F∗ OX )∗

and

F∗ OX ⊕ B ⊕ (F∗ OX )∗ ,

with the quadratic form given by the standard hyperbolic form on the two summands F∗ OX ⊕(F∗ OX )∗ and the quadratic forms on Aˆ and B introduced in sections 4.1 and 4.2. Similarly we use the exact sequence (2.2) and Proposition 4.4 (ii) to show that the corresponding spaces H 0 (X, EP ×P so(8)/p) are non-zero. Remark. One can work out the relationship between the unstable SO(7)-bundle (E, Q) and the unstable G2 -bundle EG2 constructed in [H] section 5: first, we recall that G2 ⊂ GL(6). Consider the symplectic rank-6 bundle E obtained from (E, Q) under the purely inseparable group homomorphism SO(7) → Sp(6) ⊂ GL(6). Then the bundle E has a reduction to EG2 . 6. Non-rationality of the canonical reduction of an SO(7)-bundle This section is largely inspired by the last remark of [H]. We consider the unstable SO(7)-bundle (E, Q) introduced in section 5. We denote by K the field k(t), by XK the curve X ×k K and by EK the vector bundle over XK obtained as pull-back of E under the field extension K/k. We will consider the quadratic form

K on the bundle EK defined by Q

K (x + y + x∗ ) = x, x∗ + det y + tq(x), Q where x, x∗ and y are local sections of F∗ OX , (F∗ OX )∗ and A respectively and q is the quadratic form on F∗ OX defined in (4.4). Note that by Proposition 4.4 (iii) one has q(x) = det(φ(x)).

K is non-degenerate. Proposition 6.1. (i) The quadratic form Q

K ) over XK is not (ii) The canonical reduction of the SO(7)-bundle (EK , Q defined over K, but only over the inseparable quadratic extension K = K[s]/(s2 − t). Proof. We consider the vector bundle EK over XK obtained from EK under the field extension K /K and introduce the automorphism gs of EK defined by the matrix ⎛ ⎞ IdF∗ OX 0 0 ⎝ ⎠. sφ IdA 0 0 sψ Id(F∗ OX )∗

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CHRISTIAN PAULY

We choose the generators φ ∈ Hom(F∗ OX , A) and ψ ∈ Hom(A, B −1 ) such that β(φ(x), y) = x, ψ(y) for any local sections x, y of F∗ OX and A respectively — see Proposition (4.4) (iv). Note that gs is an involution. Then we have gs (x, y, x∗ ) = (x, y + sφ(x), x∗ + sψ(y)).

K the quadratic forms on EK obtained from Q and Q

K We also denote by Q and Q under the field extensions K /k and K /K respectively. Then we have the equality

K (x, y, x∗ ), (6.1) Q(gs (x, y, x∗ )) = Q(x, y, x∗ ) + s2 q(x) = Q

K on EK differ by the automorphism gs . In i.e., the quadratic forms Q and Q

K ) determines an SO(7)-bundle

K is non-degenerate and the pair (EK , Q particular Q over the curve XK .

K ) is given Because of (6.1) the canonical reduction of the SO(7)-bundle (EK , Q by the isotropic rank-2 vector bundle α : F∗ OX → EK with α(x) = (x, sφ(x), 0) for any local section x of F∗ OX . It is clear that the inclusion α is not defined over K. References K. A. Behrend, Semi-stability of reductive group schemes over curves, Math. Ann. 301 (1995), 281–301. [BH] I. Biswas, Y. Holla, Harder-Narasimhan reduction of a principal bundle, Nagoya Math. J. 174 (2004), 201–223. [Bo] A. Borel, Linear algebraic groups, Graduate Texts in Mathematics 106, Second Edition, Springer-Verlag (1991) [D] J. Dieudonné, Algebraic homogeneous spaces over fields of characteristic two, Proc. Amer. Math. Soc. 2 (1951), 295–304. [H] J. Heinloth, Bounds for Behrend’s conjecture on the canonical reduction, Int. Math. Res. Notices (2008), Vol. 2008, article IDrnn 045. [LP] H. Lange, C. Pauly, On Frobenius-destabilized rank-2 vector bundles over curves, Comm. Math. Helv. 83 (2008), 179–209. [LS] H. Lange, U. Stuhler, Vektorb¨ undel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe, Math. Zeit. 156 (1977), 73–83. [Ram] S. Ramanan, Orthogonal and spin bundles over hyperelliptic curves, Proc. Indian Acad. Sci. 90 (1981), no. 2, 151–166. [Ra] A. Ramanathan, Moduli for principal bundles, Algebraic Geometry Proceedings, Copenhagen 1978, Lect. Notes Math. 732, Springer. [Ray] M. Raynaud, Sections des fibr´ es vectoriels sur une courbe, Bull. Soc. Math. France 110 (1982), no.1, 103–125. [Be]

´matiques, Universit´ D´ epartement de Mathe e de Montpellier II - Case Courrier 051, Place Eug` ene Bataillon, 34095 Montpellier Cedex 5, France E-mail address: [email protected]


The Atiyah-Singer Index Theorem M. S. Raghunathan Abstract. In this paper we give an account of the Atiyah-Singer Index Theorem following the ideas in the original announcement in the Bulletin of the American Mathematical Society. There are however, significant differences in the way several steps in the proof are handled.

1. Introduction This is essentially an expository account of the Atiyah-Singer Index theorem, undoubtedly one of the great theorems of the twentieth century. The theorem establishes the equality of two numbers that are associated with an elliptic linear differential operator on a smooth compact manifold, one defined with the geometric data on the operator and the other by using analysis. For the precise statement of the theorem, we refer to the next section. We give here a proof of the theorem broadly following the lines of the original proof given by Atiyah and Singer in their announcement in [2] offering however some new ways for dealing with the various steps that lead up to the final proof. Atiyah and Singer did not publish the details of the proof indicated in their paper. The details were however worked out in the seminar conducted by Palais at the Institute for Advanced Study, the notes of which are published in the Annals of Mathematics Studies series [4]. The proof given here follows in broad outline the ideas in the Bulletin announcement; however it deviates in many ways in the matter of details indicated there (the Palais seminar follows closely the scheme set out in [2] even in the matter of details). Odd-dimensional manifolds however are dealt with exactly as in [4] both the analytic and topological indices are zero for all differential operators on odd-dimensional manifolds. The first important difference is in the proof of the all-important fact that the analytic index of a linear elliptic differential operator depends only on the K-theory element defined by its symbol. The proof given here makes no use of the theory of pseudo-differential operators - there is a trade-off though, in that we need to appeal to some somewhat more refined results from topology - among others, to the fact that the odd dimensional sphere is rational homotopy equivalent to the real projective space of the same dimension under the natural map. 2010 Mathematics Subject Classification. 58J20. Key words and phrases. Elliptic operators, Index. c Mathematical 0000 (copyright Society holder) c 2010 American

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M. S. RAGHUNATHAN M.S.RAGHUNATHAN

The second point of difference is that we make use of some qualitative information on the heat kernel to prove the “bordism invariance” of the analytic index (which is a crucial ingredient of the proof) rather than results about boundary value problems in the theory of partial differential equations as is done in [4]. Specifically we need the fact that the terms in the asymptotic expansion of the heat kernel of a second order elliptic operator are determined by the local behaviour of the operator. This is essentally a result of Minakshisundaram and Pleijel [3]. It may be remarked that the heat equation plays a crucial role in many of the different proofs of the index theorem; however the use of the heat kernel in those proofs has a very different flavour from our use of it here. Atiyah and Singer define an equivalence relation involving cobordism on the set consisting of elliptic operators (on all smooth manifolds and vector bundles on them); they give the set of equivalence classes a natural ring structure. Then they show that both the analytic and topological indices are constant on equivalence classes of elliptic operators (this is where the “bordism invariance” mentioned in the last paragraph is needed) and the resulting Q-valued functions are both homomorphisms of this ring into Q. They analyse the structure of the ring and exhibit a set of generators; the index theorem for these generators is a consequence on the one hand of the Gauss-Bonnet theorem and on the other hand Hirzebruch’s signature theorem. We replace these arguments by an induction argument on the dimension of the manifold. This is achieved by viewing the indices as functions on (even) cohomology with coefficients in Q rather than on K-theory tensored with Q and make use of a theorem due to Serre [5] (as well as an idea employed by Thom in a different context). The theorem in question asserts that, if X is a finite complex of dimension n and α is a q-cohomology class with coefficients in Q with n < 2q − 1, then there is a map f from X to S q such that α is in the image of H q (S q , Q) under the map induced by f . Finally it may be remarked that the index theorem itself is formulated in [2] for elliptic pseudo-differential operators (for which too the topological and analytic indices are defined), but once the theorem for differential operators is established the general case follows from the following considerations: in the even-dimensional case, the K-theoretic symbol of an elliptic pseudo-differential operator (which determines both the indices) is the K-theoretic symbol of a suitable elliptic differential operator; the odd-dimensional case can be reduced to the even-dimensional case by forming the product with the unit circle on which there is a pseudo-differential operator for which analytic and topological indices are both 1 while both these indices behave multiplicatively with respect to the formation of the (external) tensor products of the symbols and the K¨ unneth isomorphism in K-theory. This account confines itself to differential operators. As the remarks in the last paragraph indicate this yields the theorem in the more general case of pseudodifferential operators as well. My introduction to the index theorem took place in a seminar organised by Ramanan and M.S.Narasimhan soon after the Bulletin announcement of Atiyah and Singer appeared, in which I participated. When I was thinking about that seminar (during a spell of nostalgia a few years ago) this somewhat different way of handling the proof occured to me. This paper is the result and I am happy that it appears in this volume in honour of Ramanan.

THE ATIYAH-SINGER INDEX THEOREM

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I thank Peter Newstead for reading the manuscript carefully and correcting errors. 2. The statement of the theorem 2.1. The symbol and the analytic index. Throughout this paper (except in the Appendix) we will be working with smooth manifolds and bundles. Unless otherwise specified explicitly all maps considered will be smooth. In particular sections of vector bundles will be smooth. Let M be a smooth compact closed manifold of dimension m. Let E and F be complex vector bundles on M . Recall that a linear differential operator from E to F is a C-linear map D : Γ(E) → Γ(F ) such that, for any section Φ of E, the support of D(Φ) is contained in the support of Φ. All differential operators considered in this paper will be linear; so we will drop the adjective “linear” in the sequel. If we denote by Jk (E) the bundle of k-jets of E, then, for all suitably large k, D defines and is defined by a bundle homomorphism from Jk (E) to F . The minimal k for which this homomorphism is defined is the order of D. Let T (resp. T ∗ ) be the tangent (resp. cotangent) bundle of M and S k (T ∗ ) the k-th symmetric power of T ∗ ; then one has an inclusion of S k (T ∗ ) ⊗ E in Jk (E). If now D is a differential operator of order k then the homomorphism it defines from Jk (E) to F gives by restriction a homomorphism from S k (T ∗ )⊗E to F . The diagonal inclusion of T ∗ in S k (T ∗ ) enables one to view this last homomorphism as a bundle homomorphism of the pull-back p∗ (E) of E to the pull-back p∗ (F ) of F under the natural projection p : T ∗ → M . This element of Hom(p∗ (E), p∗ (F )) which on each fibre of T ∗ is a homogeneous polynomial of degree k is the symbol, σ(D), of D. It is a basic fact from the theory of linear differential operators that the kernel (resp. cokernel) of an operator D whose symbol is an injective (resp. surjective) homomorphism outside the zero section of T ∗ is finite-dimensional. A differential operator D (from E to F ) is elliptic if and only if the symbol of D is an isomorphism outside the zero section of T ∗ . This means of course that E and F have the same rank. The analytic index a(D) of D is defined as the integer dim(kernel(D)) − dim(cokernel(D)). 2.2. The K-theoretic symbol. We fix a Riemannian metric on M and denote by B (resp. S) the unit disc (resp. sphere) subbundle of T ∗ . We denote by p the projection of T ∗ on M as well as its restriction to B. If now D is an elliptic operator from E to F its symbol σ(D) defines an isomorphism of the restrictions to S of the bundles p∗ (E) and p∗ (F ) (here for a bundle V on M , p∗ (V ) is the pull-back of V to B under p). The “difference construction” in K-theory (see [4], p.15) now yields an element σ0 (D) in the relative K-group K 0 (B, S) which we will refer to as the K-theoretic symbol of D in the sequel. The following properties of the K-theoretic symbol follow from its definition (via the difference construction). Lemma 2.1. If D : Γ(E) → Γ(F ) and D : Γ(F ) → Γ(G) are elliptic differential operators from E to F and from F to G respectively, then D D is an elliptic operator from E to G and and σ0 (D D) = σ0 (D) + σ0 (D ). Also if the operator D (resp. D ) is an elliptic operator from E (resp. E ) to F (resp. F ) and D ⊕ D is the operator from E ⊕ E to F ⊕ F defined by setting (D ⊕ D )(σ ⊕ σ ) = D(σ) ⊕ D(σ )

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for sections σ, σ respectively of E and E , then D ⊕ D is elliptic and σ0 (D ⊕ D ) = σ0 (D) + σ0 (D ). 2.3. The topological index. Now, we have a natural ring homomorphism, the Chern character (see [4], p.14) Ch, from K 0 (B, S) to the sum H even (B, S; Q) of the even-dimensional singular cohomology groups of the pair (B, S) with coefficients in Q (which in fact gives an isomorphism from K 0 (B, S)⊗Q to H even (B, S; Q)).We get thus a cohomology class Ch(σ0 (D)) in the latter group. Let T d(M ) denote the Todd class of M as well as its pull-back to H ∗ (B; Q). The cup-product gives a bilinear pairing H ∗ (B, S; Q) × H ∗ (B; Q) → H ∗ (B, S; Q) and we thus obtain a cohomology class Ch(σ0 (D)).T d(M ) in H ∗ (B, S; Q). Now the pair (B, S) has a canonical orientation and hence H2n (B, S; Z), which is isomorphic to Z, has a canonical generator. The topological index t(D) of the elliptic operator D is defined as the (rational) number obtained by evaluating Ch(σ0 (D).T d(M ) on this generator. With these definitions and notation we can state the Atiyah-Singer index theorem. Theorem 2.2. Let D be an elliptic operator on a compact manifold from a vector bundle E to a vector bundle F . Then a(D) = t(D). 2.4. Remarks. Note that the theorem implies that the analytic index of D depends only on the K-theoretic symbol of D (as this is true for t(D) by its very definition). This fact is proved as the first step in the proof of the theorem. Also note that the topological index, which we know only to be a rational number, turns out to be an integer as a consequence of the theorem. 3. Construction of some differential operators 3.1. For a vector bundle W on M and a point x in M , Wx will denote the fibre of W at X. Let E and F be vector bundles on M and D a differential operator from E to F . We introduce a Riemannian metric on M and denote by μ the Borel measure defined by it on M . We also intoduce hermitian inner products along the fibres of E and F . In the sequel these inner products on E and F as well as the inner products on T and T ∗ defined by the Riemannian metric will be denoted . With this notation, we have the notion of the adjoint D∗ of the operator: this is the unique differential operator from F to E which satisfies the following condition: for sections α of E and β of F , < D(α), β > dμ = < α, D∗ (β > dμ. M

M

That such a D∗ exists and has the same order as D is a standard fact and is proved easily by integration by parts on local charts over which the bundles E and F are trivial. Also the adjoint D ∗∗ of D∗ is D. When D is elliptic, so is D∗ and one has natural isomorphisms of kernel D (resp. cokernel D) on cokernel D∗ (resp. Kernel D∗ ). It follows that a(D) = −a(D∗ ). In particular, if F is the same as E and D is self-adjoint, i.e. D = D∗ , then a(D) = 0. The symbol of the operator Δ = D∗ D from E to itself assigns to each v in Tx∗ , x ∈ M , the automorphism σ ∗ (v)σ(v) of Ex , where for an endomorphism U of Ex , U ∗ denotes its conjugate transpose with respect to the inner product on Ex . It follows that σ(D∗ D) is at every point of S a hermitian-symmetric positive definite automorphism and we can therefore raise it to the power t for every t in the closed interval [0,1]. This yields a homotopy


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between the symbol of D∗ D and the identity automorphism of p∗ (E) over S. As this last automorphism (evidently) extends over all of B, we conclude (from the definition of the difference construction) that the K-theoretic symbol of D∗ D is zero. We have the following proposition. Proposition 3.1. Given any vector bundle E on a manifold M there is a selfadjoint elliptic operator ΔE from E to E of order 2 such that σ0 (ΔE ) is zero (and a(ΔE ) = 0). 3.2. In the light of the disscussion in the last paragraph, to prove the proposition we need only exhibit an operator D of order 1 from E to a suitable bundle F such that the symbol σ(D) is an injective bundle homomorphism outside the zero section of T ∗ – one can then take for Δ the operator D∗ D for some hermitian inner products along the fibres of E and F and a Riemannian metric on M . To construct a D of order 1, we take F to be Hom(T, E) (= T ∗ ⊗ E), the bundle of E-valued 1-forms on M . We fix a hermitian inner product on E and a Riemannian metric on E; these give rise to a natural inner product on F as well. We then take D to be the exterior differentiation with respect to a unitary connection ω on E. One sees then that for this operator Δ, the symbol associates to each v in Tx∗ , x ∈ M , the automorphism || v ||2 .Identity. 3.3. For a vector bundle V and a non-negative integer r, we denote by r.V the direct sum of r copies of V . A differential operator D from E to another vector bundle F evidently defines a differential operator r.D from r.E to r.F which on each component of r.E is the operator D from E to F , the latter considered as the corresponding component of F . It follows from the definition of the K-theoretic symbol that σ0 (r.D) = r.σ0 (D). With this notation we will now establish the following. Proposition 3.2. For a positive integer n, let λ(n) = 22n+2 . Then, given any vector bundle E on a manifold M of dimension m, there is an elliptic operator DE of order 1 from λ(m).E to itself such that a(DE ) = 0 and σ0 (DE ) = 0. 3.4. Let I : M → R2m+1 be an imbedding and I the imbedding in R2m+2 obtained by composing the standard inclusion of R2m+1 in R2m+2 with I . We have then an inclusion of T in the trivial real vector bundle on M of rank 2m + 2 and hence (using the standard inner product on R2m+2 ) an inclusion of T ∗ in the trivial real vector bundle of rank 2m + 2. Now let V denote the trivial complex vector bundle of rank 2m + 2. As I factors through the inclusion of R2m+1 in R2m+2 , we note that T ∗ is contained in a trivial (complex) vector sub-bundle V such that V is the direct sum of V and a trivial line bundle equipped with an everywhere non-zero section s. Now let x ∈ M and v ∈ Tx∗ . Define a homomorphism e(v) : Tx∗ ⊗ Λp (V )x → Λp+1 (V )x (resp. i(v) : Tx∗ ⊗ Λp (V )x → Λp−1 (V )x ) (where for a non-negative integer l, Λl (V ) is the lth exterior power of V ) as the exterior (resp. interior) multiplication by v (note that Tx∗ is a subspace of Vx and V being the trivial bundle carries a natural inner product). Let Λe (V ) (resp. Λo (V )) be the direct sum of the even (resp. odd) exterior powers of V . Then e(v) + i(v) as v varies in T ∗ defines homomorphisms T ∗ ⊗ V e → V o and T ∗ ⊗ V o → V e which we denote σ e and σ o respectively in the sequel. If now E is any vector bundle on M , σ e ⊗ (Identity) is a homomorphism from p∗ (V e ⊗ E) to p∗ (V o ⊗ E) which is an isomorphism outside the zero-section: this is because σ o ⊗(Identity)·σ e ⊗(Identity)

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is the endomorphism || v ||2 .(Identity) of p∗ (V e ⊗ E) (here v ∈ Tx∗ , x ∈ M ). Now if we introduce a hermitian inner product on E and a connection on E compatible with it, one sees that σ e ⊗ (Identity) and σ o ⊗ (Identity) can be lifted to (elliptic) ∗ differential operators DE and DE which are adjoints of each other. It follows that ∗ of DE and D gives an elliptic self-adjoint operator of order 1 the direct sum D E from Λ(V ) ⊗ E (Λ(V ) is the exterior algebra bundle of V ) to itself. Moreover t(e(v) + i(v)) + (1 − t)(e(s(x)) + i(s(x))) (v ∈ Tx∗ , x ∈ M ) with an isomorphism which extends gives a homotopy over S of the symbol of D ∗ over all of T . This proves the proposition: since V is a trivial bundle of rank 2m + 2, Λ(V ) is trivial of rank λ(m) and Λ(V ) ⊗ E is isomorphic to a direct sum of λ(m) copies of E. 4. K-theoretic symbol and the analytic index Our aim in this section is to establish the following: Theorem 4.1. Let D and D be elliptic operators from vector bundles E and E to F and F respectively. If the K-theoretic symbols of D and D are equal, then a(D) = a(D ).

