Singularities in Geometry and Topology: Strasbourg 2009

IRMA Lectures in Mathematics and Theoretical Physics 20 Edited by Christian Kassel and Vladimir G. Turaev Institut de R...

Author: Vincent Blanloeil | Toru Ohmoto (eds.) 2013

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IRMA Lectures in Mathematics and Theoretical Physics 20 Edited by Christian Kassel and Vladimir G. Turaev

Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 rue René-Descartes 67084 Strasbourg Cedex France

Chalk drawing by Tatsuo Suwa

Singularities in Geometry and Topology Strasbourg 2009 Vincent Blanlœil Toru Ohmoto Editors

Editors: Vincent Blanloeil IRMA, UFR Mathématiques et Informatique Université de Strasbourg 7, rue René Descartes 67084 Strasbourg, France

Toru Ohmoto Department of Mathematics Faculty of Science Hokkaido University Sapporo 060-