4.1. We will first prove the following much weaker statement: Proposition 4.2. Suppose given two elliptic differential operators D and D , both from E to F of the same order k, such that there is a smooth homotopy σt , t ∈ [0, 1], of isomorphisms from p∗ (E) to p∗ (F ) such that each σt is a homogeneous polynomial of degree k along the fibres of T ∗ and σ0 = σ(D), σ1 = σ(D ). Then a(D) = a(D ). In particular if D and D have the same symbol, a(D) = a(D ). 4.2. For any vector bundle W on M with a hermitian inner product and a non-negative integer k, one defines a pre-Hilbert structure on Γ(W ) with the inner productk defined as follows. Fix once and for all a finite covering {Ui | i ∈ I} of M by cooordinate open sets and an isomorphism of the restriction of W to the Ui with a trivial bundle. Fix a shrinking {Vi | i ∈ I} of {Ui | i ∈ I}. Then each section of W defines a collection F = {Fi | i ∈ I}, each Fi being a Rn -valued smooth function on Vi . For a pair F, F of sections of W , let < ∂ α Fi /∂xα , ∂ α /∂xα > . < F, F >k = i∈I, |α|≤k

The completion of this pre-Hilbert structure on Γ(W ) is denoted Hk (W ). The topological vector space structure on Hk (W ) is independent of the choice of the covering, its shrinking and the trivialisations of W over the open sets of the covering. We now fix vector bundles E and F on M . With the notation introduced above, one then sees that a differential operator D : Γ(E) → Γ(F ) of order k extends to a continuous linear map from Hk (E) to Hk (F ). When D is elliptic, there is a constant C = C(D) > 0 such that, for all σ in the orthogonal complement of kernel(D), one has || D(σ) || ≥ C(D). || σ ||. Suppose now that Dt , t ∈ [0, 1], is a smooth family of elliptic operators. Let t0 ∈ [0, 1] and let H be the orthogonal complement of kernel(Dt0 ) in Hk (E). Then there is a neighbourhood U of t0 in [0,1] and a constant C > 0 such that C(Dt ) ≥ C for all t in U . From this it follows


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easily that a(Dt ) is independent of t. Now if D and D have the same symbol, t.D + (1 − t)D , t ∈ [0, 1], provides a smooth family of elliptic operators leading to the conclusion that a(D) = a(D ). Now since the inclusion of the pull-back of S k (T ∗ ) ⊗ E to M × [0, 1] in the pull-back of J k (E) admits a right inverse vector bundle morphism, we see that, given σt as in the statement of the proposition, we can find a smooth family of (elliptic) operators Dt , t ∈ [0, 1], such that σ(Dt ) = σt . It follows now that a(D) = a(D ). 4.3. Suppose now that D is an elliptic differential operator from a vector bundle E to a vector bundle F . It is obvious that t(r.D) = r.t(D) and a(r.D) = r.a(D) for any non-negative integer r. Thus equality of a(D) and t(D) holds if it holds for a(r.D) and t(r.D) for some integer r greater than zero. Observe next that, if D is an elliptic operator from F to a vector bundle G of the same rank, then a(D D) = a(D ) + a(D) and t(D D) = t(D ) + T (D). Now suppose that D (resp. D )is an elliptic operator from E to F (resp. from E to F ) of order k (resp. k ) are such that their K-theoretic symbols are the same. We want to show that they have the same analytic index. Observe first that we may replace the operators D and D by r.D and r.D for any positive integer r. By choosing r = λ(n) as in Proposition 4 and composing with a suitable power of DF (resp. DF ), we may assume that k = k = 2l is an even integer. Let W (resp. W ) be a vector bundle such that the direct sum of E (resp. E ) and W (resp. W ) is trivial of rank q. Let ΔE and ΔE be as in Proposition 3 and let L and L denote their respective lth powers. Let D (resp. D ) be the direct sum of D (resp. D ) and L (resp. L ). Clearly D and D have the same K-theoretic symbol and we need to prove that their analytic indices are the same. 4.4. This means that we can assume that we are in the following situation. Let E denote the trivial bundle of rank q and D (resp. D ) be an elliptic operator from E to a rank-q vector bundle F (resp. F ) of even order 2l such that σ0 (D) = σ0 (D ). We have to show that a(D) = a(D ). Since E is trivial, so is F (resp F ) over S. This means that the bundles F and F are stably isomorphic: this follows from the fact that the K-theoretic symbols of D and D obtained by the diifference construction are the same. Using Proposition 4 for the trivial bundle, one finds that by forming the direct sum with a suitable number of copies of the trivial line bundle 1 equipped with a suitable power of Δ1 , we may assume that both D and D are elliptic operators from the (trivial) bundle E to the same bundle F . 4.5. That the two operators have the same K-theoretic symbol implies the following: there exists a positive integer N and an automorphism u : E ⊕1N → E ⊕ 1N such that Σ1 = (σ(D) ⊕ IdN ).u and Σ2 = σ(D ) ⊕ IdN are homotopic through a 1-paramaeter family Σt , t ∈ [0, 1] of sections of Hom(p∗ (E)⊕1N , p∗ (F )⊕1N ) over S, each Σt being an isomorphism. Σ1 and Σ2 are both symbols of differential operators D1 and D1 respectively of order k = 2l (see Proposition 3 and the discussion in 3.2) with the same K-theoretic symbols as D and D respectively; further a(D1 ) = a(D1 ) as u is a 0th order operator and therefore has index 0. Now let P denote the real projective space bundle associated to T ∗ : this is a quotient of S by the involutive automorphism which maps each unit vector in S into its negative. Let q denote the natural projection of P on M . Then since the order k of D and D is even, σ(D) and σ(D ) are pull-backs of sections σ and σ of Hom(q ∗ (E), q ∗ (F )). It is shown in the Appendix that the existence of the homotopy σt implies that, replacing D1 and D1

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by r.D1 and r.D1 , we can assume that Σ1 and Σ2 are homotopic through sections of Hom(q ∗ (E), q ∗ (F )) which are all isomorphisms of q ∗ (E) on q ∗ (F ). Equivalently, we can assume that the homotopy Σt above is such that, for v ∈ Tx∗ , x ∈ M , Σt (v) = Σt (−v). 4.6. Fix some extension of the homotopy Σt to a homotopy of sections over t . We assume that all of T ∗ of Hom(p∗ (E), p∗ (F )) and denote this extension by Σ ∗ t the Σt vanish outside a compact neighbourhood of S in T . Now convolving Σ 1/2 2 ∗ ∗ with the function (c/π) exp(−c || (.) || ) on T along the fibres of T we obtain a (c,t) such that, on every fibre of T ∗ , it admits a Taylor expansion which homotopy Σ (c,t) converges uniformly converges uniformly on compact sets. As c tends to ∞, Σ t . It follows that, for all large c, Σ (c,t) is invertible over S as on compact sets to Σ also is a suitable truncation θ(c,t) of its Taylor expansion (along the fibres). Let τ(c,t) = (θ(c,t) + θ(c,−t) )/2. Since Σt is an even function on the fibres of T ∗ , we can choose c and the truncation so that τ(c,t) is invertible over all of S and τ(c,0) (resp. (c,0) (resp.Σ ˜ (c,1) ). We replace Σ t now by such a τ(c,t) and denote the τ(c,1) ) equals Σ latter by Σt in the sequel. 4.7. It is clear that Σt is now a polynomial along the fibres of T ∗ with every homogeneous component of even degree. It can therefore be made homogeneous with the function || . ||2 on T ∗ (which is homogeneous of degree 2 along the fibres of T ∗ and is identically 1 on S). In other words we can assume that the homotopy σt is a homogeneous polynomial function along the fibres of T ∗ of fixed degree. This may mean however that the degrees of Σ0 and Σ1 have been raised. However from Proposition 4 we know that the composites of ΔF ⊕1N raised to any power with D1 and D1 have the same analytic index as well as the same K-theoretic symbol as D and D respectively. We are now in a situation where the homotopy between the symbols of D and D is through isomorphisms of p∗ (E) on p∗ (F ) all of which are homogeneous polynomials of the same even degree. Theorem 5 now follows from Proposition 6. 5. The signature operators In this section we will describe the so-called twisted signature operators. 5.1. Let M be a manifold of dimension m. We fix a Riemannian metric on M which gives rise to a Borel measure μ on M . Let ΩM denote the line bundle of exterior m-forms on M . When there is no ambiguity about the underlying manifold M , we denote ΩM simply Ω. The metric gives a canonical reduction of the structure group R∗ of the line bundle Ω to the subgroup {1, −1} so that a 2-sheeted covering of M imbeds in Ω, the two sheeted covering being trivial or nontrivial according as M is orientable or not; in the former case an orientation gives a natural trivialisation of the covering. This reduction of structure group leads also to a canonical isomorphism of Ω ⊗ Ω with the trivial bundle M × R. If the manifold is oriented, the metric gives a natural trivialisation of Ω. For an integer p with 0 ≤ p ≤ m, let Ωp denote the p-th exterior power of T ∗ ; in particular Ωm = Ω. The exterior multiplication then defines a pairing Ωp ⊗ Ωm−p → Ω which is non-degenerate and hence gives a non-degenerate pairing of Ωp with Ωm−p ⊗ Ω. Using the isomorphism of Ωm−p with its dual given by the Riemannian metric, we obtain a bundle isomorphism from Ωp to Ωm−p ⊗Ω which is denoted ∗ in the sequel.


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The natural isomorphism from Ω ⊗ Ω to the trivial bundle yields (by taking tensor product with the identity morphism of Ω) an isomorphism from Ωp ⊗ Ω to Ωm−p which will also be denoted ∗. One then finds that ∗2 on Ωp equals (−1)p(m−p) . Suppose now that α and β are two p-forms on M ; then α ∧ ∗β is a section of Ω ⊗ Ω and – as this last line bundle is naturally isomorphic to the trivial bundle – is a function on M . On the other hand the Riemannian metric gives an inner product denoted on the Ωp . One checks easily that < α, β >= α ∧ ∗β. For α, β as above we set (α, β) = < α, β > dμ M

5.2. Suppose now that E is a complex vector bundle with a hermitian inner product along its fibres. Let ω be a unitary connection on E. We then have the differential operator of order 1, namely exterior differentiation dω with respect to ω of E-valued p-forms on M – these are nothing but sections of the bundle Ωp ⊗ E: dω : Γ(Ωp ⊗ E) → Γ(Ωp+1 ⊗ E). The hermitian inner product on E gives a conjugate-linear isomorphism from E to the dual E ∗ . Forming the tensor product of this isomorphism with ∗, we obtain a conjugate-linear isomorphism from Ωp ⊗ E to Ωm−p ⊗ Ω ⊗ E ∗ which we denote also ∗: ∗ : Ωp ⊗ E → Ωm−p ⊗ Ω ⊗ E ∗ . Now, if α (resp. β) is a p-form (resp. (m − p)-form) on M with values in E (resp. E ∗ ), then using the pairing between E and E ∗ , we get a m-form α∧β with values in Ω ⊗ Ω. One checks easily that for p-forms α, β with values in E, < α, β > (the inner product on Ωp ⊗ E is the one deduced from that given by the Riemannian metric on M and the hermitian inner product on E) equals α ∧ ∗β (treated as a function via the canonical trivialisation of Ω ⊗ Ω). It follows moreover from Stokes’ theorem that, for a p-form α and a p + 1-form β, we have, setting δω = (−1)p+1 ∗ dω ∗, < dω α, β > dμ = < α, δω β > dμ. M

M

Thus we see that δω is the adjoint of dω . Let Ωe (resp. Ωo ) be the direct sum of the even (resp. odd) exterior powers of T ∗ . Then dω + δω is an elliptic differential operator (of order 1) from Ωe ⊗ E to Ωo ⊗ Ω ⊗ E. In the sequel we denote this operator DE,deRh and refer to it as the de Rham-Hodge operator for E (or the E-twisted de Rham-Hodge operator). When E is the trivial line bundle, we drop the E in the suffix and refer to it simply as the de Rham-Hodge operator. 5.3. When the dimension m of M is even, equal to 2n, say, there is another way of decomposing the bundle ΛT ∗ of (all) exterior differential forms into a direct sum of two vector bundles Ω+ and Ω− : if we define ∗ : Ωp → Ωm−p ⊗ Ω by setting ∗ (ξ) = ip(p−1) ∗ (ξ), one checks easily that ( ∗)2 = (−1)n (here i is a square root of -1 fixed once and for all). It follows that when n is even (resp. odd), that ΛT ∗ decomposes as a direct sum of two subbundles Ω+ and Ω− where Ω+ (resp. Ω− )is the eigen-subbundle of ΛT ∗ corresponding to the eigenvalue θ (resp. −θ) for ∗, θ being 1 or i according as n is even or odd. The operator dω + δω is then seen to define an elliptic operator from Ω+ ⊗ E to Ω− ⊗ E (which will be denoted DE,ω in the sequel); we call this the E-twisted signature operator on M . We have then the following crucial fact from K-theory (for a proof see [4], pp. 225–26).

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Theorem 5.1. Let T h : K 0 (M ) → K 0 (B, S) be the homomorphism defined by setting T h(E) = σ0 (DE,ω ) for a vector bundle E on M . Then T h gives an isomorphism of K 0 (M ) ⊗ Q on K 0 (B, S) ⊗ Q. Note that for a vector bundle E, T h(E) depends only on the K-theoretic class of E and is independent of the choice of the hermitian metric and the connection on E and defines a homomorphism from K 0 (M ) to K 0 (B, S). Theorems 5 and 7 together show that, to prove Theorem 2, it suffices to prove that a(DE,ω ) = t(DE,ω ). This is what will be done in the rest of this paper. 6. “Bordism Invariance” of the index We adopt the following notation in the sequel. For a vector bundle E on a manifold M , a(E) (resp.t(E)) will be the analytic (resp. topological) index of the E-twisted signature operator DE,ω . This notation is justified as the indices in question are independent of the connection ω as the K-theoretic symbol of DE,ω itself is independent of ω. We then have: Theorem 6.1. Suppose now that M and M are two smooth manifolds and E and E are vector bundles on them. Suppose further that there is a manifold with boundary W such that the boundary of W is the disjoint union of M and M and there is a vector bundle E on W such that E restricts to E (resp. E ) on M (resp. M ). Then a(E) = a(E ) and t(E) = t(E ). 6.1. For the result for the topological index which is easy to prove, see [4], p.228. For the analytic index we will make use of some results about the heat kernel essentially due to Minakshisundaram and Pleijel [3]. We recall these now. The operator DE,ω has for its adjoint the operator DE⊗Ω, ω with ω denoting the connection on E ⊗ Ω obtained from ω and the flat connection on Ω. We set E = E ⊗ ω in the sequel.Then D E, ω DE,ω (resp. DE,ω D E, ω ) is an elliptic second order operator of Ω+ ⊗E (resp.Ω− ⊗ E) to itself which we denote simply Δ+ (resp. Δ− ) in the rest of this section. For t ≥ 0, the operator exp(−tΔ+ ) (resp. exp(−tΔ− )) has a smooth kernel K + (t, x, y) (resp.K − (t, x, y)), a smooth section of Hom(p∗1 (Ω+ ⊗ E), p∗2 (Ω+ ⊗E)) (resp. Hom(p∗1 (Ω− ⊗ E), p∗2 (Ω− ⊗ E)) which depends smoothly on − the parameter Further K + (t, x, x) (resp. x, x)) has an asymptotic K (t, r−m/2 t as well. − r−m/2 + Kr−m/2 (x, x) (resp. t Kr−m/2 (x, x)) as t expnasion 0≤r≤∞ t 0≤r≤∞ + goes to zero. The analytic index of DE,ω is then equal to + T race(K0 (x, x))dμ(x) − T race(K0− (x, x))dμ(x) M

M

The crucial fact we need about the asymptotic expansion is the following. If E and E are two vector bundles with hermitian inner products and and unitary connections ω and ω respectively such that, on an open set U of M , we have a hermitian isomorphism from the restriction of E to the restriction of E which carries ω into ω , then, on U × U , all the terms in the asymptotic expansions of the two heat kernels coincide. In particular, the function T race(K0+ (x, x)) − T race(K0− (x, x)) on M (which we will denote TE in the sequel) is determined in the neighbourhood of any point by the local data on the Riemannian metric, the hermitian inner product on E and the unitary connection in that neighbourhood. TE is naturally a section of Ω ⊗ Ω and so is to be regarded as a closed m-form on M with values in Ω.


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6.2. Since any cobordism between manifolds is a composition of successive elementary cobordisms, we may assume for proving Theorem 8 that W is an elementary cobordism. In other words, there is smooth function f : W → [−1, 1] such that M = f −1 (−1), M = f −1 (1) and f has exactly one non-degenerate critical point w (of index r, say) with f (w) = 0. One then has an open neighbourhood U of w in W and a diffeomorphism F from an open disc D(4 ), > 0, of radius 4 around the origin 0 in Rm , to U with F (0) = w and such that f composed with F is the function Q defined on D(4 ) as follows: for a suitable decomposition of Rm as an orthogonal direct sum of Rr and Rs (r + s = m), denoting by x1 , x2 , ....xr (resp. y1 , y2 , .....ys ) the coordinates in Rr (resp. Rs ), one has Q(x1 , x2 ....xr , y1 , y2 , ....ys ) = yi2 − x2j . 1≤i≤s

1≤j≤r

For any c > 0, let D(c) denote the disc of radius c around the origin in Rm . Introduce a Riemannian metric g on W such that it induces the standard metric on D(3 ) under the map F . Let X be the vector field on W = W \ {w} such that < X, Y >= 0 for all vector fields Y on W with Y f = 0 and < X, X > = 1. The local 1-parameter group φt of local diffeomorphisms of W determined by X then defines for 0 < c < 3 a smooth map Φ : (M \ F (D(c))) × [0, 1] → W given by Φ(x, t) = φ2t (x) which is a diffeomorphism onto a closed subset W (c) in W ; the interior of the manifold with boundary (M \ F (D(3 ))) × [0, 1] maps onto the interior of W (c) as a subset of W . This last open set is also diffeomorphic to (M \ F (D(c))) × [0, 1] under the map that takes (x , t) to φ−2t (x ). We now replace the metric g by a metric g which coincides with g on F (D( )) and induces on (M \ F (D(2 ))) × [0, 1] a product Riemannian metric under the map Φ – note that F (D( )) and W (2 ) have disjoint closures. The vector bundle E on W when pulled back to (M \ F (D( ))) × [0, 1] is necessarily isomorphic to the pull-back of a bundle on (M \ F (D( ))) under the cartesian projection on (M \ F (D( ))). We assume the hermitian inner product to be the pull-back of one on the bundle on (M \ F (D( ))). We note that the complement of W (3 ) in W is contained in F (D(4 )) and hence E is trivial over this complement. Observe that, if ν denotes the unit inward normal field along the boundary, the interior product with ν yields an isomorphism from the bundle ΩW of m+1-forms on W restricted to M (resp.M ) to ΩM (resp. ΩM ), the bundle of m-forms on M (resp. M ), which in addition is compatible with the flat connections on these bundles. Now consder the closed ΩM -valued m-form TE on M and let α denote the pull-back of its restriction to M \ F (D(3 )) to W (3 ) (identified with (M \ F (D(3 ))) × [0, 1] via Φ). Then α restricted to Φ((M \ F (D(3 ))) × 1 = M \ F (D(3 )) is the same as the restriction of TE (on M ). Let α be a m-form on W which equals α on W (3 ), TE on M and TE on M . Applying Stokes’ theorem to the ΩW -valued m form α, we get α− α= dα. M

M

W

Now since the form α is closed, so is its pull back to W (3 ).) It follows that the integral on the right hand side equals the integral over F (D(4 )). Now there is an elementary cobordism V between the sphere S m and S r × S s admitting a Morse function g : V → [−1, 1] with exactly one non-degenerate critical point of index r admitting a neighbourhood with properties entirely analogous to the neighbourhood

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U of w in W . Let F (resp. F , F ) be the trivial bundle of rank equal to that of E. Then we see that the same argument as above shows that the integral over F (D(4 )) is equal to TF − TF S r ×S s

Sm

and this is zero as the analytic index of the signature operator for the trivial bundle on the product of two spheres is zero as can be checked easily (using the Hodge-De Rham theorem). This proves Theorem 8. 7. Proof of the Atiyah - Singer theorem In this section we complete the proof of the theorem. We first dispose of the case of odd-dimensional M in the same way as is done in [4]: we show that both the analytic and topological indices vanish. 7.1. The odd-dimensional case. When M is odd-dimensional the asymptotic expansion of the heat kernel is in odd powers of t1/2 so that for any vector bundle E on M (with a unitary connection ω), TE is zero. Hence a(DE ) = 0. On the other hand t(DE ) is zero as the K-theoretic symbol of any elliptic differential operator itself is zero. This is seen as follows. As we have seen, we may assume that the operator has even degree. But this means that the symbol and hence also the K-theoretic symbol is invariant under the bundle automorphism −(Identity) while this automorphism induces the map −(Identity) on K 0 (B, S) as the antipodal map on even-dimensional spheres is orientation reversing. 7.2. The case of spheres. When the manifold M is an even-dimensional sphere, the group K 0 (B, S) ⊗ Q is isomorphic to Q2 and is generated as a vector space over Q by D1 and the de Rham-Hodge operator. Here 1 denotes the trivial line bundle (see [4]). For the first of these two operators, it is easily checked (as was observed earlier) that both indices vanish; for the second the equality of the two operators follows from the Gauss-Bonnet theorem (see [4]). Thus the theorem holds for all spheres. 7.3. Products. If M and M are two manifolds for which the analytic and topological indices are equal the same holds for M × M . This is easily seen as follows. (i) K 0 (M × M ) ⊗ Q is isomorphic to (K 0 (M ) ⊗ Q) ⊗ (K 0 (M ) ⊗ Q) under the natural map induced by the formation of “external” tensor products and this is compatible with the K¨ unneth isomorphism in cohomology and the Chern character; note also that the isomorphism T h for M × M is the tensor product of the T h for the two factors. This implies that the topological index is multiplicative: if E (resp. E ) is a vector bundle on M (resp. M ) and p (resp. p ) is the Cartesian projection of M × M on M (resp. M ), then t(p∗ (E) ⊗ p∗ (E )) = t(E).t(E ). (ii) That a(p∗ (E) ⊗ p∗ (E )) = a(E).a(E ) follows from the fact that the twisted signature operator on (p∗ (E) ⊗ p∗ (E )) is obtained from the Eand E -twisted operators on M and M respectively by taking the tensor product of these last operators.


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Since we have proved the theorem for spheres and on the even-dimensional sphere there is a bundle E with a(E) non-zero – the one which corresponds to the de Rham-Hodge operator - it follows that the theorem holds for M if (and only if) it holds for M × S 2q . 7.4. Spherical cohomology classes. We need the following result due to Serre (see [5]): Theorem 7.1. Let X be a finite C-W complex of dimension n and α an element of H q (X, Q). Assume that n < 2q − 1. Then there is a continuous map f : X → S q and an element α0 in H q (S q , Q) such that f ∗ (α0 ) = α. 7.5. Conclusion of the proof. Since the Chern character is an isomorphism from K 0 (M ) ⊗Q to H even (M ; Q), we may view the analytic and topological indices as homomorphisms from H even (M ; Q) to Q. Suppose now that the theorem holds for all manifolds of (even) dimension less than m, the dimension of M . Let α be an element of H q (M ; Q) with q an even integer greater than or equal to zero. We need to prove that a(α) = t(α). If q = 0, this is the Hirzebruch signature theorem (as is shown by an explicit computation of the topological index – the analytic index, it is easy to show, is the signature of the manifold). Assume then that q ≥ 2. Let l be an even integer such that m + q < 2(q + l). Let β be a generator of H l (S l ; Q) and let γ be the class in H q+l (M × S l ; Q) which corresponds to α ⊗ β under the K¨ unneth isomorphism. It suffices to show that a(γ) = t(γ) since the theorem holds for spheres, a(β) = t(β) is non-zero and one has a(γ) = a(α).a(β) and t(γ) = t(α).t(β). Now, in view of Serre’s theorem, there is a smooth map f : M × S l → S q+l such that γ is the pull-back of a (q + l)-cohomology class η of S q+l under f . Now M × S l is the boundary of W = M × Dl+1 and we can extend f to a smooth map F of W into the disc Dq+l+1 . Let Z be an interior point of the q + l + 1 disc which is not a critical value of F . Let U be an open disc containing Z such that the closure of U is disjoint from the set of critical values. Then the inverse image of Dq+l+1 \ U under F gives a cobordism W between M × S l and the inverse image N of the boundary of U in Dq+l+1 . This last manifold is diffeomorphic to the product of a manifold M of dimension m + l + 1 − (q + l + 1) = m − q < m and a sphere of dimension q + l. If γ (resp. γ ) denotes the inverse image of η in the cohomology of W (resp. N ), by the bordism invariance theorem a(γ) = a(γ ) and t(γ) = t(γ ). On the other hand, since dim M < dim M , the theorem holds for M and hence also for its product N with S l . Hence a(γ) = t(γ). The theorem follows. 8. Appendix Let X be a finite CW-complex of dimension N and W a real vector bundle on X of rank m equipped with an inner product. Let B (resp. S) be the unit disc (resp. sphere) bundle in W . Let denote the antipodal map along the fibres of S. Let P denote the real projective space bundle associated to W : P is the same as the quotient of S obtained by identifying each point s of S with (s). Let p (resp. q) be the projection of S (resp. P ) on M and u the natural map from S to P so that q · u = p. Let E and F be complex vector bundles on X of rank n and ES and FS (resp. EP and FP ), their pull-backs to S (resp. P ). Let σt , t ∈ [0, 1], be a

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homotopy of sections of the bundle Iso(ES , FS ) of isomorphisms between fibres of ES and FS over S. With this notation we have the following. Proposition 8.1. Assume that σ0 and σ1 are lifts of sections τ0 and τ1 of Iso(EP , FP ). In other words σ0 and σ1 are invariant under the antipodal map of S. Let ν(n) = 2(m−1).n . Then there is a family θ(t,s) , (t, s) ∈ [0, 1]×[0, 1] of sections of Iso(ν(n).ES , ν(n).FS ) depending continuously on (t, s) such that θ(t,0) = ν.σt , θ(0,s) and θ(1,s) are independent of s, and θ(t,1) is -invariant. We argue by induction on the dimension of X. When dim X = 0, the assertion is obvious. Assume then that it is proved for all finite complexes of dimension less than n = dim X. Let Y denote the (n−1)-skeleton of X; then X is obtained by attaching the disjoint union A = 1≤i≤r Di of n-discs through a map φ of its boundary ∂A = 1≤i≤r ∂Di in Y . Fix trivialisations of the pull-backs of W , E and F to A. We may then regard the pull-backs of σt (resp. τ0 , τ1 ) to A as GL(N, C)-valued functions on A × S m−1 (resp. ∂A × P m−1 ) which we denote by ft (resp. g0 , g1 ) (here P m−1 is the (m − 1)-dimensional real projective space). By the induction hypothesis, there is a family σ(t, s) of sections of Iso(ES , FS ) over the inverse image SY of Y in S with σ(t,0) = ν(n − 1).σt , σ(0,s) = ν(n − 1).σ0 , σ(1,s) = ν(n − 1).σ1 and σ(t,1) invariant under . This means that there is a family f(t,s) of maps from ∂A × S to GL(ν(n − 1)N, C) such that f(t,0) = ν(n − 1).ft , f(t,1) is -invariant, f(0,s) = g0 and f(1,s) = g1 . Now using the homotopy extension property, one sees that there is a family F(t,s) of maps from A × S to GL(ν(n − 1)N, C) such that F(t,s) = f(t,s) on ∂A × S, F(0,s) = f0 and F(1,s) = f1 . Let ht = F(t,1) . Then for each i with 1 ≤ i ≤ r, ht , t ∈ [0, 1], defines a map from the boundary of Di × P m−1 × [0, 1] (which can be identified with S n+1 × P m−1 ) to GL(ν(n − 1)N, C). Equivalently we have a map H from P m−1 to Ωp (GL(ν(n − 1)N, C), the p-th free loop space of GL(ν(n − 1)N, C), which when composed with the projection u of S m−1 on P m−1 factors through the space of all maps from Dn+1 to GL(ν(n − 1)N, C). Now homotopy classes of maps from P m−1 to Ωp (GL(ν(n − 1)N, C)) can be identified with the group K −p−1 (P m−1 ) so that H defines an element [H] in that group. Moreover the image of [H] in K −p−1 (S m−1 ) under the map u∗ induced by u is trivial. Since the kernel of the map u∗ : K −p−1 (P m−1 ) → K −p−1 (S m−1 ) is annihilated by 2m−1 (this is seen easily as a consequence of the Atiyah-Hirzebruch spectral sequence for K-theory [1] and the fact that the kernel of the cohomology map from P m−1 to S m−1 is a direct sum of m − 2 copies of Z/2), the proposition follows. References [1] M.F.Atiyah and F.Hirzebruch, Vector bundles and homogeneous spaces, 1961 Proc. Sympos. Pure Math., Vol. III pp 7–38, American Mathematical Society, Providence, RI. [2] M.F.Atiyah and I.M.Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Mah. Soc. 69 (1963), 422–433. [3] S.Minakshisundaram and A.Pleijel, Some properties of the eigenfunctions of the Laplaceoperator on Riemannian manifolds, Canadian J. Math. 1 (1949), 242–256. [4] Richard S.Palais, Seminar on the Atiyah-Singer index theorem, Annals of Mathematics Studies, No. 57, Princeton University Press, Princeton N. J. 1965. [5] J.-P.Serre, Groupes d’homotopie et classes des groupes abeliens, Ann. of Math. (2) 58 (1953), 258–294. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India E-mail address: [email protected]


Spin(7) Instantons and the Hodge Conjecture for Certain Abelian Four-folds: a Modest Proposal T. R. Ramadas Dedicated to S. Ramanan on the occasion of his seventieth birthday.

Abstract. The Hodge Conjecture is equivalent to a statement about conditions under which a complex vector bundle on a smooth complex projective variety admits a holomorphic structure. In the case of abelian four-folds, recent work in gauge theory suggests an approach using Spin(7) instantons. I advertise a class of examples due to Mumford where this approach could be tested. I construct explicit smooth vector bundles - which can in fact be constructed in terms of of smooth line bundles - whose Chern characters are given Hodge classes. An instanton connection on these vector bundles would endow them with a holomorphic structure and thus prove that these classes are algebraic. I use complex multiplication to exhibit Cayley cycles representing the given Hodge classes. I find alternate complex structures with respect to which the given bundles are holomorphic, and close with a suggestion (due to G. Tian) as to how this may possibly be put to use.

1. Introduction Let X be a smooth complex projective variety of dimension n, and c a rational (p, p) cohomology class (0 < p < n). The Hodge Conjecture is that H: there exist finitely many (reduced, irreducible) (n − p)-dimensional subvarieties Yi and rational numbers ai such that c = i ai [Yi ], where [Yi ] is the (rational) cohomology class dual to Yi . That is, c is dual to a rational algebraic cycle. This is equivalent to V: there exists a holomorphic vector bundle E such that its Chern character ch(E) is equal to a rational multiple of c modulo (classes of) rational algebraic cycles. The second statement implies the first because the Chern character of a holomorphic (and therefore algebraic) bundle factors through the Chow ring of algebraic 2010 Mathematics Subject Classification. 32J25, 14C30, 70S15. It is a pleasure to thank Bobby Acharya, who told me of Spin(7) instantons and gave me a copy of Christopher Lewis’ thesis. I am also grateful to Dominic Joyce, Nigel Hitchin, M.S. Narasimhan, and Gang Tian for very helpful comments. c 2010 American c Mathematical 0000 (copyright Society holder)

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varieties. The converse also holds. In fact, as Narasimhan pointed out to me, it is known ([M]) that the rational Chow ring is generated by stable vector bundles. Let X, c be as above. By a theorem of Atiyah-Hirzebruch ([A-H], page 19), the Chern character map ch : K 0 (X) ⊗ Q → H even (X, Q) is a bijection, where K 0 (X) is the Grothendieck group of (topological/smooth) vector bundles on X. Thus we are assured of the existence of a smooth bundle E and and an integer n > 0 such that ch(E) = rank(E) + nc. A possible strategy to show that a given class c is algebraic suggests itself – find a suitable such bundle E and then exhibit a holomorphic structure on it. This note is written to argue that recent progress in mathematical gauge theory, and in particular the work of G. Tian and C. Lewis, makes this worth pursuing, at least in the case of certain abelian four-folds. Such an approach to the Hodge Conjecture for the case of Calabi-Yau four-folds is surely known to the experts (and this has been confirmed to me), but I have only been able to locate some coy references. Claire Voisin ([V]), following a similar approach, has much more definitive negative results in the case of non-algebraic tori. Before proceeding, let us note that the known “easy” cases of the Hodge conjecture are proved essentially by the above method. First, given an integral class c ∈ H 2 (X, Z), a smooth hermitian line bundle L exists with (first) Chern class equal to c. Given any real 2-form Ω representing c there exists a unitary connection on L with curvature −2πiΩ. If c is a (1, 1) class, it can be represented by an Ω which is (1, 1). The corresponding connection defines a holomorphic structure on L. If c is an integral (n − 1, n − 1) class, the strong Lefschetz theorem exhibits the dual class as a rational linear combination of complete intersections. What follows is the result of much trial and error and computations - which I either only sketch or omit altogether - using Mathematica; the notebooks are available on request. (I used an exterior algebra package of Sotirios Bonanos, available from http://www.inp.demokritos.gr/~sbonano/. ) 2. Mumford’s examples We consider Hodge classes on certain abelian four-folds. These examples are due to Mumford ([P]). It is best to start with some preliminary algebraic number theory. If F is an algebraic number field, with degree F = d, the ring of algebraic integers Λ ≡ oF is a free Z-module of rank d which generates F as a Q-vector space. If V denotes the real vector space R ⊗Q F , then Λ ⊂ V is a lattice and Xr = V /Λ is a real d-torus. ¯ ⊂ C. (That is, L is the smallest Let L denote the Galois saturation of F in Q subfield Galois over Q and containing any (and therefore all) embeddings of F .) Then G = Gal(L/Q) acts transitively on the set E of embeddings ι : L → Q by (g, ι) → g(ι) = g ◦ ι (g ∈ G, ι ∈ E), and the image by ι is the fixed field of the stabiliser of ι. Further, the map ¯ ⊗Q F → Q ¯E Q ¯ vector spaces. given by 1 ⊗ x → (ι(x))E is an isomorphism of Q Turning to the real torus Xr : ¯ =Q ¯ E; (1) we have natural isomorphisms H1 (Xr , Z) = Λ and H1 (Xr , Q) 1 ¯ (2) H (Xr , Q) has basis {dtι }E , where dtι is induced by the projection to the ¯ E. ιth factor from Q

INSTANTONS AND THE HODGE CONJECTURE

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In what follows we will identify the real or complex cohomology of Xr with the corresponding spaces of translation-invariant forms on Xr . We will need the following result, whose proof is straightforward. Proposition 2.1. A one-form ω = ι ωι dtι represents a rational class iff the coefficients ωι belong to L and satisfy the equivariance ωg(ι) = g(ωι ), g ∈ G

Similarly, a two-form φ = ι,κ φι,κ dtι ∧ dtκ (with the coefficients antisymmetric functions of the two indices) represents a rational class iff φg(ι),g(κ) = g(φι,κ ), g ∈ G Suppose now that the embeddings E occur in complex conjugate pairs - E = E E , with each ι ∈ E corresponding to ¯ι ∈ E . Then the map V = R ⊗Q F (→ C ⊗Q F ∼ CE ) → CE

is an isomorphism of real vector spaces and induces a (translation-invariant) complex structure on Xr , which becomes a complex torus, which we will denote simply X. We turn now to specifics. Let P = ax4 +bx2 +cx+d be an irreducible polynomial with rational coefficients and all roots x1 , x2 , x3 , x4 real. We will suppose that the roots are numbered such that x1 > x2 > x3 > x4 . Let L1 /Q be the splitting field L1 = Q[x1 , x2 , x3 , x4 ] ⊂ R. We suppose that P is chosen such that the Galois group is S4 . This is equivalent to demanding that [L1 : Q] = 24. We set L ≡ L1 [i]. This is a Galois extension of Q, with Galois group S4 × {e, ρ}, where ρ is complex conjugation. Consider a cube, with vertices labeled as in the figure: 3

2

4

1

1b

2b

4b

3b

Let G denote the group of symmetries of the cube. We have the exact sequence: 1 → {e, ρ} → G → S4 → 1

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where now ρ denotes inversion, and S4 is the group of permutations of the four diagonals. Splitting this, identifying S4 with (special orthogonal) rotations implementing the corresponding permutation of diagonals. we get an identification G ∼ S4 × {e, ρ} = Gal(L/Q) Let H denote the stabiliser of the vertex 1, F the corresponding fixed field, and ϕ1 : F → L → C the corresponding embedding. The left cosets of H can be identified with the vertices of the cube, as well as embeddings of F in C. We label the latter ϕj , ϕ¯j (j = 1, 2, 3, 4). Note that the field F is invariant under complex conjugation, which therefore acts on it with fixed field F1 . Clearly, F1 = Q[x1 ] . We set D = (x1 − x2 )(x1 − x3 )(x1 − x4 )(x2 − x3 )(x2 − x4 )(x3 − x4 ) Given our ordering of the roots, D > 0. Note that iD ∈ F , F = F1 [iD], and Δ ≡ D2 is a rational number. We will assume that (after multiplying all the xi by a common natural number if necessary) Δ is an integer (and so D is an algebraic integer). We will repeatedly use the fact that the Galois conjugates of iD ∈ F are given by (2.1)

φj (iD) = −(−1)j iD φ¯j (iD) = (−1)j iD

In our case Xr is a real 8-torus. The embeddings ϕi : F → C induce R-linear maps zi : V → C, such that z = (z1 , z2 , z3 , z4 ) is an isomorphism of R-vector spaces V → C4 . We let X denote the complex manifold V /Λ obtained thus. Note that if a ∈ oF , multiplication by a is a Q-linear map F → F which induces a R-linear map V → V taking the lattice Λ to itself. If z(a) = (a1 , a2 , a3 , a4 ), and u ∈ V with z(u) = (z1 , z2 , z3 , z4 ) we also have z(au) = (a1 z1 , a2 z2 , a3 z3 , a4 z4 ), so that we see that this induces an analytic map (in fact an isogeny) X → X. In other words, oF acts on X by “complex multiplication”. As a complex torus, X is certainly Kähler, and we shall see below that it is algebraic. What is relevant for our purposes is that it is possible to describe explicitly the Hodge decomposition as well as the rational structure of the complex cohomology of X. Let T (for “top”) denote the set of indices {1, 2, 3, 4} and B (for “bottom”) the indices {¯ 1, ¯ 2, ¯ 3, ¯4}. (The corresponding vertices are denoted 1b, etc. in the figure.) Proposition 2.2. A basis of H p,q is labeled by subsets P ⊂ T , Q ⊂ B, with |P | = p, and |Q| = q, and given by the translation-invariant forms dz P d¯ z Q , where P for example, if P = {i, j}, with i < j we set dz = dzi ∧ dzj , and if Q = {¯i, ¯j} zi ∧ d¯ zj . A basis of the rational cohomology HQr (again with i < j), we set d¯ z Q = d¯ is labelled by pairs (R, χ) where • R is an orbit of G in the set of sequences μ ≡ (μ1 , . . . , μr ) of distinct elements in T ∪ B, and • χ runs over a Q-basis of HR , the space of G-equivariant maps R → L, satisfying χ(μσ(1) , . . . , μσ(r) ) = sign(σ)χ(μ1 , . . . , μr ) for any permutation σ such that μ, μσ ∈ R.


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The corresponding classes are given by the forms χ(μ)dz μ μ∈R

We use the notation dz = dzμ1 ∧· · ·∧dzμr , with the convention that dz¯1 = d¯ z1 , etc. μ

It is useful to note the following Lemma 2.3. Given R, the Q-dimension of HR is |R|/r!. Note that if r = 2p, a rational class as above is of type (p, p) iff the orbit consists of sequences with elements equally divided between the top and bottom faces of the cube. In particular, the rational (1, 1) classes correspond to the G-orbit of the sequence (1, ¯ 1). Since in this case HR has dimension 4, we see that the Neron-Severi group has rank 4. Consider now the orbit of the sequence (1, 3, ¯2, ¯4). This corresponds to a twodimensional space M of rational (2, 2) classes, which have the property that these are not products of rational (1, 1) classes. It is easy to check that but for (the Q-span of) these, rational (2, 2) classes are generated by products of rational (1, 1) classes. Proposition 2.4. A Q-basis of M is given by the classes • M = D(dz1 d¯ z2 dz3 d¯ z4 + d¯ z1 dz2 d¯ z3 dz4 ) • M = i(dz1 d¯ z2 dz3 d¯ z4 − d¯ z1 dz2 d¯ z3 dz4 ) So the Hodge conjecture in this case would be that : the classes M and M are algebraic. We will use complex multiplication in an essential way later; here I illustrate its use by showing how it can be used to halve our work. Consider multiplication by the algebraic integer a = 1 + iD ∈ oF . This induces a (covering) map πa : X → X and one easily computes: (2.2)

πa∗ M = ((1 − Δ)2 − 4Δ)M + 4(1 − Δ)ΔM πa∗ M = ((1 − Δ)2 − 4Δ)M − 4(1 − Δ)M

This proves Proposition 2.5. Algebraicity of either one of M or M implies that of the other. Before moving on, we find a positive rational (1, 1) form ω on X, which will show that it is projective. Let μ1 ∈ F1 (to be chosen in a moment) and consider the form iD ω= (μ1 dz1 d¯ z1 − μ2 dz2 d¯ z2 + μ3 dz3 d¯ z3 − μ4 dz4 d¯ z4 ) Δ where μi are Galois conjugates. Clearly this is a rational (1, 1) form, and it will be positive provided (−1)j+1 μj > 0. For example, we can take μ1 = (x1 − x2 )(x1 − x3 )(x1 − x4 ), and we will do so. With this choice the holomorphic four-form θ ≡ (1/D)dz1 dz2 dz3 dz4 satisfies (2.3)

ω4 = θ ∧ θ¯ 4!

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3. Expressing M, M in terms of Chern characters Consider the G-orbit of (1, 3). The corresponding subspace of HQ2 is spanned by the classes of the form A1 = a13 (x1 − x3 )dz1 dz3 + .... where a13 belongs to the fixed field of the subgroup of G that leaves the set of vertices {1, 3} invariant, and this coefficient determines the others in the sum by Galois covariance. We introduce the notation Ta = a13 a¯2¯4 (x1 − x3 )(x2 − x4 ) − a1¯2 a3¯4 (x1 − x2 )(x3 − x4 ) + a1¯4 a3¯2 (x1 − x4 )(x3 − x2 ) Squaring A1 , we get A21 =2a13 a24 (x1 − x3 )(x2 − x4 )dz1 dz3 dz2 dz4 + .. z1 dz2 dz1 dz3 + ... +2a¯12 a13 (x1 − x2 )(x1 − x3 )d¯ z2 dz2 d¯ z1 + .. +2a1¯2 a2¯1 (x1 − x2 )(x2 − x1 )dz1 d¯ z2 d¯ z4 + .. +2Ta dz1 dz3 d¯ If we make the replacement a13 icDa13 (c an integer introduced for later use in §7), we get a class A2 , such that A22 /(c2 Δ) =2a13 a24 (x1 − x3 )(x2 − x4 )dz1 dz3 dz2 dz4 + .. z1 dz2 dz1 dz3 + ... +2a¯12 a13 (x1 − x2 )(x1 − x3 )d¯ z2 dz2 d¯ z1 +2a1¯2 a2¯1 (x1 − x2 )(x2 − x1 )dz1 d¯ −2Ta dz1 dz3 d¯ z2 d¯ z4 − .. Suppose now that the classes Ai are integral. (This is easily arranged by clearing denominators.) Let Li (i = 1, 2) be the line bundle with Chern class Ai . Proposition 3.1. Let Vi = Li ⊕ L−1 i , i = 1, 2. Then 2

ch(V1c

Δ

V2 ) = 4c2 Δ(Ta dz1 dz3 d¯ z2 d¯ z4 + ..)

where the equality is modulo (rational) 0- and 8-forms. We have the freedom to choose the coefficient a13 , which by Galois covariance determines the other coefficients, and hence the above classes. We now make the choice a13 = h3 where for later use we introduce the notation h2 = (x1 x2 + x3 x4 ) (3.1)

h3 = (x1 x3 + x2 x4 ) h4 = (x1 x4 + x2 x3 )

Then Ta = −D, and we get Theorem 3.2. With the above choice, 2

ch(V1c

Δ

V2 ) = 4c2 ΔM

where the equality is modulo (rational) 0- and 8-forms.


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2

The virtual bundle V1c Δ V2 has the properties: c1 = 0, and c2 ∧ ω 2 = 0, where ω is the rational Kähler class defined at the end of §2. (This is because the Mi , as can be easily seen, are orthogonal to ω.) This will not do for reasons to do with the Bogomolov inequality, but this can be fixed because of a minor miracle: Proposition 3.3. With the above choices, 1 A21 ∧ ω = −2iΔ dz1 d¯ z1 dz2 d¯ z2 dz3 d¯ z3 + ... μ4 In particular, A21 ∧ ω is a (rational) (3,3) form. For later use, we also record Proposition 3.4. With the above choices, A1 ∧ ω 3 = 0 A2 ∧ ω 3 = 0 We will suppose that kω (for some positive integer k) is an integral class, and let Lkω denote a (holomorphic, in fact ample) line bundle with this Chern class. The following is a easy consequence of 3.3. −1 ˆ Theorem 3.5. Let Vˆ1 = L1 ⊗ Lkω ⊕ L−1 and set 1 ⊗ Lkω , and V2 = L2 ⊕ L2 c2 Δ ˆ ˆ E = V1 V2 . Then

ch(E) = 2c2 Δkω + 4c2 ΔM + k2 c2 Δω 2 where the equality is modulo (rational) 0-, (3,3)- and 8-forms. In particular, this (difference) bundle E satisfies the “Bogomolov inequality”: < c2 ω 2 > −

2Δ − 3 c2 Δ < c21 ω 2 > = 2 k2 < ω4 > 4(Δ − 1) c Δ−1 >0

The symbol < .. > stands for integration against the fundamental class. We use the quote marks since we are not (yet!) talking of a holomorphic bundle E. Since the virtual bundle has positive rank, we are justified, up to some non-canonical choices, in dropping the qualifiers “virtual”/“difference”. Remark 3.6. We have concentrated on the Hodge class M in this section; it is possible, with slight modifications to the above expressions, to find a smooth bundle E whose Chern character similarly contains the Hodge class M . 4. Spin(7) instantons In this section we recall the definition of Spin(7) instantons ([B-K-S], [T]), specialised to the case of a K¨ ahler four-fold X with trivial canonical bundle KX . We fix a Ricci-flat K¨ ahler form ω, and let θ denote a trivialisation of KX satisfying (2.3). We define a (complex antilinear) endomorphism : Ω(0,2) → Ω(0,2) , by |α|2 θ = α ∧ α We have 2 = 1, so we can decompose the bundle into a self-dual and anti-self-dual part: (0,2) (0,2) Ω(0,2) = Ω+ ⊕ Ω−

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Let E be a hermitian (C ∞ ) vector bundle on X. A Spin(7) instanton is a hermitian connection A on E, whose curvature F satisfies (0,2)

F+

= 0, ΛF = 0

Here Λ denotes as usual contraction with the K¨ ahler form. A crucial point is the following ([T],[L]): Proposition 4.1. The L2 -norm of the curvature of a Spin(7) instanton satis (0,2) fies ||F− ||22 = T r(F ∧ F ) ∧ θ¯ In particular, if the invariant on the right vanishes, a Spin(7) instanton is equivalent to a holomorphic structure on E together with a Hermite-Einstein connection. Clearly, such a bundle would be poly-stable, and hence (or directly from the Hermite-Einstein condition) satisfy the Bogomolov inequality: (4.1)

c2 (E).ω 2 ≥

r−1 c1 (E)2 .ω 2 2r

where r denotes the rank of E. Now that we have embedded the problem of construction a holomorphic structure on E in a broader context – that of constructing an instanton connection – one can envisage deforming the complex structure in such a way that c2 (E) ∧ θ¯ = 0 and still hope to have the moduli space of semi-stable holomorphic structures on E deform as the moduli space of instanton connections. There are several possible approaches to the construction of such a connection. (1) Exhibit an instanton by glueing. (2) The fact that the bundles are exhibited as a difference of two vector bundles, each of which is in turn a sum of explicit line bundles, suggests the use of monads, possibly combined with a twistor construction. This would involve a matrix of sections of line bundles. A third idea, suggested to me by G. Tian, is pursued in the last section of this paper. 5. Calibrations; Cayley submanifolds In his thesis, C. Lewis [L] shows how (in one particular case) one can construct an instanton by glueing around a suitable Cayley submanifold. (See also [B].) We define these terms below, and then exhibit some relevant Cayley cycles that arise in our context. (References are [H-L], and [J]; but we follow the conventions of [T].) Definition 5.1. Let M be a Riemannian manifold. A closed l-form φ is said to be a calibration if for every oriented tangent l-plane ξ, we have φ|ξ ≤ volξ where volξ is the (Riemannian) volume form. Given a calibration φ, an oriented submanifold N is said to be calibrated if φ restricts to N as the Riemannian volume form.


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It is easy to see that a calibrated submanifold is minimal. Two examples are relevant. First, if M is Kähler, with K¨ ahler form ω, for any integer p ≥ 1, the p form ωp! is a calibration, and the calibrated submanifolds are precisely the complex submanifolds. The case that concerns us is that of a four-fold X with trivial canonical bundle KX . We fix an integral Ricci-flat Kähler form ω, and let θ denote a trivialisation of KX with normalisation as in (2.3). Then 4Re(θ) is a second calibration, and the calibrated submanifolds are called Special Lagrangian submanifolds. There is a “linear combination” of the two, defined by the form Ω=

w2 + 4Re(θ) 2

which defines the Cayley calibration. The corresponding calibrated manifolds are called Cayley manifolds. Any smooth complex surface (on which the second term will restrict to zero) or any Special Lagrangian submanifold (on which the first term will vanish) furnish examples. In fact, the Cayley cycles we deal with will be of the latter kind. Cayley manifolds are not easy to find. We will use the following result (Proposition 8.4.8 of [J]): Proposition 5.2. Let X be as above, and σ : X → X an anti-holomorphic ¯ Then the fixed point set is a Special Laisometric involution such that σ ∗ θ = θ. grangian submanifold. We return to the constructions of our paper. Recall that the field F is invariant under complex conjugation, which therefore acts on it with fixed field F1 . This ¯(u), where, if z = induces an involution σˆ1 : V → V such that z(σˆ1 (u)) = z ¯ = (¯ (z1 , z2 , z3 , z4 ), we set z z1 , z¯2 , z¯3 , z¯4 ). The induced involution σ1 : X → X has fixed locus which we will denote Y . Note that σ satisfies the conditions of the previous Proposition and therefore Y is Special Lagrangian. Theorem 5.3. There exist (rational) Cayley cycles representing the Hodge classes Mi . Proof. Recall the isogeny πa : X → X, given by multiplication by the algebraic integer a = 1 + iD. It is easy to check πa∗ ω = (1 + Δ)ω πa∗ θ = (1 + Δ)2 θ We will also need a second isogeny πb , where b = iD, which satisfies πb∗ ω = Δω πb∗ θ = Δ2 θ These equations guarantee the maps πa , πb take Cayley cycles to Cayley cycles (possibly introducing singularities.) We have the following table giving the action of the above isogenies on fourforms of various types (all the forms in the list are eigenvectors):

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T. R. RAMADAS

Form dz1 dz2 dz3 dz4 dz1 d¯ z1 dz2 dz3 dz1 d¯ z1 dz2 dz4 d¯ z1 dz2 dz3 dz4 dz1 d¯ z1 dz2 d¯ z2 dz1 d¯ z1 dz2 d¯ z3 dz1 dz2 d¯ z3 d¯ z4 dz1 d¯ z2 dz3 d¯ z4 d¯ z1 dz2 d¯ z3 dz4

eigenvalue of πa∗ (1 + Δ)2 (1 + Δ)2 (1 + Δ)(1 − iD)2 (1 + Δ)(1 − iD)2 (1 + Δ)2 (1 + Δ)(1 − iD)2 (1 + Δ)2 (1 + iD)4 (1 − iD)4

eigenvalue of πb∗ Δ2 Δ2 −Δ2 −Δ2 Δ2 −Δ2 Δ2 Δ2 Δ2

“multiplicity” 2×1 2×8 2×4 2×4 6 2 × 12 4 1 1

(We list only forms of type (4,0), (3,1) and (2,2), omitting types that are related to the ones in the list by conjugation. The term “multiplicity” refers to the number of forms of a given type, not the multiplicity of eigenvalues.) Consider the operator Φa = (πa∗ − (1 + Δ)2 )(πb∗ + Δ2 ) From the list it follows that the space M ⊗Q C (spanned by the Mi ) is the sum of the eigenspaces of Φa corresponding to the non-zero eigenvalues. We have (using (2.2)) Φ∗a M = −8Δ2 [2ΔM + (1 − Δ)M ] Φ∗a M = −8Δ2 [−(1 − Δ)ΔM + 2ΔM ] Next, note that the Cayley cycle Y defined above satisfies < Y, M > = 2Dδ < Y, M > = 0 Here denotes the integration pairing of cycles and forms, and δ denotes the co-volume of the lattice oF1 ⊂ F1 ⊗Q R. By standard facts in algebraic number theory, δ is a rational multiple of D; so the above pairings are rational, as they had better be. We now consider the Cayley cycle Ca = (πa − (1 + Δ)2 )(πb + Δ2 )Y By construction Ca is orthogonal to all the forms in the above list except the Mi . Its pairings with these are as follows: < Ca , M > = −32Δ3 Dδ < Ca , M > = −16Δ2 (1 − Δ)Dδ Let now a ¯ = (1 − iD), and repeat the above construction with operators Φa¯ , etc. Φ∗a¯ M = −8Δ2 [2ΔM − (1 − Δ)M ] Φ∗a¯ M = −8Δ2 [(1 − Δ)ΔM + 2ΔM ] This gives a cycle Ca¯ satisfying < Ca¯ , M > = −32Δ3 Dδ < Ca¯ , M > = 16Δ2 (1 − Δ)Dδ Clearly the theorem is proved.


165 11

Remark 5.4. The above result, though suggestive, does not take us far. This is because the above “Cayley cycle” is not effective, but in fact a linear combination of SL subvarieties with both positive and negative coefficients. (D. Joyce has pointed out that this must be the case given that it represents a (2, 2) class.) To make matters worse, a theorem of G. Tian (Theorem 4.3.3 of [T]) states that blow-up loci of Hermite-Yang-Mills connections are effective holomorphic integral cycles consisting of complex subvarieties of codimension two. So any glueing will call for very new techniques. 6. Adapted complex structures In this section we seek translation-invariant complex structures on the eighttorus V /Λ such that the classes Ai are of type (1, 1) w.r.to these complex structures, and therefore define holomorphic structures on the line bundles Li . The original motivation was to exploit twistor techniques for the construction of instantons, but we postpone discussion of possible uses of this investigation to the last section. Consider a linear change of coordinates of the form z1 = w1 + α¯12 w ¯2 + α¯14 w ¯4 z3 = w3 + α¯32 w ¯2 + α¯34 w ¯4 z2 = w2 + α ˜ ¯21 w ¯1 + α ˜ ¯23 w ¯3 z4 = w4 + α ˜ ¯41 w ¯1 + α ˜ ¯43 w ¯3 We collect the coefficients into 2 × 2 matrices α and α ˜ as follows: α¯12 α¯14 α= α¯32 α¯34 and α ˜=

α ˜ ¯21 α ˜ ¯41

α ˜ ¯23 α ˜ ¯43

and rewrite the above change of coordinates as follows: z1 w1 w ¯2 = +α ¯ z3 w3 w ¯4 w2 ¯1 z2 ¯˜ w = +α z4 w4 w ¯3 A long but straightforward computation shows Ai will be of type (1, 1) provided: h3 (x1 − x3 )(α¯12 α¯34 − α¯14 α¯32 ) +h4 (x1 − x4 )α¯12 − h2 (x1 − x2 )α¯14 +h2 (x3 − x4 )α¯32 − h4 (x3 − x2 )α¯34 +h3 (x2 − x4 ) = 0 and h3 (x2 − x4 )(α ˜ ¯21 α ˜ ¯43 − α ˜ ¯23 α ˜ ¯41 ) ˜ ¯43 − h2 (x3 − x4 )α ˜ ¯41 +h4 (x1 − x4 )α +h2 (x1 − x2 )α ˜ ¯23 − h4 (x3 − x2 )α ˜ ¯21 +h3 (x1 − x3 ) = 0

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T. R. RAMADAS

To rewrite these conditions in a more compact form, we introduce some notation: (1) Given a 2 × 2 matrix A: a11 a12 A= a21 a22 let

Aˆ =

(6.1)

a22 −a21

−a12 a11

(If A is nonsingular, Aˆ = (det A)A−1 .) (2) Define the symmetric bilinear form on the space of 2 × 2 matrices ˆ = det (A + B) − det A − det B < A, B >= T r(AB) (3) Let

−h4 (x2 − x3 ) h2 (x3 − x4 ) H= −h2 (x1 − x2 ) −h4 (x1 − x4 )

−h4 (x1 − x4 ) −h2 (x3 − x4 ) h2 (x1 − x2 ) −h4 (x2 − x3 ) The conditions on α and α ˜ can now be rewritten: so that

ˆ = H

(6.2)

< α, H >= h3 (x2 − x4 ) + h3 (x1 − x3 )det α

and (6.3)

ˆ >= h3 (x1 − x3 ) + h3 (x2 − x4 )det α = h3 (x2 − x4 ) + h3 (x1 − x3 )det α

and

x 1 − x3 ˆ¯ α x2 − x4 (α ˆ is defined as in (6.1).) Then the forms Ai are of type (1, 1) w.r.to the wi . Further, if α satisfies x 1 − x4 α¯34 = + α ¯¯ x2 − x3 12 (6.6) x 3 − x4 α¯14 = − α ¯¯ x1 − x3 32 then ω remains of type (1, 1). α ˜=

If α satisfies (6.6), the condition (6.5) becomes x1 − x4 x 3 − x4 |α¯12 |2 + |α¯ |2 ) + h3 (x2 − x4 ) h3 (x1 − x3 ))( x − x x1 − x2 32 (6.7) 2 3 +h4 (x1 − x4 )(α¯12 + α ¯ ¯12 ) + h2 (x3 − x4 )(α¯32 + α ¯ ¯32 ) = 0 The space of solutions J˜ is clearly an 3-dimensional ellipsoid in the two-dimensional complex vector space with co-ordinates (α¯12 , α¯32 ). The condition (6.4) becomes: x1 − x4 x 3 − x4 |α¯ |2 + |α¯ |2 = 0 x2 − x3 12 x1 − x2 32 which corresponds to removing the affine hyperplane H given by h3 (x2 − x4 ) + h4 (x1 − x4 )(α¯12 + α ¯ ¯12 ) + h2 (x3 − x4 )(α¯32 + α ¯ ¯32 ) = 0

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T. R. RAMADAS

We have therefore to consider J = J˜ \H, which is the union of two open three-discs. A particular choice of α has remarkable properties. Let 4 2 ) −(x3 − x4 )(1 − 2h ) (x2 − x3 )(1 − 2h 1 ∗ h h 3 3 α = 2 4 (x1 − x4 )(1 − 2h x1 − x3 (x1 − x2 )(1 − 2h h3 ) h3 ) Theorem 6.2. With this choice, we have (x1 x3 + x2 x4 )2 A1 =(x1 − x2 )2 (x2 − x3 )2 (x1 − x4 )2 (x3 − x4 )dw1 dw ¯2 4 −(x1 − x2 )2 (x2 − x3 )2 (x1 − x4 )(x3 − x4 )2 dw2 dw ¯3 +(x1 − x2 )(x2 − x3 )2 (x1 − x4 )2 (x3 − x4 )2 dw3 dw ¯4 (6.8)

+(x1 − x2 )2 (x2 − x3 )(x1 − x4 )2 (x3 − x4 )2 dw4 dw ¯1 +(x1 − x2 )2 (x2 − x3 )2 (x1 − x4 )2 (x3 − x4 )dw ¯1 dw2 −(x1 − x2 )2 (x2 − x3 )2 (x1 − x4 )(x3 − x4 )2 dw ¯2 dw3 +(x1 − x2 )(x2 − x3 )2 (x1 − x4 )2 (x3 − x4 )2 dw ¯3 dw4 +(x1 − x2 )2 (x2 − x3 )(x1 − x4 )2 (x3 − x4 )2 dw ¯4 dw1 (x1 x3 + x2 x4 )2 A2 =(x1 − x2 )2 (x2 − x3 )2 (x1 − x4 )2 (x3 − x4 )dw1 dw ¯2 4iD +(x1 − x2 )2 (x2 − x3 )2 (x1 − x4 )(x3 − x4 )2 dw2 dw ¯3 +(x1 − x2 )(x2 − x3 )2 (x1 − x4 )2 (x3 − x4 )2 dw3 dw ¯4

(6.9)

−(x1 − x2 )2 (x2 − x3 )(x1 − x4 )2 (x3 − x4 )2 dw4 dw ¯1 −(x1 − x2 )2 (x2 − x3 )2 (x1 − x4 )2 (x3 − x4 )dw ¯1 dw2 −(x1 − x2 )2 (x2 − x3 )2 (x1 − x4 )(x3 − x4 )2 dw ¯2 dw3 −(x1 − x2 )(x2 − x3 )2 (x1 − x4 )2 (x3 − x4 )2 dw ¯3 dw4 +(x1 − x2 )2 (x2 − x3 )(x1 − x4 )2 (x3 − x4 )2 dw ¯4 dw1

(6.10) Δ(x1 x3 + x2 x4 )2 ω 4iD = −{(x1 − x2 )2 (x1 − x3 )(x2 − x3 )(x1 − x4 )2 (x3 − x4 )dw1 dw ¯1 +(x1 − x2 )2 (x2 − x3 )2 (x1 − x4 )(x2 − x4 )(x3 − x4 )dw2 dw ¯2 +(x1 − x2 )(x1 − x3 )(x2 − x3 )2 (x1 − x4 )(x3 − x4 )2 dw3 dw ¯3 +(x1 − x2 )(x2 − x3 )(x1 − x4 )2 (x2 − x4 )(x3 − x4 )2 dw4 dw ¯4 } In particular, −ω is a K¨ ahler form and the corresponding complex structure makes Xr an abelian variety. Remark 6.3. It is convenient to consider the consider the conjugate complex structure (w.r.to which holomorphic co-ordinates are the w ¯i . This has the property that the forms Ai and ω are of type (1,1), and in addition, ω is K¨ ahler. We let X denote the corresponding abelian variety.


169 15

7. A strategy Attempts to invoke twistor methods have not been successful so far. For example, N. Hitchin pointed out that results of M. Verbitsky make hyperk¨ ahler twistor spaces quite unsuitable. G. Tian made the following suggestion: construct instantons by deformation (using, say, the continuity method) from a situation when they are known to exist. In fact, the complex structure described in the Remark 6.3 provides such a starting point. I close with a brief justification for this claim. With respect to the above complex structure, the bundles Vî defined in the statement of Theorem 3.5 are holomorphic, and furthermore (using the ampleness of ω), the constant k can be chosen large enough that Vˆ2 can be embedded as a sub2 bundle of Vˆ1c Δ . The quotient bundle can be identified with the difference bundle E, which therefore has a holomorphic structure depending on the above embedding; we now show that it is possible to arrange that E, endowed with this structure, is polystable. (By stability we shall mean μ-stability w.r.to the polarisation ω.) −1 ˆ Let us start by recalling that Vˆ1 = L1 ⊗ Lkω ⊕ L−1 1 ⊗ Lkω , and V2 = L2 ⊕ L2 . Choose a large enough integer k1 such that Lk1 ω that is very ample, and let C be a general curve cut out by three sections of this line bundle. It follows from −1 Proposition 3.4 that d ≡ degree L−1 2 ⊗ L1 ⊗ Lkω |C = degree L2 ⊗ L1 ⊗ Lkω |C = 3 4 degree Lkω |C = kk1 < ω >, and will assume that k is chosen such that d > 2genus(C) = 3k14 < ω 4 > +2. We next make the following assumption: dim H 0 (C, Lkω |C ) = c2 Δ

(7.1)

which we will return to below. Let W denote a subspace of H 0 (X , L−1 2 ⊗L1 ⊗Lkω ), chosen such that • the restriction map W → H 0 (C, L−1 2 ⊗ L1 ⊗ Lkω |C ) is an isomorphism, and • W is base-point free. Consider now the evaluation map E : W ⊗ OX → L−1 2 ⊗ L1 ⊗ Lkω , and let F be the kernel; by construction F fits in the exact sequence 0 → F → W ⊗ OX → L−1 2 ⊗ L1 ⊗ Lkω → 0 . By Butler’s Theorem ([Bu]), the restriction of F to C is stable, and this proves that F itself is stable. We next choose a subspace U of H 0 (X , L2 ⊗ L−1 1 ⊗ Lkω ) with similar properties and obtain a second stable bundle G that fits in the sequence 0 → G → U ⊗ OX → L2 ⊗ L−1 1 ⊗ Lkω → 0 Dualising, tensoring by suitable line bundles and adding the two sequences, we get 2

0 → Vˆ2 → Vˆ1c

Δ

→ Fˆ ⊗ L1 ⊗ Lkω ⊕ Gˆ ⊗ L−1 1 ⊗ Lkω → 0

where Fˆ denotes the dual of F and Gˆ denotes the dual of G, and we have used the assumption (7.1), namely, dim W = dim U = c2 Δ. Repeatedly using Proposition 3.4 we see that the two summands in the last sum have the same slope. Consider now the assumption (7.1). By Riemann-Roch, this is equivalent to: (kk13 − (3/2)k14 ) < ω 4 >= c2 Δ This is solved by taking k=(

c2 Δ 3k1 )/ < ω 4 > + k13 2

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T. R. RAMADAS

This is where the choice of c comes in - we choose c and k1 such that k is an integer (and large enough). Once this is done Theorem 7.1. The bundle E (on X ) can be given a holomorphic structure such that it is polystable. The above application of Butler’s theorem is inspired by its use in [M]. By Donaldson-Uhlenbeck-Yau, such a bundle would admit a Hermite-Einstein metric and therefore a Spin(7) instanton. References M.F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, Prec. Sympos. Pure Math. 3 (1961), 7–38. [A-O] B.S. Acharya and M. O’Loughlin, Selfduality in d=8 dimensional euclidean gravity, Phys. Rev. D 55 (1997), 4521–4524. [B-K-S] L. Baulieu, H. Kanno and I.M. Singer, Special quantum field theories in eight and other dimensions, Commun. Math. Phys. 194 (1998), 149–175. [B6] K. Becker, M. Becker, D.R. Morrison, H. Ooguri, Y. Oz, Y. Zheng, Supersymmetric cycles in exceptional holonomy manifolds and Calabi-Yau four-folds, Nuclear Physics B, 480 (1996), 225-238. [B] S. Brendle, Complex anti-self-dual instantons and Cayley submanifolds, math.DG/0302094, 2003. [Bu] D.C. Butler, Normal generation of vector bundles over a curve, J. Differential Geom. 39 (1994), 1–34. [D-T] S. K. Donaldson and R. P. Thomas, Gauge theory in higher dimensions, in “The Geometric Universe; Science, Geometry, And The Work Of Roger Penrose”, Oxford University Press, (1998). [H-L] F. R. Harvey and H. B. Lawson, Jr, Calibrated geometries. Acta Mathematica, 148 (1982), 47–157. [J] D.C. Joyce, Riemannian Holonomy Groups and Calibrated Geometry, Oxford Graduate Texts in Mathematics (2007). [L] C. Lewis, Spin(7) Instantons, Oxford University D.Phil. thesis (1998). [P] H. Pohlmann, Algebraic cycles on abelian varieties of complex multiplication type, Ann. of Math. 88 (1968), 161–180. [M] E. C. Mistretta, Stable vector bundles as generators of the Chow ring, Geom. Dedicata 117 (2006), 203–213. [S] A. Sommese, Quaternionic manifolds, Math. Ann. 212 (1975), 191–214. [T] G. Tian, Gauge theory and calibrated geometry I, Ann. of Math. 151 (2000), 193–268. [V] C. Voisin, Some aspects of the Hodge conjecture, Japanese Journal of Mathematics 2 2 (2007), 261–296.

[A-H]

Abdus Salam I.C.T.P., 11 Strada Costiera, Trieste 34014, Italy E-mail address: [email protected]


Remarks on Parabolic Structures C. S. Seshadri Abstract. Let Y be a smooth projective curve with an action of a finite group π and X = Y mod π. Let G be a reductive group and P −→ Y a (π − G) bundle i.e. P is a principal G-bundle on Y such that the action of π on Y lifts to an action on P commuting with the canonical action of G on P . The question is to give an equivalent description of (π − G) bundles on Y by suitable objects on X. When G is the full linear group, the answer is given by principal G-bundles on X, endowed with what are called parabolic structures at the ramification points of X. This notion of parabolic structures generalizes in an obvious manner, to the case of principal G-bundles on X when G is a general reductive. However, in the case of a general reductive group, the required description by objects turns out to be subtler than this obvious generalization and gets connected with Bruhat-Tits theory.

1. Introduction Let Y be a smooth projective curve (defined over C) with an (effective) action of a finite group π, X the smooth projective curve X = Y mod π and p : Y −→ X the canonical quotient morphism. Let G be a reductive algebraic group and P a (π − G) bundle on Y i.e. P is a principal G-bundle over Y and the action of π on Y lifts to an action on P commuting with the (right) action of G on P . The question is to give an equivalent description of the moduli problem of (π − G) bundles on Y by objects on X. Recall that if G is the full linear group, this required description is given by principal G-bundles on X having parabolic structures at the ramification points of X i.e. the points of X over which p is ramified. In fact P is equivalently described by a π-vector bundle V on X and if W = pπ∗ (V ) is the invariant direct image of V , the parabolic structure is described by giving flags of fixed type on the fibres of W at the ramification points of X ([6]). In the general case the description by objects on X (given in Remark 2.13 below) turns out to be somewhat subtler and it gets intimately connected with Bruhat-Tits theory (see [3]). For example, it does not seem to be possible, in general, to associate naturally to P a principal G-bundle on X (see Case II, §3 below; also the connection with vector bundles with orthogonal and symplectic structures having singularties at the ramification points in X); however, the moduli stack of these objects seems to be related to the moduli stack of principal G-bundles by a “Hecke correspondence” (see Remark 2.13 below 2010 Mathematics Subject Classification. 14D20, 14H60. c Mathematical 0000 (copyright Society holder) c 2010 American

1 171

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C. S. SESHADRI SESHADRI C.S.

and [7]). In this talk we shall illustrate this phenomenon for the case of classical groups. These objects appear in the study of compactifications of moduli spaces of principal G-bundles on nodal curves (see [9]). They are also related to the work [8], which we came to know recently. See also the more recent preprint [5]. These remarks have taken shape in the course of numerous discussions with V. Balaji and are inspired by the classical work of A. Weil [11]. In a subsequent article a description of the above mentioned objects on X in terms of Bruhat-Tits theory and a correspondence between these objects and (π − G)-bundles, will be given. Recall that the moduli spaces of (π − G)-bundles on Y have been constructed by Tannakian methods (see [1], [2]). 2. (π − G) bundles on Y and lattices on X Let x ∈ X be a ramification point of p : Y −→ X. Let Ux denote the group of (π − G) automorphisms of the restriction of P to a sufficiently small π-invariant neighbourhood of p−1 (x). We call Ux the unit group at x. The main point is to describe this as an object on X. Let y be a point of Y lying over a ramification point x ∈ X. Let Γ be the isotropy subgroup of π at y. We can find sufficiently small neighbourhoods E and D of y and x respectively such that E is Γ-invariant and D = E mod Γ. We can suppose that E and D are discs with centres (0) which correspond to y and x respectively. Let τ be a coordinate function on E so that z = τ r is a coordinate function on D, where r is the order of the cyclic group Γ. The restriction of P to E is a (Γ − G) bundle and one sees that Ux is just the group of (Γ − G) automorphisms of P over E, which identifies with the trivial bundle E × G (G action given by multiplication on the right). Recall the fact that the Γ-action on E ×G is given by a representation ρ : Γ −→ G (see [11], [4], [10]) i.e. γ · (u, g) = (γu, ρ(γ)g), u ∈ E, γ ∈ Γ.

(2.1) Let φ0 ∈ Ux i.e. (2.2)

φ0 : E × G −→ E × G is a (Γ − G) automorphism.

We see that φ0 (u, g) = (u, φ(u)g) where φ : E −→ G is a regular map satisfying the following: (2.3)

φ(γ · u)

= ρ(γ)φ(u)ρ(γ)−1 , u ∈ E.

We see then that the functions φ as above identify with the elements of Ux . Let us fix a maximal torus T of G. Now T is a product of one dimensional tori. We represent it in the diagonal form as follows (or if we wish we could embed G in a full linear group and work with matrices) ⎡ (2.4)

⎢ T =⎢ ⎣

t1

0 . .

0

tm

⎤ ⎥ ⎥. ⎦

PARABOLIC BUNDLES

173 3

Then we can suppose that the representation ρ of Γ in G factors through T (by a suitable conjugation). Fix a generator γ in Γ. We can suppose that the coordinate function τ of E is so chosen that γ · τ = ζτ , where ζ is a (primitive) rth of unity. Then ρ takes the form ⎡ ⎢ ρ(γ) = ⎢ ⎣

(2.5)

ζ a1

0 . .

0

⎤ ⎥ ⎥ with ai ∈ Z. ⎦

ζ am

We can suppose that |ai | < r for all i (or even 0 ≤ ai < r) and take αi = ai /r, so that |αi | < 1.

(2.6)

We denote the function τ ai by z αi (z = τ r ). Define the “meromorphic” (or “rational”) map Δ : E −→ T (morphism on the punctured disc E \ (0)): ⎡ a ⎤ ⎡ α ⎤ τ 1 0 0 z 1 ⎢ ⎥ ⎢ ⎥ . . ⎥=⎢ ⎥ Δ=⎢ (2.7) ⎣ ⎦ ⎣ ⎦ . . am αm 0 τ 0 z Then we have (2.8)

Δ(γu) = ρ(γ)Δ(u),

u∈E

where Δ can be taken as a function Δ : E −→ G (through T → G). Now define the function ψ : E −→ G by (2.9)

ψ = Δ−1 φΔ

That is, φ = ΔψΔ−1 . Then we see that ψ(γu) = ψ(u) so that ψ goes down to a function D −→ G and we denote this by the same letter ψ. We see that ψ is a meromorphic map. We see then that the unit group Ux can be identified with the set of meromorphic maps ψ : D −→ G defined above. It is customary to work with complete local rings. Let Oy (resp. Ox ) denote the completion of the local ring of Y at y (resp. of X at x). Then φ defines an element of G(Oy ) (the group of Oy valued points of G). We write simply φ ∈ G(Oy ). Note that the action of Γ on E × G induces a canonical action of Γ on G(Oy ) (can be deduced from (2.3)) and then the unit group Ux is just G(Oy )Γ . Let Kx (resp. Ky ) be the quotient field of Ox (resp. Oy ). We see that ψ ∈ G(Kx ). Then by the map φ −→ ψ, we can identify Ux with a subgroup of G(Kx ); however, the choice of Δ satisfying (2.8) is not unique and if we choose another, we see easily that the resulting one is a conjugate subgroup in G(Kx ). Thus we have (see Remark 2.12 below):

(2.10)

Ux determines a lattice or bounded subgroup in G(Kx ), welldetermined upto conjugacy. We denote this by the same letter Ux .

Let us now take G = GL(m). Then we can write φ = ||φij (τ )||, ψ = ||ψij (z)||, 1 ≤ i, j ≤ m (as matrices). Then (2.9) takes the form

174 4

(2.11)


φij (τ ) = ψij (z)z αi −αj .

We can suppose that 0 ≤ α1 ≤ α2 ≤ · · · ≤ αm < 1. Since |αi − αj | < 1, we deduce easily that ψij are regular i.e. Ux ⊂ G(Ox ). To see this suppose that ψij is not regular. Then considered as a function in τ (z = τ r ), ψij has a pole of order ≥ r, whereas z αi −αj could have only a zero of order strictly less than r (as a function in τ ). But φij (τ ) is regular, which leads to a contradiction. Besides, we see also that ψij (0) is a lower triangular matrix. This fact gives the parabolic (or rather quasi-parabolic) structure on the vector bundle pπ∗ (V ) = W , V being the π-vector bundle associated to the principal bundle P (for details see (§1 of [6]), also ([11] esp. p.194). Remark 2.12. Let L be a p-adic local field and A the ring of integers in L (A is a complete local ring and L is its quotient field). Then G(L) is locally compact (we have to make suitable hypothesis on the field of definition for G). Then G(A) is a maximal compact subgroup of G(L) and unlike the real case, all the maximal compact subgroups need not be conjugate. Compact subgroups of G(L) are often referred to as lattices and the maximal ones as maximal lattices. In the geometric case that we are considering, G(Kx ) is not locally compact and the notion of bounded subgroups generalises that of compact subgroups (following Bruhat-Tits [3]). A subgroup of H of G(Kx ) is said to be bounded if the “order of pole” of an element of H is bounded by a universal constant (taking an embedding of H in a full-linear group so that an element of H is represented by a matrix with entries in Kx , this notion makes sense). We observe that Ux is a bounded subgroup of G(Kx ) in this sense. This is an easy consequence of (2.9) noting that φ is regular. Of course G(Ox ) is a bounded subgroup; in fact it is a maximal bounded subgroup of G(Kx ). Let j : G(Ox ) −→ G be the “evaluation map” i.e. the map G(Ox ) −→ G(C) induced by the canonical map Ox −→ residue field which is C in our case. We denote G(C) by G. The inverse image by j of a Borel subgroup in G is an example of an “Iwahori subgroup” in G(Kx ). More generally an Iwahori subgroup of G(Kx ) is, by definition, a conjugate of this particular Iwahori subgroup. A parahoric subgroup of G(Kx ) is, by definition, a subgroup which contains an Iwahori subgroup. Fix an Iwahori subgroup I. Then one knows that there are only a finite number r of maximal bounded subgroups of G(Kx ) containing I and any maximal bounded subgroup is a conjugate of precisely one containing I. If G is the full linear group, then one knows that r = 1, so that a maximal bounded subgroup of G(Kx ) is always a conjugate of G(Ox ). This also explains why Ux is contained in G(Ox ) in this case. Remark 2.13. We shall now describe (π − G) bundles on Y as suitable objects on X. For simplicity we suppose that x is the only ramification point and that G is semi-simple. The description of (π−G) bundles on Y by objects on X is given in (2.26) below and generalises a well-known description of principal G-bundles as a double coset space (or an equivalent adèlic description). The point of this well-known description is that the restriction of a principal G-bundle to D, as well as X \ {x}, is trivial and hence it is described by a single transition function i.e. a map D \ (0) −→ G

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such that isomorphism classes of G-bundles are interpreted as equivalence classes of transition functions. This argument generalises in a fairly easy manner to the case of (π − G) bundles on Y with, for example, a transition function being interpreted as a Γ-equivariant map E \ (0) −→ G. The details are given below. Two (π − G) bundles on Y are said to be locally isomorphic at x if they are isomorphic as (π − G) bundles over p−1 (D) = V1 , D a sufficiently small disc neighbourhood of x as above. We see that two such bundles are locally isomorphic at x if and only if their restrictions to E are isomorphic as Γ-bundles. Recall that E is a Γ-invariant neighbourhood of y, y being a point of Y lying over x. We can suppose that V1 is a disjoint union of discs of the form E i.e. each disc contains a unique point of Y lying over x. Hence if P is a (π − G) bundle on Y , we call the restriction of P to E, which is a (Γ − G) bundle, as the local invariant of P at x. We shall now fix this local invariant i.e. we fix y and suppose that the restriction P to E is the (Γ − G) bundle with the action of Γ on E × G given as in (2.1) i.e. (2.14)

P |E E × G with the action of Γ given by γ · (u, g) = (γu, ρ(γ)g), u ∈ E and g ∈ G

ρ : Γ −→ G being a representation. We can also say that the local invariant is given by the representation ρ (fixing y) of the isotropy subgroup of π at y. We denote M (ρ) by the moduli space (or rather moduli stack) of isomorphism classes of (π − G) bundles with local invariant ρ. Let X1 denote the complement of x in X and Y1 = p−1 (X1 ). Now π acts freely on Y1 so that the restriction of P to Y1 goes down to a principal G-bundle on X1 which is trivial (in the algebraic sense) since G is semi-simple. Hence we have: (2.15)

P |Y1 Y1 × G with the action of π given by γ · (u, g) = (γu, g), γ ∈ π and u ∈ Y1 .

Let us call the restriction of P to V1 (resp. Y1 ) by P1 (resp. P2 ). We note that the restriction of P1 to E is just the one described in (2.14) and P2 is the one described in (2.15). The (π − G) bundle P is given by a “transition function” i.e. a (π − G) isomorphism (holomorphic): (2.16)

θ : P2 |V1 ∩Y1 −→ P1 |V1 ∩Y1 .

Now if s is a π-invariant (i.e. π-equivariant) rational section of P over Y (which exists, since for example; the restriction of P to Y1 descends to the trivial G-bundle on X1 in the algebraic sense). Then if s1 (resp. s2 ) is the restriction of s to a section of P1 over V1 (resp. a section of P2 over Y1 ), note that by the isomorphism θ in (2.16) above, s2 is taken to s1 . Besides by (2.15) above, s2 defines a morphism Y1 −→ Y1 × G given by u −→ (u, s∗2 (u)) so that s∗2 defines a rational (in the algebrogeometric sense) map Y −→ G which is regular on Y1 . Of course s∗2 is π-invariant i.e. s∗2 (γ · u) = s∗2 (u), γ in Γ. Thus we have (i) θ(s2 ) = s1 over V1 ∩ Y1 (2.17) (ii) s2 defines (through the identification (2.15)) a π-invariant rational map s∗2 : Y −→ G which is regular on Y1 . Observe that if Q is any (π −G) bundle in M (ρ), then Q|V1 P1 and Q|Y1 P2 as (π − G) bundles by (2.14) and (2.15) above (noting that Q|V1 is determined by

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Q|E ). Thus Q is defined by an isomorphism as in (2.16) above. Let us denote it by φ. Then P is (π − G) isomorphic to Q if and only if we have the following: (2.18)

λθ μ=φ λ a (π − G) automorphism of P1 μ a (π − G) automorphism of P2 .

Observe that μ is given by a morphism (using (2.15)):

(2.19)

Y1 × G −→ Y1 × G, (u, g) −→ (u, μ∗ (u)g), μ∗ (γ · u) = μ(u), γ ∈ π i.e. μ∗ goes down to morphism X1 −→ G.

Note that the punctured disc E ∗ = E \ (0) is contained in V1 ∩ Y1 and the (π − G) isomorphism θ is completely characterized by its restriction to E ∗ . We observe by (2.15) that the restriction of P2 to E ∗ is the (Γ − G) bundle E ∗ × G over E ∗ with the action of Γ given by (2.20)

γ : E ∗ × G −→ E ∗ × G, γ ∈ Γ γ(u, g) = (γu, g).

The restriction of P1 to E ∗ is the (Γ − G) bundle E ∗ × G on E ∗ with the action of Γ given by (2.21)

γ : E ∗ × G −→ E ∗ × G γ(u, g) = (γu, ρ(γ)g), γ ∈ Γ.

Then the restriction of θ to E ∗ is then a (Γ − G) isomorphism of the bundle in (2.20) with the one of (2.21). We see easily that θ is defined by:

(2.22)

E ∗ × G −→ E ∗ × G (u, g) −→ (u, θ ∗ (u)g) θ ∗ (γ · u) = ρ(γ)θ ∗ (u) θ ∗ : E ∗ −→ G.

Observe that the map Δ, as in (2.8), has the property Δ(γ · u) = ρ(γ)Δ(x). Thus we can write θ ∗ = Δθ ∗∗ such that θ ∗∗ (γu) = θ ∗∗ (u) i.e. θ ∗∗ goes (2.23) down to a regular map D∗ −→ G, D∗ = D \ (0). We claim that θ ∗∗ extends to a meromorphic map E −→ G and hence by 2.23 (2.24) descends to a meromorphic map D −→ G which is regular on D∗ . This is an easy consequence of (2.17) as follows. The restriction of s1 to E defines a regular map E −→ E × G, u −→ (u, s∗1 (u)), where s∗1 : E −→ G is a regular map. The restriction of s∗2 to E ∗ is regular and extends to a meromorphic map E −→ G by the property (ii) of (2.17). Further (i) of (2.17) translates to the relation θ ∗ = s∗1 · (s∗2 )−1 , so that θ ∗ is meromorphic in E and hence θ ∗∗ is also meromorphic in E, which proves (2.24). Now the (π −G) automorphisms of P1 identify with the (Γ−G) automorphisms of the restriction of P1 to E i.e. the (Γ − G) bundle on E as in (2.14). Thus λ identifies with an element λ∗ of the unit group at x (i.e. an element as in (2.2)

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above, before its identification as a subgroup of G(Kx )). Thus the equivalence relation (2.18) translates into the following form: λ∗ (Δθ ∗∗ )μ∗ = Δφ∗∗ i.e. (Δ−1 λ∗ Δ)θ ∗∗ μ∗∗ = φ∗∗ .

(2.25)

Let us denote by G(Kxh ) the set of germs of regular maps D∗ −→ G which extend to meromorphic maps D −→ G. We have an inclusion G(Kxh ) ⊂ G(Kx ). Set Uxh = Ux ∩ G(Kxh ) i.e. we work in the holomorphic and meromorphic set up instead of the “formal” set up. Observe that θ ∗∗ ∈ G(Kxh ) and (Δ−1 λ∗ Δ) ∈ Uxh . Further μ∗ is a regular map X1 −→ G extending to a meromorphic map X −→ G so that μ∗ is a rational map i.e. μ∗ ∈ G(X1 ). Thus from (2.25) we deduce the following: The moduli stack M (ρ) identifies with the double coset space (2.26) h U h \G(Kx ) /G(X ). 1

x

In fact, it can be shown that the moduli stack M (ρ) can be identified with (i) Ux \G(Kx ) /G(X1 ) or in adèlic language as follows: (ii) C \G(A) /G(K) p∈X

p

where A denotes group of the adèles, K the function field of X, Cp = Op for p = x and Cx = Ux . The moduli stack M (ρ) depends only on the unit group Ux and not on the explicit nature of the representation ρ. We can then denote M (ρ) as M (Ux ). The moduli stack M (Ux ) is to be understood as the moduli stack with “quasi-parabolic structures” in the sense of [6], where the representation ρ gives weights to define parabolic structures and define stability and semistability, leading to the construction of moduli spaces of semistable parabolic vector bundles as projective varieties. Note that the moduli stack M (Ux ) makes sense, say for any bounded subgroup Ux of G(Kx ). It is likely that there are quasi-projective schemes associated to these moduli stacks (under suitable hypotheses). These could be called moduli stacks with level structures. Suppose that we have an inclusion Ux ⊂ Ux of bounded subgroups of G(Kx ). Then we have a canonical morphism of stacks M (Ux ) −→ M (Ux ). Note that if Ux = G(Ox ), then the moduli stack M (Ux ) is just the moduli stack M of principal G-bundles on X. If we have Ux ⊂ G(Ox ) then we have a canonical morphism of stacks M (Ux ) −→ M i.e. we have an underlying principal G-bundle to a (π − G) bundle in the case Ux ⊂ G(Ox ). As we saw above this happens in the case of the full linear group. Suppose that Ux is a parahoric subgroup in G(Kx ). Then by a suitable conjugation we can suppose that we have inclusions I ⊂ Ux and I ⊂ G(Ox ), I being an Iwahori subgroup. This leads to the following diagram of morphisms of stacks: M (I) M M (Ux )

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i.e. M and M (Ux ) are related by a “Hecke correspondence” (see [7]). We shall see examples of these in §3 below. It seems likely that the moduli stacks M (Ux ) coming from (π − G) bundles (as above) are related to M by “Hecke correspondences which are good” in the sense that there are quasi-projective schemes associated to these correspondences. 3. The case of G = SO(2n) or Sp(2n) Let us now consider the case when G = SO(2n) (resp. Sp(2n)). Define

0 J ∗ (respectively J = J 0 =

0 −J

J 0

where J is the (n × n)–matrix with 1’s on the antidiagonal and zero elsewhere: ⎤ ⎡ 0 1 ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ J =⎢ . ⎥ . ⎥ ⎢ ⎦ ⎣ . 1 0 n×n We take SO(2n) (resp. Sp(2n)) as the subgroup of GL(2n) which leaves J ∗ invariant i.e. formed by S in GL(2n) such that SJ ∗ t S = J ∗ . Then a maximal torus T of G can be taken in the diagonal form ⎡ ⎤ t1 0 ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ tn ⎥. (3.1) T =⎢ −1 ⎢ ⎥ tn ⎢ ⎥ ⎣ ⎦ . −1 0 t1 One knows that G ∩ B + (resp. G ∩ B − ) is again a Borel subgroup in G, B + = (resp. B − ) = upper (resp. lower) triangular matrices in GL(2n). Let us consider the case where the ai and the associated αi in (2.6) can be taken in the following form: (3.2)

−1 < −α1 ≤ −α2 ≤ · · · ≤ −αn ≤ αn ≤ αn−1 ≤ · · · ≤ α1 < 1,

with (0 ≤ αi < 1) We write β1 = −α1 , β2 = −α2 , · · · βn = −αn , βn+1 = αn , · · · , β2n = α1 . Thus we have in this case diagonal matrices: ⎡ (3.3)

⎢ A=⎢ ⎣

z −α1

0 . .

0

z −αn

⎤

⎡

⎥ ⎢ ⎥=⎢ ⎦ ⎣

z β1

0 . .

0

z βn

⎤ ⎥ ⎥, ⎦

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⎡ ⎢ B=⎢ ⎣

(3.4)

z αn

0 . .

0

⎤

⎡

⎥ ⎢ ⎥=⎢ ⎦ ⎣

z α1 Δ=

0 . .

A 0

0 B

⎤ ⎥ ⎥. ⎦

z β2n

0

Let

z βn+1

Thus we get (3.5)

φ = ΔψΔ−1 i.e. φij (τ ) = ψij (z)z βi −βj

and φ, ψ and Δ leave J ∗ invariant. Let I denote the subgroup of G(Ox ) such that by the evaluation map G(Ox ) −→ G, I is the inverse image of the Borel subgroup induced by the lower triangular matrices in GL(2n). Then I is an Iwahori subgroup in G(Kx ). We note that |βi − βj | < 2 and ψij can have at most a pole of order 1. We need to consider three cases: Case I: |βi − βj | < 1 Case II: |βi − βj | ≤ 1 and the value 1 is attained Case III: |βi − βj | > 1 for some i, j In Case I, by the same argument as for the full linear group (βi ’s are increasing), we deduce that ψij (z) are regular and ψij (0) is lower triangular. (βi ’s are increasing). In Case II, the ψij (z) can have poles of order 1 and in Case III, the order of the pole is at most 1 but they can have zeroes of order 2. Thus we get, in Case I, G(Ox ) ⊃ Ux ⊃ I In Case II, we have G(Kx ) ⊃ Ux ⊃ I but Ux is not contained in G(Ox ). We observe that, in the cases I and II, Ux is a parahoric subgroup of G(Kx ), but in case II, Ux is not even conjugate (in G(Kx )) to a subgroup of G(Ox ), in particular the maximal lattice containing Ux is not conjugate to the maximal lattice G(Ox ). In Case III, in contrast, we have of course G(Kx ) ⊃ Ux but Ux does not contain I. It is not clear whether Ux is a parahoric subgroup of G(Kx ). Let M, M (I) and M (Ux ) be the moduli stacks of (π − G) bundles on Y with local invariants defined by the unit group G(Ox ), I and Ux respectively (see towards the end of Remark 2.13). Now we have the following: Case I: |βi − βj | < 1 We have a morphism of stacks: (3.6)

M (I) −→ M (Ux ) −→ M

(where M = M (G(Ox )) - set of isomorphism classes of G-bundles on X). Note that this is similar to the case of the full linear group. An object in M (Ux ) is to be viewed as having a parabolic (strictly speaking a quasi-parabolic) structure over the image of this object in M , which is again a G-bundle on X.

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Case II: |βi − βj | ≤ 1 and the value 1 is attained We have morphisms of stacks M (I) M

(3.7)

M (Ux )

induced by the inclusions I ⊂ G(Ox ) and I ⊂ Ux . The above diagram means that M and M (Ux ) are related by a “Hecke correspondence” (see [7]). There seems to be no natural way of associating a G-bundle on X to an object in M (Ux ) in this case II. Let αi be as in (3.2) above and let us define δi , 0 ≤ δi < 1, 1 ≤ i ≤ 2n, as follows: (3.8)

{δi = 1 − αi if αi = 0 and δi = 0 if αi = 0, 1 ≤ i ≤ n, δ2n+1−j = αj , 1 ≤ j ≤ n}.

Set

⎡ ⎢ A1 = ⎢ ⎣

(3.9)

z δ1

0 . .

⎢ B1 = ⎢ ⎣

(3.10)

⎥ ⎥ ⎦

z δn

0 ⎡

⎤

z δn+1

⎤

0

⎥ ⎥ ⎦

. . z δ2n

0

Now let Δ1 =

A1 0

0 B1

Define θ as follows: (3.11)

δi −δj . φ = Δ1 θΔ−1 1 ⇐⇒ φij (z) = θij (z)z

We see that

θ = L · ψ · L−1 with L = Δ−1 1 ·Δ and θ leaves invariant the form Q = LJ ∗ L i.e θ Q tθ = Q observing that Lt = L. Since |δi − δj | < 1 we find that θij (z) are regular functions i.e. θ ∈ GL(2n)(Ox ) (as in (2.11) above, the order of the δi ’s do not play a role at this point). Let k = #{i|αi = 0}. We find that −1

z Ik×k 0 L= 0 I2n−k×2n−k We see that Q is a quadratic (resp. symplectic) form over Kx with a singularity at x. Let Uxk be the unit group of Q i.e. consisting of all the elements of GL(2n)(Ox ), which leave Q invariant. We see that, by conjugation in GL(2n)(Kx ), Ux can be identified with a subgroup of Uxk and coincides with Uxk for the choice αi = 1/2,

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1 ≤ i ≤ k. The groups Uxk seem to be maximal parahoric and mutually nonconjugate. The (π −G) bundle P can be equivalently described by a rank 2n vector bundle V on X, endowed with a π-invariant, non-degenerate quadratic (resp. symplectic) form. Then if W = pπ∗ (V ), W acquires a quadratic (resp. symplectic) form which is non-degenerate outside x and described by Q locally at x. Thus the moduli stack of (π − G) bundles is described equivalently as the moduli stack of vector bundles W on X with a non-degenerate quadratic (resp. symplectic) form outside x and with singularity at x of type given by Q as above. 4. The case when G = SL(n) Let us now take G = SL(n). Fix k, 1 ≤ k ≤ n − 1. Then we can take the choice of αi or δi (as in (3.2) and (3.8) above) as follows: (4.1) (4.2)

(n − k) n = · · · = αn = k/n

α1 = · · · = αk = − αk+1

or equivalently δ1 = · · · = δn = k/n. We denote the corresponding unit group by Uxk . For k = 0, we take Ux0 = SL(n)(Ox ). These groups are maximal parahoric. Then we see that Uxk ⊂ SL(n)(Kx ) (1 ≤ k ≤ n − 1), and takes the form

zB A Ux = z −1 C D where the entries of A, B, C, D are in Ox , and A is a (k × k)-matrix, and D an (n − k) × (n − k)-matrix. We see that the lattices Uxk are not conjugate in SL(n)(Kx ). They are maximal parahorics. The moduli space M (Uxk ) is equivalently described as the moduli space of vector bundles on X of rank n whose determinant is a fixed line bundle of degree k (in particular for 1 ≤ k ≤ n − 1, they are not SL(n) bundles on X) and they are all related by Hecke correspondences (see [7]). For general {αi }, the moduli space can then be viewed as a moduli space with quasi-parabolic structures in the usual sense on a vector bundle on X. References [1] V. Balaji, I. Biswas and D.S. Nagaraj, Principal bundles over projective manifolds with parabolic structure over a divisor. Tˆ ohuku Math. Jour. 53 (2001), 337–368. [2] V. Balaji, I. Biswas and D.S. Nagaraj, Ramified G-bundles as parabolic bundles, Jour. Ramanujan Math Society, 18, No 2 (2003) 123-138. [3] F. Bruhat and J. Tits, Groupes r´ eductifs sur un corps local II, Sch´ emas en groupes, Existence ´ d’une donn´ ee radicielle valu´ ee, Inst. Hautes Eludes Sci. Publ. Math. 60 (1984), p.197–376. [4] A. Grothendieck, Sur la m´ emoire de Weil “G´ en´ eralisation des fonctions ab´ eliennes”, S´ eminaire Bourbaki, Expos´ e 141, (1956-57). [5] J. Heinloth, Uniformization of G–bundles, arXiv: 0711.4450. [6] V. Mehta and C.S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann 248 (1980), 205–239. [7] M.S. Narasimhan and S. Ramanan - The papers [13], [18] and [19] in the Collected papers of M.S. Narasimhan - Hindustan Book Agency. [8] G. Pappas and M. Rapoport, Some questions about G-bundles on curves, Mathematisches Institut der Universitat Bonn, November 2008.

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[9] C.S. Seshadri. Moduli spaces of torsion free sheaves on nodal curves and generalisations - I, to appear in the volume dedicated to P.E. Newstead. [10] C. Teleman and C. Woodward, Parabolic bundles, products of conjugacy classes, and quantum cohomology, Annales Institut Fourier, Vol. 53, (2003), no. 3, p. 713-748. [11] A. Weil - G´ en´ eralisations des fonctions ab´ eliennes - Collected papers, Vol. I, Springer-Verlag. Chennai Mathematical Institute, SIPCOT IT Park, Padur Post, Siruseri 603103, Tamil Nadu, India E-mail address: [email protected]


Iterated Destabilizing Modifications for Vector Bundles with Connection Carlos Simpson Abstract. Given a vector bundle with integrable connection (V, ∇) on a curve, if V is not itself semistable as a vector bundle then we can iterate a construction involving modification by the destabilizing subobject to obtain a Hodge-like filtration F p which satisfies Griffiths transversality. The associated graded Higgs bundle is the limit of (V, t∇) under the de Rham to Dolbeault degeneration. We get a stratification of the moduli space of connections, with as minimal stratum the space of opers. The strata have fibrations whose fibers are Lagrangian subspaces of the moduli space.

1. Introduction Suppose X is a smooth projective curve over C. Starting with a rank r vector bundle with integrable holomorphic connection (V, ∇), if V is semistable as a vector bundle, we get a point in the moduli space U(X, r) of semistable vector bundles of rank r and degree 0 on X. Let MDR (X, r) denote the moduli space of vector bundles with integrable connection of rank r. The open subset G0 where the underlying vector bundle is itself semistable thus has a fibration MDR (X, r) ⊃ G0 → U(X, r). The fiber over a point [V ] ∈ U(X, r) (say a stable bundle) is the space of connections on V , hence it is a principal homogeneous space on H 0 (End(V ) ⊗ Ω1X ) ∼ = H 1 (End(V ))∗ = TV∗ U(X, r). So, the above fibration is a twisted form of the cotangent bundle TV∗ U(X, r) → U(X, r). At points where the bundle V is not semistable, we will extend G0 to a stratification of MDR (X, r) by locally closed subsets Gα . If V is not semistable, let H ⊂ V be the maximal destabilizing subsheaf. Recall that H is a subsheaf whose slope μ(H) = deg(H)/rk(H) is maximal, and among such subsheaves H has maximal rank. It is unique, and is a strict subbundle so the quotient V /H is also a bundle. 2010 Mathematics Subject Classification. Primary 14H60; Secondary 14D07, 32G34. Key words and phrases. Connection, Deformation, Higgs bundle, Moduli space, Oper, Reductive group. This research is partially supported by ANR grants BLAN08-1-309225 (SEDIGA) and BLAN08-3-352054 (G-FIB). c Mathematical 0000 (copyright Society holder) c 2010 American

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The connection induces an algebraic map θ = ∇ : H → (V /H) ⊗ Ω1X . Define a Higgs bundle (E, θ) by setting E 1 := H, E 0 := V /H, E := E 0 ⊕ E 1 , and θ is the above map. It is a “system of Hodge bundles”, that is a Higgs bundle fixed by the C∗ action. If (E, θ) is a semistable Higgs bundle, the process stops. If not, we can continue by again choosing H ⊂ (E, θ) the maximal destabilizing sub-Higgs-bundle, then using H to further modify the filtration according to the formula (3.2) below. At the end of §3 the proof of our first main Theorem 2.5 consists in showing that this recursive process stops at a Griffiths-transverse filtration of (V, ∇) such that the associated graded Higgs bundle is semistable. Classically filtered objects (V, ∇, F • ) arose from variations of Hodge structure. In case an irreducible connection supports a VHS, our iterative procedure constructs the Hodge filtration F • starting from (V, ∇). In general the filtration F • given by Theorem 2.5 is not unique, see Proposition 4.3. The associated-graded Higgs bundle (E = GrF• (V ), θ) is unique up to Sequivalence, as follows from the following limit point interpretation. In terms of the nonabelian Hodge filtration [54] MHod → A1 the above process gives a way, described in §4, of calculating the limit point limt→0 (tλ, V, t∇). The limit is a point in one of the connected components of the fixed point set of the Gm action on the moduli space of Higgs bundles MH . Looking at where the limit lands gives the stratification by Gα ⊂ MDR . Existence of the limit is a generalization to MDR of properness of the Hitchin map for MH . The interpretation in terms of Griffiths-transverse filtrations was pointed out briefly in [54]. Given the S-equivalence class, it makes sense to say whether (E, θ) is stable or not. Proposition 4.3 shows that the filtration F • is unique up to shifting indices, if and only if (E, θ) is stable. In §6, the nonuniqueness of the filtration is related to a wall-crossing phenomenon in the parabolic case. In the present paper, after describing the explicit and geometric construction of the limit point by iterating the destabilizing modification construction, we consider various aspects of the resulting stratification. For example, we conjecture that the stratification is nested, i.e. smaller strata are contained in the closures of bigger ones. This can be shown for bundles of rank 2. A calculation in deformation theory shows that the set Lq ⊂ MDR of points (V, ∇) such that limt→0 (V, t∇) = q, is a lagrangian subspace for the natural symplectic form. We conjecture that these subspaces are closed and form a nice foliation 7.4. We mention in §6 that the same theory works for the parabolic or orbifold cases, and point out a new phenomenon there: the biggest open generic stratum no longer necessarily corresponds to unitary bundles. The Hodge type of the generic stratum varies with the choice of parabolic weights, with constancy over polytopes. At the end of the paper we do some theoretical work (which was missing from [54]) necessary for proving the existence of limit points in the case of principal bundles. All along the way, we try to identify natural questions for further study. It is a great pleasure to dedicate this paper to Professor Ramanan. I would like to thank him for the numerous conversations we have had over the years, starting from my time as a graduate student, in which he explained his insightful points of

DESTABILIZING MODIFICATIONS

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view on everything connected to vector bundles. These ideas are infused throughout the paper. I would also like to thank Jonathan Aidan, Daniel Bertrand, Philip Boalch, David Dumas, Jaya Iyer, Ludmil Katzarkov, Bruno Klingler, Vladimir Kostov, Anatoly Libgober, Ania Otwinowska, Tony Pantev, Claude Sabbah, and Szilard Szabo for interesting communications related to the subjects of this paper. 2. Griffiths transverse filtrations Suppose X is a smooth projective curve, and V is a vector bundle with integrable holomorphic connection ∇ : V → V ⊗OX Ω1X . A Griffiths transverse filtration is a decreasing filtration of V by strict subbundles V = F0 ⊃ F1 ⊃ F2 ··· ⊃ Fk = 0 which satisfies the Griffiths transversality condition ∇ : F p → F p−1 ⊗OX Ω1X . p p p p+1 . Define the OX In this case put E := GrF (V ) := p E with E := F /F linear map θ : E p → E p−1 ⊗OX Ω1X using ∇. Precisely, if e is a section of E p , lift it to a section f of F p and note that by the transversality condition, ∇f is a section of F p−1 ⊗OX Ω1X . Define θ(e) to be the projection of ∇(f ) into E p−1 ⊗OX Ω1X . Again by the transversality condition, θ(e) is independent of the choice of lifting f . If a is a section of OX then ∇(af ) = a∇(f ) + f ⊗ da but the second term projects to zero in E p−1 ⊗OX Ω1X , so θ(ae) = aθ(e), that is θ is OX -linear. We call (E, θ) the associated-graded Higgs bundle corresponding to (V, ∇, F • ). Griffiths-transverse filtrations are the first main piece of structure of variations of Hodge structure, and in that context the map θ is known as the “Kodaira-Spencer map”. This kind of filtration of a bundle with connection was generalized to the notion of “good filtration” for D-modules, and has appeared in many places. A complex variation of Hodge structure consists of a (V, ∇, F • ) such that furthermore there exists a ∇-flat hermitian complex form which is nondegenerate on each piece of the filtration, and with a certain alternating positivity property (if we split the filtration by an orthogonal decomposition, then the form should have sign (−1)p on the piece splitting F p /F p+1 ). For a VHS, the associated-graded Higgs bundle (E, θ) is semistable. The historical variation of Hodge structure picture is motivation for considering the filtrations and Kodaira-Spencer maps, however we don’t use the polarization which is not a complex holomorphic object. Instead, we concentrate on the semistability condition. Definition 2.1. We say that (V, ∇, F • ) is gr-semistable (resp. gr-stable) if the associated-graded Higgs bundle (E, θ) is semistable (resp. stable) as a Higgs bundle. The Higgs bundle (E, θ) is a fixed point of the C∗ action, which is equivalent to saying that we have a structure of system of Hodge bundles [52, 4.1], i.e. a decomposition E = p E p with θ : E p → E p−1 ⊗OX Ω1X .

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Remark 2.2. If (E, θ) is a system of Hodge bundles, then it is semistable as a Higgs bundle if and only if it is semistable as a system of Hodge bundles. In particular, if it is not a semistable Higgs bundle then the maximal destabilizing p subobject H ⊂ E is itself a system of Hodge bundles, that is H = H with H p := H ∩ E p . Indeed, if (E, θ) is not semistable, it is easy to see by uniqueness of the maximal destabilizing subsheaf that H must be preserved by the C∗ action. Such objects have appeared in geometric Langlands theory under the name “opers”: Example 2.3. An oper is a vector bundle with integrable connection and Griffiths transverse filtration (V, ∇, F • ) such that F • is a full flag, i.e. E p = GrFp (V ) are line bundles, and ∼ =

θ : E p → E p−1 ⊗OX Ω1X are isomorphisms. If g ≥ 1 then an oper is gr-semistable. This motivates the following definition and terminology. Definition 2.4. A partial oper is a vector bundle with integrable connection and Griffiths-transverse filtration (V, ∇, F • ) which is gr-semistable. Every integrable connection supports at least one partial oper structure. Theorem 2.5. Suppose (V, ∇) is a vector bundle with integrable connection on a smooth projective curve X. Then there exists a gr-semistable Griffiths-transverse filtration giving a partial oper structure (V, ∇, F • ). The proof will be given in the next section.

3. Construction of a gr-semistable filtration

If (E = E p , θ) is a system of Hodge bundles, for any k let E [k] denote the system of Hodge bundles with Hodge index shifted, so that (E [k] )p := E p−k . Let (V, ∇) be fixed. Suppose we are given a Griffiths-transverse filtration F • such that (GrF (V ), θ) is not a semistable Higgs bundle. Choose H to be the maximal destabilizing subobject, which is a sub-system of Hodge bundles of GrF (V ). Thus H= H p , H p ⊂ GrFp (V ) = F p V /F p+1 V. Note that the H p are strict subbundles here. We can consider H p−1 ⊂ V /F p V which is again a strict subbundle. Define a new filtration G• of V by V /F p V p . (3.1) G := ker V → H p−1 The condition θ(H p ) ⊂ H p−1 ⊗OX Ω1X means that the new filtration G• is again Griffiths-transverse. We have exact sequences p 0 → GrFp (V )/H p → GrG (V ) → H p−1 → 0


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which, added all together, can be written as an exact sequence of systems of Hodge bundles (3.2)

0 → GrF (V )/H → GrG (V ) → H [1] → 0.

We would like to show that the process of starting with a filtration F • and replacing it with the modified filtration G• stops after a finite number of steps, at a gr-semistable filtration. As long as the result is still not gr-semistable, we can choose a maximal destabilizing subobject and continue. To show that the process stops, we will define a collection of invariants which decrease in lexicographic order. For a system of Hodge bundles E, let β(E) denote the slope of the maximal destabilizing subobject. Let ρ(E) denote the rank of the maximal destabilizing subobject. Define the center of gravity to be rk(E p ) · p ζ(E) := . rk(E) This measures the average location of the Hodge indexing. In particular, suppose U = E [k] . Then U p = E p−k so prk(E p−k ) (p + k)rk(E p ) prk(U p ) = = = ζ(E) + k. ζ(U ) = rk(U ) rk(E) rk(E) This gives the formula (3.3)

ζ(E [k] ) = ζ(E) + k.

Now for any non-semistable system of Hodge bundles E, let H denote the maximal destabilizing subobject and put γ(E) := ζ(E/H) − ζ(H). This normalizes things so that γ(E [k] ) = γ(E). Denote βF := β(GrF (V ), θ), ρF := ρ(GrF (V ), θ), and γF := γ(GrF (V ), θ). Lemma 3.1. In the process F • → G• described above, and assuming that G• is also not gr-semistable, then the triple of invariants (β, ρ, γ) decreases strictly in the lexicographic ordering. In other words, (βG , ρG , γG ) is strictly smaller than (βF , ρF , γF ). Proof. Use the exact sequence (3.2) and the formula (3.3). If K denotes the maximal destabilizing subobject of GrG (V ) then by (3.2), slope(K) ≤ slope(H). u If equality then the map K → H [1] is injective, otherwise the kernel would be semistable of the same slope and couldn’t map to GrF (V )/H. Thus rk(K) ≤ rk(H), and again in case of equality then u would be an isomorphism, therefore ζ(K) = ζ(H) + 1 by 3.3 but ζ(GrG (V )/K) = ζ(GrF (V )/H) so γG < γF . In order to show that the γ(E) remain bounded, observe the following. Lemma 3.2. If (V, ∇) is an irreducible connection, and F • is a Griffithstransverse filtration, then there are no gaps in the E p = GrFp (V ), that is there are no p < p < p such that E p = 0 but E p = 0 and E p = 0. Proof. If there were such a gap, then by Griffiths transversality the piece F p = F would be a nontrivial subbundle preserved by the connection, contradicting irreducibility of (V, ∇). p+1

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Lemma 3.3. Suppose (V, ∇) is an irreducible connection. For all E = GrF (V ) coming from non-gr-semistable Griffiths-transverse filtrations of our fixed (V, ∇), each of the invariants β(E), ρ(E) and γ(E) can take on only finitely many values. 3.2.

Proof. For the slope and rank this is clear. For γ(E) it follows from Lemma

Proof of Theorem 2.5. Assume first of all that (V, ∇) is irreducible. Under the operation F • → G• , the triple of invariants (β, ρ, γ) (which takes only finitely many values by Lemma 3.3) decreases strictly in the lexicographic ordering by Lemma 3.1 until we get to a gr-semistable filtration. For a general (V, ∇), glue together the filtrations provided by the previous paragraph on the semisimple subquotients of its Jordan-H¨ older filtration. This can be done after possibly shifting the indexing of the filtrations. Question 3.4. Understand what would happen if we tried to do the above procedure in the case dim(X) ≥ 2 where the destabilizing subobjects could be torsionfree sheaves but not reflexive. 4. Interpretation in terms of the nonabelian Hodge filtration Consider the “nonabelian Hodge filtration” moduli space MHod (X, r) = {(λ, V, ∇), ∇ : V → V ⊗ Ω1X , ∇(ae) = a∇(e) + λd(a)e} with its map λ : MHod (X, r) → A1 , such that: —λ−1 (0) = MH is the Hitchin moduli space of semistable Higgs bundles of rank r and degree 0; and —λ−1 (1) = MDR is the moduli space of integrable connections of rank r. The group Gm acts on MHod over its action on A1 , via the formula t·(λ, V, ∇) = (tλ, V, t∇). Therefore all of the fixed points have to lie over λ = 0, that is they are in MH . We can write Pα (MH )Gm = α

as a union of connected pieces. There is a unique piece denoted P0 along which all Gm -orbits of MH are incoming. As discussed in §5.4 it corresponds to the moduli of semistable bundles with θ = 0. For the other pieces, certain invariants such as the ranks and degrees of the Hodge bundles serve to distinguish components, however the decomposition into connected components Pα could be finer. Lemma 4.1. For any y ∈ MHod , the limit limt→0 t · y exists, and is in one of the Pα . The limit is the associated-graded Higgs bundle of any gr-semistable Griffiths-transverse filtration on the connection corresponding to y. Proof. An abstract proof was given in [54]. For λ(y) = 0 in which case we may assume λ(y) = 1 i.e. y ∈ MDR , the convergence can also be viewed as a corollary of Theorem 2.5. Indeed, y corresponds to a vector bundle with integrable connection (V, ∇) and if we choose a gr-semistable Griffiths-transverse filtration F • then the limit can be calculated as lim (V, λ∇) = (GrF (V ), θ).

λ→0

This can be seen as follows. The Rees construction gives a locally free sheaf ξ(V, F ) := λ−p F p V ⊗ OX×A1 ⊂ V ⊗ OX×Gm .


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over X×A1 and by Griffiths transversality the product λ∇ extends to a λ-connection on ξ(V, F ) in the X-direction (here λ denotes the coordinate on A1 ). This family provides a morphism A1 → MHod compatible with the Gm -action and having limit point (ξ(V, F ), λ∇)|λ=0 = (GrF (V ), θ). If λ(y) = 0 i.e. y ∈ MH then a construction similar to that of Theorem 2.5 gives a calculation of the limit. Alternatively, on the moduli space of Higgs bundles it is easy to see from the properness of the Hitchin map [28] that the limit exists. We next note that the limit is unique. Corollary 4.2. If F • and G• are two gr-semistable filtrations for the same (V, ∇) then the Higgs bundles (GrF (V ), θF ) and (GrG (V ), θG ) are S-equivalent, that is the associated-graded polystable objects corresponding to their Jordan-H¨ older filtrations, are isomorphic. Proof. The moduli space is a separated scheme whose points correspond to S-equivalence classes of objects. Now, given that the limiting Higgs bundle is unique, we can use it to measure whether the partial oper structure will be unique or not: Proposition 4.3. Suppose (V, ∇) is a vector bundle with integrable connection and let (E, θ) be the unique polystable Higgs bundle in the S-equivalence class of the limit. Then the gr-semistable Griffiths transverse filtration for (V, ∇) is unique up to translation of indices, if and only if (E, θ) is stable. Proof. Consider the Rees families (ξ(V, F ), λ∇) and (ξ(V, G), λ∇) on X × A1 constructed for 4.1 above. These are vector bundles with ΛHod -connections on X × A1 (see [53, I, p. 87], [54]). For any t ∈ A1 the space of morphisms from (ξ(V, F ), λ∇)t to (ξ(V, G), λ∇)t is the zero-th hypercohomology on X × {t} of a complex on X × A1 flat over OA1 , obtained by the Rees construction applied to the de Rham complex of V ∗ ⊗ V using the filtrations F and G. As is classically wellknown, see [42] for example, there is a complex of finite rank bundles over A1 whose fiber over t calculates the space of morphisms, so the dimension is semicontinuous in t. For t = 0 the dimension is at least 1, containing the identity of (V, ∇). If the unique limiting Higgs bundle (E, θ) is stable then the dimension of the space of morphisms at t = 0 is equal to 1, showing that the dimension is always 1. It follows that the Hom between these, relative to the base A1 , is a rank one locally free sheaf over A1 with action of Gm , and this relative Hom commutes with base change. After appropriately shifting one of the filtrations we get a Gm -invariant section which translates back to equality of the filtrations. On the other hand, if the limiting Higgs bundle is not stable, we can choose a sub-system of Hodge bundles and apply the construction (3.1) to change the filtration. The exact sequence (3.2) shows that the new filtration is different from the old. 5. The oper stratification As is generally the case for a Gm -action, the map y → limt→0 t · y is a constructible map from MHod to the fixed point set α Pα .

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Proposition 5.1. For any α, the subset Gα ⊂ MDR (X, r) consisting of all points y such that limt→0 t · y ∈ Pα is locally closed. These partition the moduli space into the oper stratification MDR (X, r) = Gα . α

Furthermore, for any point p ∈ Pα (which corresponds to an S-equivalence class of systems of Hodge bundles), the set Lp ⊂ MDR (X, r) of points y with limt→0 t·y = p, is a locally closed subscheme (given its reduced subscheme structure). Proof. This is the classical Bialynicki-Birula theory [31]: the moduli space can be embedded Gm -equivariantly in PN with a linear action; the stratification and fibrations are induced by those of PN but with refinement of the Pα into connected components. For the moduli of Higgs bundles we have a similar stratification with strata

α ⊂ MH (X, r) again defined as the sets of points such that limt→0 t · y ∈ denoted G Pα . 5.1. Opers. The uniformizing Higgs bundles [28] are of the form E = E 1 ⊕ −1 , such that E 0 = L ⊕ L a direct sum of two line bundles with E 0 = L ∼ = L ⊗ KX θ : E 1 → E 0 ⊗ Ω1X is an isomorphism. The space of these is connected, determined by the choice of L ∈ P icg−1 (X). For bundles of rank r, one of the connected components Pα is the space of systems of Hodge bundles of the form E 0 ⊕ · · · ⊕ E r where the E i are line bundles, ∼ = with θ : E i → E i−1 ⊗ Ω1X . Any such E is isomorphic to a symmetric power r−1 ∼ E = Sym (F ) for F = L ⊕ L of rank 2 as above. These systems of Hodge bundles are rigid up to tensoring with a line bundle, indeed once E 0 is chosen the ⊗i remaining pieces are determined by E i = E 0 ⊗ KX . Thus, the set of them forms a connected component Pα where α is just a notation for its indexing element. The determinant map Pα → P ic0 (X) is finite. The classical moduli space of GL(r)-opers [5] [20] is the subset Gα defined in Proposition 5.1 corresponding to the space Pα of symmetric powers of uniformizing Higgs bundles. The stratum of classical opers Gα is closed, because the corresponding stratum

α is closed in MH . It also has minimal dimension among the strata, as can be G seen from Lemma 7.3 below. We conjecture that it is the unique closed stratum and the unique stratum of minimal dimension. These are easy to see in the case of rank 2, see §7.3. 5.2. Variations of Hodge structure. If (V, ∇, F • , ·, ·) is a polarized complex VHS then the underlying Hodge filtration F • (which is Griffiths-transverse by definition) is gr-semistable. The Higgs bundle (GrF (V ), θ) is the one which corresponds to (V, ∇) by the nonabelian Hodge correspondence. This implies that if (V, ∇) is a VHS with irreducible monodromy representation then it is gr-stable. In this case the filtration F • is unique and the process of iterating the destabilizing modification described in §3 provides a construction of the Hodge filtration starting from just the bundle with its connection (V, ∇). For any stratum Gα as in Proposition 5.1, let GVHS ⊂ MDR (X, r) be the real α analytic moduli space of polarized complex variations of Hodge structure whose


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underlying filtered bundle is in Gα . We have a diagram of real analytic varieties GVHS α ∼ =

real

→

Gα .

Pα Under the nonabelian Hodge identification ν : MDR (X, r) ∼ = MH (X, r) the space GVHS of variations of Hodge structure is equal to P and the diagonal isomorphism α α in the above diagram is the identity when viewed in this way.

α under the The other points of Gα don’t necessarily correspond to points of G nonabelian Hodge identification ν, and indeed it seems reasonable to make the Conjecture 5.2.

α ). GVHS = Gα ∩ ν −1 (G α

In a similar vein, let MB (X, r)R denote the real subspace of representations which go into some possibly indefinite unitary group U (p, q). Lemma 5.3. Restricting to the subset of smooth points, GVHS is a connected α component of Gα ∩ MB (X, r)R . The proof will be given in §7.1 below. On the other hand, it is easy to see that there are other connected components too, for example when p, q > 0 a general representation π1 → U (p, q) will still correspond to a stable vector bundle, so it gives a point in G0 ∩ MB (X, r)R which is not a unitary representation. These points probably correspond to twistor-like sections of Hitchin’s twistor space, but which don’t correspond to preferred sections. Question 5.4. How can we distinguish GVHS among the connected components α of Gα ∩ MB (X, r)R ? 5.3. Families of VHS. We now explain one possible motivation for looking at the stratification by the Gα , related to Lemma 5.3. Suppose given a smooth projective family of curves f : X → Y over a base which is allowed to be quasiprojective. Let Xy := f −1 (y) be the fiber over a point y ∈ Y . Then π1 (Y, y) acts on MB (Xy , r). The fixed points are the representations which come from global representations on the total space X of the fibration (approximately, up to considerations involving the center of the group and so on). In the de Rham point of view there is a “connection” on the relative de Rham moduli space MDR (X/Y, r) → Y . This can be called the “nonabelian Gauss-Manin connection” but is also known as the system of “isomonodromic deformation” equations, or Painlevé VI for the universal family of 4-pointed P1 ’s. The fixed points of the above action on the Betti space correspond to global horizontal sections of the n.a.GM connection. These have been studied by many authors. The global horizontal sections will often be rigid (hence VHS’s) and in any case can be deformed, as horizontal sections, to VHS’s. Let ρ(y) ∈ MB (Xy , r) denote a global horizontal section which is globally a VHS; then we will have ρ(y) ∈ GVHS (Xy ) α for all y in a neighborhood of an initial point y0 . The combinatorial data corresponding to the stratum α will be invariant as y moves; let’s assume that we can say that the index sets of our stratifications remain locally constant as a function of y, and that ρ(y) stays in the “same stratum”—although that would require further

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proof. Then we have, in particular, ρ(y) ∈ Gα (y). The stratum Gα (y) will not be fully invariant by the n.a.GM connection [26], so ρ(y) has to lie in the subset of points which, when displaced by the n.a.GM connection, remain in the same (i.e. corresponding) stratum. We get a system of equations constraining the point ρ(y). One can hope that in certain special cases, these equations have only isolated points as solutions in each fiber Gα (y). A program for finding examples of global horizontal sections ρ would be to look for these isolated points and then verify if the family of such points is horizontal. 5.4. The open stratum. At the other end of the range of possible dimensions of strata, is the unique open stratum G0 in each component of the moduli space. When X is a smooth compact curve of genus ≥ 2, the open stratum consists of connections of the form (V, ∂ + A) where V is a polystable vector bundle, ∂ is the unique flat unitary connection, and A ∈ H 0 (End(V ) ⊗ Ω1X ). The Griffithstransverse filtration is trivial, and the corresponding stratum of systems of Hodge bundles is P0 = U(X, r), the moduli space of semistable vector bundles on X. The

0 = T ∗ P0 and G0 is a principal T ∗ P0 Higgs stratum is just the cotangent bundle G VHS torsor over P0 . The space G0 is just the space of unitary representations. The situation becomes more interesting when we look at parabolic bundles or bundles on an orbifold. 6. The parabolic or orbifold cases Let (X, {x1 , . . . , xk }) be a curve with some marked points, and fix semisimple unitary conjugacy classes C1 , . . . , Ck ⊂ GL(r). We can consider the various moduli spaces of representations, Higgs bundles, connections, or λ-connections on U := X − {x1 , . . . , xk } with logarithmic structure at the points xi and corresponding monodromies contained in the conjugacy classes Ci respectively. These objects correspond to parabolic vector bundles with real parabolic weights and λ-connection ∇ respecting the parabolic filtration and inducing the appropriate multiple of the identity on each graded piece. At λ = 0 the residues of the Higgs field are zero since we are assuming that the conjugacy classes are unitary, whereas for λ = 0 the relationship between parabolic weights, residue and conjugacy classes is as described in [51] [41]. If in addition the conjugacy classes are assumed to be of finite order, the parabolic weights should be rational, and our objects may then be viewed as lying on an orbicurve or Deligne-Mumford stack X with ramification orders mi corresponding to the common denominators of the weights at xi [7] [13] [30]. Denote by MH (X ; C1 , . . . , Ck ) ⊂ MHod (X ; C1 , . . . , Ck ) ⊃ MDR (X ; C1 , . . . , Ck ) etc the resulting moduli spaces. One could further assume for simplicity that there exists a global projective etale Galois covering Z → X with Galois group G. The map Z → X is a ramified Galois covering such that the numbers of branches in the ramification points above xi are always equal to mi . Local systems or other objects on X are the same thing as G-equivariant objects on Z. This enables an easy construction of the moduli stacks and spaces. More general constructions of moduli spaces of parabolic objects can be found in [58] [57] [43] [37] [32] [29] [15]. The correspondence between Higgs bundles and local systems works in this case, as can easily be seen by pulling back to the covering Z (although the analysis can also be done directly). All of the other related structures also work the same


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way. The C∗ action preserves the conjugacy classes since they are assumed to be unitary. There is a proper Hitchin fibration on the space of Higgs bundles [19] [58] [37], and similarly the nonabelian Hodge filtration satisfies the positive weight property saying that limits exist as in Lemma 4.1. A first classical case appearing already in the paper of Narasimhan and Seshadri [45] is when there is a single parabolic point x1 and the residual conjugacy class of the connection is that of the scalar matrix d/r. Orbifold bundles of this type correspond to usual bundles on X of degree d, which can have projectively flat connections. This led to moduli spaces in which all semistable points are stable when (r, d) = 1. The semistable ⇒ stable phenomenon occurs fairly generally in the orbifold or parabolic setting. The Cj have to satisfy the equation det(Cj ) = 1. However, if the eigenvalues of the individual blocks are chosen sufficiently generally, it often happens that there is no sub-collection of 0 < r < r eigenvalues at each point, such that the product over all points is equal to 1. We call this Kostov’s genericity condition. It implies that all representations are automatically irreducible, or in terms of vector bundles with connection or Higgs bundles, semistable objects are automatically stable. This situation is particularly relevant for our present discussion: Lemma 6.1. Suppose the conjugacy classes C1 , . . . , Ck satisfy Kostov’s genericity condition, then any vector bundle with connection (V, ∇) ∈ MDR (X ; C1 , . . . , Ck ) has a unique partial oper structure, the filtration being unique up to shifting the indices. Also, in this case the moduli stacks M• (X ; C1 , . . . , Ck ) are smooth, and they are Gm -gerbs over the corresponding moduli spaces M• (X ; C1 , . . . , Ck ). Proof. If all semistable objects are automatically stable, then the same is true for the associated-graded Higgs bundles. Proposition 4.3, which works equally well in the orbifold case, implies that the filtration is unique up to a shift. For the statement about moduli stacks, stable objects admit only scalar automorphisms, and the GIT construction of the moduli spaces of stable objects yields etale-locally fine moduli, which is enough to see the gerb statement. For smoothness, the obstructions land in the trace-free part of H 2 which vanishes by duality. Note that when there is a single point and the conjugacy class is scalar multiplication by a primitive r-th root of unity, this is Kostov-generic, and we are exactly in the original situation considered by Narasimhan and Seshadri corresponding to bundles of degree coprime to the rank. Corollary 6.2. If the conjugacy classes satisfy Kostov’s genericity condition, then MHod (X ; C1 , . . . , Ck ) is smooth over A1 , and the fixed point sets of the

α → Pα are smooth Gm -action Pα are smooth. The projections Gα → Pα and G fibrations topologically equivalent to the normal bundle NGα /Pα → Pα . Proof. The fixed-point sets are smooth by the theory of Bialynicki-Birula. Let GHod ⊂ MHod (X ; C1 , . . . , Ck ) denote the subset of points whose limit lies in α

α . By the action of Pα . It is smooth, and the fiber over λ = 0 is the smooth G Hod 1 Hod Gm , the map Gα → A is smooth everywhere. Let Gα,V HS ⊂ GHod denote α the subspace of points which correspond to variations of Hodge structure. The

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preferred-section trivialization gives 1 ∼ GHod α,V HS = Pα × A

compatibly with the Gm -action. Let T ⊂ GHod be a tubular neighborhood of GHod α α,V HS , and denote by T0 the fiber over λ = 0. We can suppose that ∂T0 is transverse to the vector field defined by the action of R∗>0 ⊂ Gm . Consequently, there is such that ∂T is transverse to this vector field for any |λ| < 2. Choose a trivialization F : ∂T0 × A1 ∼ = ∂T compatible with the map to A1 . Then define a map R : R∗>0 × ∂T0 → Gα , R(t, x) := t · F (x, t−1 ). This is a diffeomorphism in the region t > −1 /2. On the other hand, let T be the fiber over λ = and consider the map −1 : T → Gα . This glues in with the map R defined on the region t ≥ −1 to give a topological trivialization of Gα . Everything can be done relative to Pα so we get a homeomorphism between Gα → Pα and the

α → Pα . normal bundle of Pα in Gα . The same happens for G 6.1. The Hodge type of the open stratum. The open stratum consists of variations of Hodge structure of a certain type. This type can change as we move the conjugacy classes. For example, the set of vectors of conjugacy classes (C1 , . . . , Ck ) for k points in P1 , for which there exists a unitary representation, is defined by some inequalities on the logarithms of the eigenvalues [1] [6] [8] [15] [56]. If we fix a collection of partitions at each point, then the space of unitary conjugacy classes corresponding to those partitions is a real simplex, and the set of finite-order conjugacy classes is the set of rational points therein. The set of non-Kostov-generic points is a union of hyperplanes, as can be seen from the identification between residues and parabolic weights, so the set of Kostov-generic points decomposes into a union of open chambers bounded by pieces of hyperplanes (in particular, the chambers are polytopes). Suppose that we know that the varieties M• (X ; C1 , . . . , Ck ) are connected— this is the case, for example, if at least one of the conjugacy classes has distinct eigenvalues [33]. Then there is a single open stratum. If the genericity condition is satisfied so all objects are stable, then the partial oper structure at a general point is unique, and its Hodge type (up to translation) depends only on the conjugacy classes. Furthermore, given a Kostov-generic collection of unitary conjugacy classes (C1 , . . . , Ck ) and a parabolic system of Hodge bundles with these conjugacy classes, if we perturb slightly the parabolic weights then the object remains stable. This shows that the Hodge type for the unique open stratum is constant on the Kostovgeneric chambers. A limit argument should show that if we fix a particular Hodge type {hp,q }, then the set of vectors of conjugacy classes (C1 , . . . , Ck ) for which the points in the open stratum admit a partial oper structure with given hp,q , is closed. We don’t do that proof here, though—let’s just assume it’s true. It certainly holds for the unitary case h0,0 = r. This closed set is then a closed polytope. For the unitary case, it is the polytope found and studied by Boden, Hu, Yokogawa [14] [15], Biswas [8] [9], Agnihotri, Woodward [1], Belkale [6]. The boundary is a union of pieces of the non-Kostov-generic hyperplanes.


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The fact that the Hodge type varies in a way which is locally constant over polytopes fits into the general philosophy which comes out of the results of Libgober [36] and Budur [17]. There is a direct relationship in the rigid case: by Katz’s algorithm the unique point in a rigid moduli space can be expressed motivically, and the Hodge type is locally constant on polytopes by the theory of [36] [17]. There seems to be a further interesting phenomenon going on at the boundary of these polytopes: the complex variations of Hodge structure corresponding to the fixed point limits of points in the open stratum, become automatically reducible. In the unitary case this is pointed out in several places in the references (see for example Remark (4) after Theorem 7 on page 75 of [6], or also the last phrase of Theorem 3.23 of [9]): irreducible unitary representations exist only in the interior of the polytope. A sketch of proof in general is to say that if there is a stable system of Hodge bundles with given parabolic structure then the parabolic structure can be perturbed keeping stability and keeping the same Hodge type (it is the same argument as was used above at the interior points of the chambers, which is indeed essentially the same as in the references). In other words, at the walls between the chambers, points in the open stratum become gr-semistable but not gr-stable. Now, the non-uniqueness of the partial oper structure, Proposition 4.3, is exactly what allows the Hodge type to jump. For generic (C1 , . . . , Ck ) where the moduli space is smooth, the space of complex variations of Hodge structure corresponding to the open stratum is a real form of the Betti moduli space (extend to the parabolic case the proof of Lemma 5.3 in §7.1 below). It is defined algebraically in MB (X; C1 , . . . , Ck ) as a connected component of the space of representations into some U (p, q). However, at boundary points (C1 , . . . , Ck ) between the different Hodge polytopes this subspace is concentrated on the singular locus of reducible representations (whereas in most cases there still exist irreducible representations). The open stratum is presented as a conic fibration over a fixed point set inside the singular locus. The reader will be convinced that this kind of thing can happen by looking at the case of the real circle x2 +y 2 = t: for t > 0 it is a real form of the complex quadratic curve, but at t = 0 the real points are just the singularity of two crossing complex lines. It is an interesting question to understand what the degeneration of the real form of the smooth space looks like for the case of MB (X; C1 , . . . , Ck ). 7. Deformation theory The deformation theory follows [11] [16] [39] and others. The discussion extends without further mention to the case where X is an orbifold. Further work is needed [59] [16] for more general parabolic cases corresponding for example to non-unitary conjugacy classes of local monodromy operators. Suppose (V, ∇) is an irreducible connection. Consider the complex End(V ) ⊗ Ω•X := [End(V ) → End(V ) ⊗ Ω1X ]. d

The space of infinitesimal deformations, or the tangent space to the moduli stack MDR at (V, ∇), is Def (V, ∇) = H1 (End(V ) ⊗ Ω•X ). Given a Griffiths-transverse filtration F • we get a decreasing filtration of the complex End(V ) ⊗ Ω•X defined by F p (End(V )) := {ϕ ∈ End(V ), ϕ : F q V → F p+q V },

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F p (End(V ) ⊗ Ω1X ) := F p−1 (End(V )) ⊗ Ω1X . The F p (End(V ) ⊗ Ω•X ) are subcomplexes, so we can take the spectral sequence of the filtered complex (End(V ) ⊗ Ω•X , F • ). It induces a filtration F • Def (V, ∇) and the spectral sequence is (7.1)

Hi (GrFp (End(V ) ⊗ Ω•X )) = E1p,i−p ⇒ GrFp Hi (End(V ) ⊗ Ω•X ).

The obstruction theory is controlled by the H2 of the trace-free part, so if (V, ∇) is irreducible then the deformations are unobstructed, there are no problems with extra automorphisms, and the moduli space is smooth. The limiting system of Hodge bundles (GrF (V ), θ) has its own deformation theory in the world of Higgs bundles. The complex End(GrF (V )) ⊗ Ω•X is made as above, but the differentials using θ are OX -linear maps. The tangent space to the moduli stack MH at (GrF (V ), θ) is calculated as Def (GrF (V ), θ) = H1 (End(GrF (V )) ⊗ Ω•X ). The complex End(GrF (V )) ⊗ Ω•X has a direct sum decomposition, and indeed it is the associated-graded of the previous one: End(GrF (V )) ⊗ Ω•X = GrF (End(V ) ⊗ Ω•X ). This tells us that the E1•,• term of the previous spectral sequence corresponds to the deformation theory of (GrF (V ), θ). Lemma 7.1. If (GrF (V ), θ) is stable then the spectral sequence (7.1) degenerates at E1 with

θ (7.2) GrFp (Def (V, ∇)) = H1 GrFp (End(V )) → GrFp−1 (End(V )) ⊗ Ω•X . Let T (Gα /Pα ) denote the relative tangent bundle of the fibration Gα → Pα where it is smooth. Then the two middle subspaces of the filtration at p = 0, 1 are interpreted in a geometric way as F 0 Def (V, ∇) = T (Gα )(V,∇) ,

and F 1 Def (V, ∇) = T (Gα /Pα )(V,∇) .

Proof. Since (GrF (V ), θ) is stable, H0 (GrF (End(V ) ⊗ Ω•X )) = C,

H2 (GrF (End(V ) ⊗ Ω•X )) = C

and these are the same as at the limit of the spectral sequence. By invariance of the Euler characteristic the same must be true for H1 so the spectral sequence degenerates. The geometric interpretation may be seen by looking at a cocycle description: F 0 Def (V, ∇) is the space of deformations of the bundle with its filtration (V, F • , ∇). Following out 4.1, this is the tangent space to Gα ; while F 1 Def (V, ∇) is the space of deformations of (V, F • , ∇) inducing the trivial deformation on the associated-graded so it is the tangent space to the fiber of the projection Gα → Pα . 7.1. At a variation of Hodge structure. Suppose (V, ∇) ∈ GVHS is an α irreducible complex variation of Hodge structure. Then End(V ) is a real VHS of weight 0, independent of the scalar choice of polarization on V . The hypercohomology Def (V, ∇) = H1 (End(V ) ⊗ Ω•X ) is a real Hodge structure of weight 1, with the same Hodge filtration as defined above and which we write as HR ⊂ Def (V, ∇) ∼ Hk,1−k . =

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This decomposition splits the Hodge filtration and is naturally isomorphic to the decomposition (7.2). The real subbundle of End(V ) is defined to consist of matrices which infinitesimally preserve the polarization form, so the real subspace HR is the space of deformations of the monodromy representation globally preserving the polarization form, i.e. as a representation in the real group U (p, q). In other words it is the tangent space to MB (X, r)R in the notations of Lemma 5.3. and Gα ∩ MB (X, r)R have the same tanProof of Lemma 5.3. Both GVHS α k,1−k gent spaces at (V, ∇). The tangent space of Gα is the sum F 0 = . k≥0 H 0 1,0 0,1 The intersection of F with HR is (H ⊕ H )R which is the tangent space of GVHS . α The symmetry of Hodge numbers extends to other points too: Corollary 7.2. For any gr-stable point (V, ∇), not necessarily a VHS, let F • be the unique (up to shift) partial oper structure. Then the induced filtration on the tangent space of the moduli stack at (V, ∇) satisfies dim GrFp Def (V, ∇) = dim GrF1−p Def (V, ∇). Proof. Let (V , ∇ ) denote the VHS corresponding to the system of Hodge bundles (GrF (V ), θ). Note that (GrF (V ), θ ) ∼ = (GrF (V ), θ), so p p 1 ∼ Gr Def (V, ∇) = H (End(GrF (V )) ⊗ Ω• ) ∼ = Gr Def (V , ∇ ). X

F

F

∼ Therefore we can write = Hp,1−p (End(V ) ⊗ Ω•X ), and these spaces are associated to a real Hodge structure of weight one, so we get the claimed symmetry of dimensions. GrFp Def (V, ∇)

In the Hitchin moduli space the tangent space at a point (E, θ) ∈ Pα decomposes as a direct sum under the Gm -action, and this decomposition is compatible with the filtration, so it gives a splitting: p T (MH )(E,θ) ∼ GrF T (MH )(E,θ) ∼ Hp,1−p . = = The tangent space to the fixed point set is GrF0 T (MH )(E,θ) = T (Pα )(E,θ) , whereas the pieces p < 0 may be identified as the “outgoing” directions for the Gm -action. If Pα = P0 is the lowest piece corresponding to the open stratum G0 ⊂ M then there are no outgoing directions, so the terms of Hodge type (p, 1 − p) vanish for p < 0. By symmetry they vanish for p > 1, which says that GrFp T (MH )(E,θ) = 0 unless p = 0, 1; the same holds for MDR . Furthermore, the two pieces are dual by the symplectic form (see below), so we can identify H0,1 = Gr 0 T (MH )(E,θ) ∼ = T (P0 )(E,θ) , F

∼ H = = T ∗ (P0 )(E,θ) . We even have an isomorphism of fibrations

0 G T ∗ P0 ∼ ↓ = ↓ P0 P0 , 1,0

GrF1 T (MH )(E,θ)

whereas G0 → P0 is a twisted form of the same fibration (it is a torsor). These observations are elementary and classical when X is a smooth projective curve, however they also hold in the parabolic or orbifold case when the weights are generic so that all points are gr-stable.

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This includes cases where there are no stable parabolic vector bundles, but the lowest stratum P0 corresponds to variations of Hodge structure of nontrivial Hodge type. 7.2. Lagrangian property for the fibers of the projections Gα → Pα . There is a natural symplectic form on the tangent bundle to the moduli space over the open subset of stable points, given in cohomological terms [11, §4] as the cup product followed by Lie bracket [ , ]◦∪

H1 (End(V ) ⊗ Ω•X ) ⊗ H1 (End(V ) ⊗ Ω•X ) −→ H2 (End(V ) ⊗ Ω•X ) = C. In the moduli space of connections on a smooth projective curve, the open stratum G0 fibers over the moduli space P0 of semistable vector bundles. In the

0 ∼ Hitchin moduli space, under the isomorphism G = T ∗ P0 , the symplectic form is equal to the standard one [11], in particular the fibers are lagrangian. It has been noticed by several authors [24] [16] [55] [35] [2] that the fibers of the projection G0 → P0 (over stable points) are similarly lagrangian subspaces of the moduli space of connections. We point out that the lagrangian property extends to the fibers in all strata, at gr-stable points. Lemma 7.3. Suppose p = (E, θ) ∈ Pα is a stable system of Hodge bundles. Then the fiber Lp of the projection Gα → Pα over p is a lagrangian subspace of MDR . Proof. Fix a point (V, ∇) which is gr-stable, hence also stable itself. By Lemma 7.1, the tangent space to the fiber of Gα /Pα is the subspace F 1 H1 ⊂ Def (V, ∇). In degree two, F 1 H2 (End(V ) ⊗ Ω•X ) is the full hypercohomology space H2 = C whereas F 2 H2 = 0. On the other hand, the Hodge filtration is compatible with cup product, so ∪ : F 1 H1 ⊗ F 1 H1 → F 2 H2 = 0, which is to say that F 1 H1 ⊂ Def (V, ∇) is an isotropic subspace for the symplectic form. The symmetry property Corollary 7.2 readily implies that F 1 H1 has half the dimension, so it is lagrangian. A natural problem is to understand the relationship between the lagrangian fibers on different strata. Question 7.4. Do the lagrangian fibers of the projections fit together into a smooth lagrangian foliation with closed leaves? This is of course true within a given stratum Gα ; does it remain true as α varies? A heuristic argument for why this might be the case is that if a fiber of Gα → Pα were not closed in MDR , this might lead to a projective curve in MDR which cannot exist since MB is affine. A similar closedness or properness property was proven in [55] for the case of Fuchsian equations. On the other hand, the analogous statement for MH is not true: the fibers of

α → Pα definitely do sometimes have nontrivial closures in MH , indeed any fiber G contained in the compact nilpotent cone will be non-closed.


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If the answer to Question 7.4 is affirmative, it could be useful for the context of geometric Langlands, where the full moduli stack BunGL(r) might profitably be replaced by the algebraic space of leaves of the foliation (see Arinkin’s work [3]). They share the open set of semistable bundles. This could help in the ramified case [23]—when there are no semistable bundles, it would seem logical to consider the lowest stratum P0 parametrizing variations of Hodge structure. 7.3. Nestedness of the stratifications. We are given a stratification or disjoint decomposition into locally closed subsets M = α Gα . Say that it is nested if there is a partial order on the index set such that Gα − Gα = Gβ . β 1, Pe is the space of systems of Hodge bundles which are direct sums of line bundles of degrees e and −e. A deformation theory argument will show that starting with a point in Ge we can deform it to a family of λ-connections which go into the next stratum Ge−1 . At e = 0, the space P0 is the moduli space of rank two semistable vector bundles on X, which is sometimes denoted UX (2), and is known to be an irreducible variety. For e > 0 the space Pe parametrizes Higgs bundles of the form E = E 0 ⊕ E 1,

θ : E 1 → E 0 ⊗ Ω1X ,

where E 0 and E 1 are line bundles of degrees −e and e respectively. We require e > 0 because a Higgs bundle of the above form with deg(E i ) = e = 0 would be semistable but not stable, in the same S-equivalence class as the polystable vector bundle E 0 ⊕ E 1 which is a point in P0 . The map θ is a section of the line bundle (E 1 )∗ ⊗E 0 ⊗Ω1X of degree 2g−2−2e, so e ≤ g−1 and in the case of equality, θ is an isomorphism. Let D be the divisor of θ; it is an effective divisor of degree 2g −2−2e, and the space of such is Sym2g−2−2e (X). Given D and E 1 ∈ P ice (X) then E is determined by E 0 = E 1 ⊗ T X ⊗ OX (D). Thus, Pe ∼ = Sym2g−2−2e (X) × P ice (X) which is a smooth irreducible variety. All points are stable.

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The strata of the oper stratification are enumerated as Ge ⊂ MDR (X, 2) for

e . The upper 0 ≤ e ≤ g − 1; the corresponding strata of MH (X, 2) are denoted G (smallest) stratum e = g − 1 is the case of classical opers. To show the nested property, it suffices to show that Ge ⊂ Ge−1 for e ≥ 2. Note that for e = 1 this is automatic since G0 is an open dense subset so G0 = MDR . We will consider a point (V, ∇) ∈ Ge and deform it to a family (Vt , ∇t ) ∈ Ge−1 for t = 0, with limit (V, ∇) as t → 0. Having (V, ∇, F • ) ∈ Ge means that there is an exact sequence (7.3)

0 → E1 → V → E0 → 0

with E 1 = F 1 and E 0 = F 0 /F 1 . The limit point in Pe is E = E 1 ⊕ E 0 with θ := ∇ : E 1 → E 0 ⊗ Ω1X . Choose a point p ∈ X and let L := E 1 (−p). Let ϕ : L → V be the inclusion. Consider the functor of deformations of (V, ∇, L, ϕ). It is controlled by the hypercohomology of the complex C • := End(V ) ⊕ Hom(L, L) → End(V ) ⊗ Ω1X ⊕ Hom(L, V ) where the differential is the matrix

∇ −◦ϕ

0 ϕ◦−

.

There is a long exact sequence . . . → Hi (Hom(L, L) → Hom(L, V )) → Hi (C • ) → Hi (End(V ) ⊗ Ω•X ) → . . . . The fact that our filtration is gr-stable implies that the spectral sequence associated to the filtered complex (End(V ) ⊗ Ω•X , F • ) degenerates at E1•,• . The VHS on End(GrF (V )) has Hodge types (1, −1) + (0, 0) + (−1, 1) so its H1 has types (2, −1) + . . . + (−1, 2). The associated-graded of H1 (End(V ) ⊗ Ω•X ) has pieces GrF2 H1 = H 0 (Hom(E 0 , E 1 ) ⊗ Ω1X ), GrF1 H1 = H1 (Hom(E 0 , E 1 ) → (Hom(E 0 , E 0 ) ⊕ Hom(E 1 , E 1 )) ⊗ Ω1X ), GrF0 H1 = H1 (Hom(E 1 , E 1 ) ⊕ Hom(E 0 , E 0 ) → Hom(E 1 , E 0 ) ⊗ Ω1X ), GrF−1 H1 = H 1 (Hom(E 1 , E 0 )). The degeneration gives a surjection H1 → → GrF−1 H1 , in other words H1 (End(V ) ⊗ Ω•X ) → H 1 (Hom(E 1 , E 0 )) → 0. The definition L = E 1 (−p) gives an exact sequence of the form (7.4)

0 → Hom(E 1 , E 0 ) → Hom(L, E 0 ) → Cp → 0.

Note that deg(L) > 0 but deg(E 0 ) < 0 so H 0 (Hom(L, E 0 )) = 0, hence the long exact sequence for (7.4) becomes 0 → C → H 1 (Hom(E 1 , E 0 )) → H 1 (Hom(L, E 0 )) → 0. Putting these two together we conclude that the map H1 (End(V ) ⊗ Ω•X ) → H 1 (Hom(L, E 0 )) is surjective, but there is an element in its kernel which maps to something nonzero in H 1 (Hom(E 1 , E 0 )). Another exact sequence like (7.4) gives a surjection H 1 (OX ) = H 1 (Hom(L, L)) → H 1 (Hom(L, E 1 )) → 0.


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The long exact sequence for Hom from L to the short exact sequence (7.3) gives H 1 (Hom(L, E 1 )) → H 1 (Hom(L, V )) → H 1 (Hom(L, E 0 )) → 0. Therefore the map (7.5)

H 1 (Hom(L, L)) ⊕ H1 (End(V ) ⊗ Ω•X ) → H 1 (Hom(L, V ))

is surjective but not an isomorphism. This gives vanishing of the obstruction theory for deforming (V, ∇, L, ϕ). And the map (7.5) has an element in its kernel which maps to something nonzero in H 1 (Hom(E 1 , E 0 )). Hence there is a deformation of (V, ∇, L, ϕ) which doesn’t extend to a deformation of E 1 ⊂ V . A one-parameter family of (Vt , ∇t , Lt , ϕt ) thus gives the deformation from (V, ∇) into the next lowest stratum Ge−1 where Lt of degree e − 1 will be the destabilizing subsheaf. 8. Principal objects Ramanan would want us to consider also principal bundles for arbitrary reductive structure group G. Notice that the category of partial oper structures, while posessing tensor product and dual operations, is not tannakian because morphisms of filtered objects need not be strict. The passage to principal objects should be done by hand. One can construct the various moduli spaces of principal bundles with connection, principal Higgs bundles, and the nonabelian Hodge moduli space of principal bundles with λ-connection MH (X, G) ⊂ MHod (X, G) ⊃ ↓ ↓ {0} ⊂ A1 ⊃

MDR (X, G) ↓ . {1}

In this section we indicate how to prove that the limit limt→0 t · p exists for any p ∈ MHod (X, G). The discussion using harmonic bundles will be more technical than the previous sections of the paper. Embedding G → GL(r), it suffices to show that the map of moduli spaces MHod (X, G) → MHod (X, GL(r)) is finite and then apply Lemma 4.1. Such a finiteness statement was considered for MH and MDR in [53] along with the construction of the homeomorphism between these two spaces. The same statements were mentionned for MHod in [54], however the discussion there was inadequate. The main issue is to prove that the map from the moduli space of framed harmonic bundles to the GIT moduli space MHod (X, G) is proper. The distinct arguments at λ = 0 and λ = 1 given in [53] don’t immediately generalize to intermediate values of λ. This is somewhat similar to—and inspired by—the convergence questions treated recently by Mochizuki in [41]. We give an argument based on the topology of the moduli space together with its Hitchin map; it would be interesting to have a more direct argument with explicit estimates. Fix a basepoint x ∈ X, and let RHod (X, x, G) be the parameter variety for (λ, P, ∇, ζ) where λ ∈ A1 , P is a principal G-bundle with λ-connection ∇ (such that (P, ∇) is semistable with vanishing Chern classes) and ζ : G ∼ = Px is a frame for P at the point x. The group G acts and RHod (X, x, G) → MHod (X, G) is a universal categorical quotient. Indeed, RHod (X, x, G) is constructed first as a closed subscheme of RHod (X, x, GL(r)) for a closed embedding G ⊂ GL(r) by the same construction as in [53, II, §9], which treated the Higgs case. This was based

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on [53, I,§4] which treats the general Λ-module case so the discussion transposes to MHod . One can choose a GL(r)-linearized line bundle on RHod (X, x, GL(r)) for which every point is semistable, which by restriction gives a G-linearized line bundle on RHod (X, x, G) for which every point is semistable. Mumford’s theory then gives the universal categorical quotient. Fix a compact real form J ⊂ G. For any λ ∈ A1 a harmonic metric on a principal bundle with λ-connection (P, ∇) is a C ∞ reduction of structure group h ⊂ P from G to J satisfying some equations (see [41] for example) which interpolate between the Yang-Mills-Higgs equations at λ = 0 and the harmonic map equations J at λ = 1. Let RHod (X, x, G) ⊂ RHod (X, x, G) denote the subset of (λ, P, ∇, ζ) such that there exists a harmonic metric h compatible with the frame ζ at x, in other words ζ : J → hx . This condition fixes h uniquely. J Lemma 8.1. The map RHod (X, x, G) → MHod (X, G) is proper, and induces homeomorphisms J MH (X, G) × A1 ∼ (X, x, G)/J ∼ = MHod (X, G). = RHod

Proof. The moduli space of harmonic λ-connections has a natural product J structure RHod (X, x, G) ∼ = Har J (X, G) × A1 where Har J (X, G) is the space of framed harmonic G-bundles; and the topological quotient by the action of J on the framing is again a product (Har J (X, G)/J) × A1 . The second homeomorphism follows from the properness statement, by the discussion in [53]. There the properness was proven at λ = 0, 1. At λ = 0 we get Har J (X, G)/J ∼ = MH (X, G), which gives the first homeomorphism. At λ = 1 we get Har J (X, G)/J ∼ = MDR (X, G). J Fix G ⊂ GL(r) such that J ⊂ U (r), then RHod (X, x, G) is a closed subset of U(r) RHod (X, x, GL(r)). Using this, one can show that the lemma for GL(r) implies the lemma for G. Suppose now G = GL(r), with compact subgroup J = U (r). The map on the J subset of stable points RHod (X, x, r)s → MHod (X, r)s is proper, using the fact that s MHod (X, r) is a fine moduli space, plus the main estimate for the construction of Hermitian-Einstein harmonic metric solutions by the method of Donaldson’s functional. This estimate is explained for the case of λ-connections in [41]. Given a polystable point (λ, V, ∇) ∈ MHod (X, r) we can associate a polystable Higgs bundle (E, θ) in the preferred section corresponding to the harmonic bundle associated to any harmonic metric on (V, ∇). The (E, θ) is unique up to isomorphism, in particular the value of the Hitchin map Ψ = det(θ − t) ∈ CN is well-defined. This gives a set-theoretically defined map Ψ : MHod (X, r) → CN . J For a sequence of points ρi ∈ RHod (X, x, r), there is a convergent subsequence if and only if the sequence Ψ[ρi ] contains a bounded subsequence. Hence, in order J to prove properness of the map RHod (X, x, r) → MHod (X, r) it suffices to prove that the function Ψ is locally bounded on MHod (X, r). This is obvious on the fiber λ = 0 where the Hitchin map Ψ is an algebraic map. On the fiber λ = 1, an argument using the characterization of harmonic metrics as ones which minimize the energy θ2L2 again shows that Ψ is locally bounded. In particular, the lemma holds over λ = 0 and λ = 1, indeed this was the proof of [53] for the homeomorphism MH (X, r) ∼ = MDR (X, r). Properness over the set of stable points means that the map MHod (X, r)s → MH (X, r)s is continuous, hence Ψ is continuous over MHod (X, r)s . Similarly, it is continuous on any stratum obtained by fixing the type of the decomposition of


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a polystable object into isotypical components. A corollary is that for any point [(λ, V, ∇)] ∈ MHod (X, r), the function C∗ t → Ψ(tλ, V, t∇) is continuous. Suppose we have a sequence of points pi → q converging in MHod (X, r), but where |Ψ(pi )| → ∞. Assume that they are all in the same fiber Mλ over a fixed value 0 = λ ∈ A1 . The points λ−1 pi converge to λ−1 q in MDR (X, r), so (by the energy argument referred to above) we have a bound |Ψ(λ−1 pi )| ≤ C1 . Fix a curve segment γ ⊂ C joining λ−1 to 1 but not passing through 0. The function t → Ψ(tq) is continuous by the previous paragraph, so there is a bound |Ψ(tq)| ≤ C2 for t ∈ γ. On the other hand, again by the continuity of the previous paragraph, for any constant C > C1 there exists a sequence of points ti ∈ γ such that |Ψ(ti pi )| = C. Possibly going to a subsequence, we can assume that ti pi → q as a limit of harmonic bundles. Continuity of the Hitchin map on MH says that |Ψ(q )| = C. The map from the space of harmonic bundles to MHod is continuous so the limit ti pi → q also holds in MHod . On the other hand, we can assume ti → t in γ (again possibly after going to a subsequence), which gives ti pi → tq in MHod . Separatedness of the scheme MHod implies that the topological space is Hausdorff, so tq = q . If C > C2 this contradicts the bound |Ψ(tq)| ≤ C2 for t ∈ γ. We obtain a contradiction to the assumption |Ψ(pi )| → ∞, so we have proven that |Ψ(pi )| is locally bounded. In the fiber over each fixed λ ∈ A1 , this shows properness, hence the homeomorphism statements, hence that Ψ is continuous. Now using the connectedness and separatedness properties of MHod , an argument similar to that of the previous paragraph will allow us to show boundedness of Ψ globally over MHod without restricting to a single fiber. Suppose pi → q in MHod over a convergent sequence λi → λ ∈ A1 , but with |Ψ(pi )| → ∞. Fix a preferred section σ : A1 → MHod and we may assume that σ(λi ) is connected to pi by a path γi : [0, 1] → Mλi . If necessary replacing X by a sufficiently high genus covering, we can view MHod as a family of connected normal varieties. Thus we can assume that the paths γi converge to a path γ connecting q to σ(λ) in the fiber Mλ . Fix a constant C > supt |Ψ(γ(t))|, in particular also C > |Ψ(σ(λi ))|, indeed the Ψ(σ(λi )) are all the same because σ was a preferred section. For large values of i we have |Ψ(σ(λi ))| = |Ψ(γi (0))| < C < |Ψ(γi (1))| = |Ψ(pi )|. Continuity of Ψ in each fiber Mλi shows that there are ti ∈ [0, 1] with |Ψ(γi (ti ))| = C. Going to a subsequence we get convergence of the harmonic bundles associated to the points γi (ti ), keeping the same norm of the Hitchin map. Thus γi (ti ) → q with |Ψ(q )| = C. For a further subsequence, ti → t and as in the previous argument, separatedness of the moduli space implies that q = γ(t), contradicting the choice of C. This proves that Ψ is locally bounded, which in turn implies properness of the map in the first statement of the lemma for GL(r), to complete the proof. Suppose now given an injective group homomorphism between reductive groups G → H. Choose compact real forms J ⊂ G and K ⊂ H such that the homomorphism is compatible: J → H. This gives a diagram J RHod (X, x, G) → ↓ MHod (X, G) →

K RHod (X, x, H) ↓ MHod (X, H)

where the vertical maps are proper by the previous lemma. The upper horizontal map is a closed embedding, indeed we can choose H → GL(r) which also induces

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G → GL(r), and the schemes RHod (X, x, G) and RHod (X, x, H) are by construction closed subschemes of RHod (X, x, GL(r)) [53]. The subsets of framings compatible with the harmonic metrics are closed, so the upper horizontal map is an inclusion compatible with closed embeddings into RHod (X, x, GL(r)), hence it is a closed embedding. In particular, it is proper. This implies that the bottom map is proper. Corollary 8.2. Given a group homomorphism with finite kernel between reductive groups G → H the resulting map on moduli spaces MHod (X, G) → MHod (X, H) is finite. Proof. If G → H is injective, the above argument shows that the map on moduli spaces is proper. It is quasi-finite [53] so it is finite. Then the same argument as in [53] yields the same statement in the case of a group homomorphism with finite kernel between reductive groups. Corollary 8.3. Suppose G is a reductive group. Then for any point p ∈ MHod (X, G) the limit point limt→0 t · p exists and is unique in the fixed point set MH (X, G)Gm . Hence we get a stratification of MDR (X, G) just as in Proposition 5.1. Proof. Choose G → GL(r); apply Lemma 4.1 together with the finiteness of Corollary 8.2 to get existence of the limit. Uniqueness follows from separatedness of the moduli space [53]. The stratification is defined in the same way as in 5.1. The above arguments prove that the limit points exist, however it would be good to have a geometric construction analogous to what we did in §2, §3. This should involve a principal-bundle approach to the instability flag [48]. Question 8.4. How to give an explicit description of the limiting points in terms of Griffiths-transverse parabolic reductions in the case of principal G-bundles? We obtain the oper stratification of MDR (X, G) just as in Proposition 5.1. The smallest stratum consisting of G-opers is treated in much detail in [5]. It would be good to generalize the other elements of our discussion §§5, 6, 7 to the principal bundle case too. The theory of parabolic structures would hit the same complications mentionned by Seshadri in the present conference. The theory of deformations should follow [11]. References [1] S. Agnihotri, C. Woodward, Eigenvalues of products of unitary matrices and quantum Schubert calculus, Math. Res. Lett. 5 (1998), 817–836. [2] J. Aidan, Doctoral dissertation, preliminary version (June 2008). [3] D. Arinkin, Orthogonality of natural sheaves on moduli stacks of SL(2)-bundles with connections on P1 minus 4 points. Selecta Mathematica 7 (2001), 213–239. [4] V. Balaji, I. Biswas, D. Nagaraj, Principal bundles over projective manifolds with parabolic structure over a divisor, Tohoku Math. J. 53 (2001), 337–367. [5] A. Beilinson, V. Drinfeld, Opers, Preprint arXiv:math/0501398. [6] P. Belkale, Local systems on P1 − S for S a finite set, Compositio Math. 129 (2001), 67–86. [7] I. Biswas, Parabolic bundles as orbifold bundles, Duke Math. J., 88 (1997), 305–325. [8] I. Biswas, A criterion for the existence of a parabolic stable bundle of rank 2 over the projective plane, Int. J. Math. 9 (1998), 523–533. [9] I. Biswas, On the existence of unitary flat connections over the punctured sphere with given local monodromy around the puncture, Asian J. Math. 3 (1999), 333–344.


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[10] I. Biswas, T. Gomez, A Torelli theorem for the moduli space of Higgs bundles on a curve, The Quarterly Journal of Mathematics 54 (2003), 159–169. [11] I. Biswas, S. Ramanan, An Infinitesimal study of the moduli of Hitchin pairs, Journal of the London Math. Soc. 49 (1994), 219–231. [12] P. Boalch, From Klein to Painlev´ e via Fourier, Laplace and Jimbo, Proc. London Math. Soc. 90 (2005), 167–208. [13] H. Boden, Representations of orbifold groups and parabolic bundles, Comm. Math. Helv. 66 (1991), 389–447. [14] H. Boden, Y. Hu, Variations of moduli of parabolic bundles, Math. Ann. 301 (1995), 539–559. [15] H. Boden, K. Yokogawa, Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs over smooth curves, I, Internat. J. Math. 7 (1996), 573–598. [16] F. Bottacin, Symplectic geometry on moduli spaces of stable pairs, Ann. Sci. Ecole Norm. Sup. 28 (1995) 391–433. [17] N. Budur, Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers, Preprint arXiv:math/0610382. [18] K. Corlette, C. Simpson, On the classification of rank-two representations of quasiprojective fundamental groups, Compositio Math. 144 (2008), 1271–1331. [19] R. Donagi, E. Markman, Spectral covers, algebraically completely integrable, hamiltonian systems, and moduli of bundles, Integrable Systems and Quantum Groups (Montecatini Terme 1993), Springer L.N.M. 1620 (1996), 1–119. [20] E. Frenkel, Wakimoto modules, opers and the center at the critical level, Adv. in Math. 195 (2005), 297–404. [21] O. Garcia-Prada, P. Gothen, V. Munoz, Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, Memoirs Amer. Math. Soc. 187 (2007). [22] O. Garcia-Prada, S. Ramanan, Twisted Higgs bundles and the fundamental group of compact K¨ ahler manifolds, Math. Res. Lett. 7 (2000), 517–535. [23] S. Gukov, E. Witten, Gauge Theory, ramification, and the geometric Langlands program, Preprint arXiv:hep-th/0612073v1. [24] R. Hain, On a generalizaton of Hilbert’s 21st problem, Ann. Sci. E.N.S. 19 (1986), 609–627. [25] T. Hausel, Compactification of moduli of Higgs bundles, J. Reine Angew. Math. 503 (1998), 169–192. [26] V. Heu-Berlinger, D´ eformations isomonodromiques des connexions de rang 2 sur les courbes, Doctoral dissertation, Rennes (2008). [27] N. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91–114. [28] N. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), 59–126. [29] M. Inaba, K. Iwasaki, Masa-Hiko Saito, Moduli of stable parabolic connections, RiemannHilbert correspondence and geometry of Painlev´ e equation of type VI, Part I, Publ. Res. Inst. Math. Sci. 42 (2006), 987–1089. [30] J. Iyer, C. Simpson, A relation between the parabolic Chern characters of the de Rham bundles, Math. Ann. 338 (2007), 347–383. [31] F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Princeton University Press (1984). [32] H. Konno, Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface, J. Math. Soc. Japan 45 (1993), 253–276. [33] V. Kostov, The connectedness of some varieties and the Deligne-Simpson problem, J. of Dynamical and Control Systems 11 (2005), 125–155. [34] V. Kostov, The Deligne-Simpson problem for zero index of rigidity, Perspectives In Complex Analysis, Differential Geometry And Mathematical Physics, St Constantin (Bulgaria), 2000, World Scientific (2001), 1–20. [35] S. Lekaus, Unipotent flat bundles and Higgs bundles over compact K¨ ahler manifolds, Trans. Amer. Math. Soc. 357 (2005) 4647–4659. [36] A. Libgober, Problems in topology of the complements to plane singular curves, Singularities in geometry and topology, World Sci. Publ. (2007), 370–387. [37] M. Logares, J. Martens, Moduli of parabolic Higgs bundles and Atiyah algebroids, Preprint math.AG/0811.0817. [38] F. X. Machu, Monodromy of a class of logarithmic connections on an elliptic curve, Symmetry, Integrability and Geometry: Methods and Applications 3 (2007), 31pp.

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[39] E. Markman, Spectral Curves and Integrable Systems, Compositio Math. 93 (1994), 255–290. [40] M. Maruyama, K. Yokogawa, Moduli of parabolic stable sheaves, Math. Ann. 293 (1992), 77–99. [41] T. Mochizuki, Kobayashi-Hitchin correspondence for tame harmonic bundles II, Geometry and Topology 13 (2009) 359–455. [42] D. Mumford, Abelian Varieties. Tata Institute of Fundamental Research Studies in Mathematics (1970). [43] H. Nakajima, Hyper-K¨ ahler structures on moduli spaces of parabolic Higgs bundles on Riemann surfaces, Moduli of vector bundles (Sanda, Kyoto, 1994), Lect. Notes Pure Appl. Math. 179 (1996), 199–208. [44] M.S. Narasimhan, S. Ramanan, Deformations of the moduli space of vector bundles over an algebraic curve, Ann. Math. 101 (1975), 391–417. [45] M.S. Narasimhan, C. Seshadri, Stable and unitary bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540–564. [46] N. Nitsure, Moduli space of semistable pairs on a curve, Proc. London Math. Soc. 62 (1991), 275–300. [47] Nitin Nitsure, Moduli of semistable logarithmic connections, Journal of the A.M.S. 6 (1993), 597–609. [48] S.Ramanan, A.Ramanathan, Some remarks on the instability flag, Tohoku Math. J. 36 (1984), 269–291. [49] C. Seshadri, Moduli of π-vector bundles over an algebraic curve, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome (1970) 139–260. [50] A. Schmitt, Projective moduli for Hitchin pairs, Int. J. of Math. 1 (1998), 107–118. [51] C. Simpson, Harmonic bundles on noncompact curves. J.A.M.S. 3 (1990), 713–770. [52] C. Simpson, Higgs bundles and local systems, Publ. Math. I.H.E.S. 75 (1992), 5–95. [53] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I,II, Publ. Math. I.H.E.S. 79, 80 (1994, 1995). [54] C. Simpson, The Hodge filtration on nonabelian cohomology, Algebraic geometry—Santa Cruz 1995, , Proc. Sympos. Pure Math., 62, Part 2, A.M.S. (1997), 217–281. [55] S. Szabo, Deformations of Fuchsian equations and logarithmic connections, Preprint math.AG/0703230. [56] C. Teleman, C. Woodward, Parabolic bundles, products of conjugacy classes and GromovWitten invariants, Ann. Inst. Fourier 53 (2003), 713–748. [57] M. Thaddeus, Variation of moduli of parabolic Higgs bundles, J. Reine Angew. Math. 547 (2002), 1–14. [58] K. Yokogawa, Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves, J. Math. Kyoto Univ. 33 (1993), 451–504. [59] K. Yokogawa, Infinitesimal deformation of parabolic Higgs sheaves, Internat. J. Math 6 (1995), 125–148. CNRS, Laboratoire J. A. Dieudonn´ e, UMR 6621 Universit´ e de Nice-Sophia Antipolis, 06108 Nice, Cedex 2, France E-mail address: [email protected] URL: http://math.unice.fr/∼carlos/

This volume contains a collection of papers from the Conference on Vector Bundles held at Miraflores de la Sierra, Madrid, Spain on June 16–20, 2008, which honored S. Ramanan on his 70th birthday. The main areas covered in this volume are vector bundles, parabolic bundles, abelian varieties, Hilbert schemes, contact structures, index theory, Hodge theory, and geometric invariant theory. Professor Ramanan has made important contributions in all of these areas.

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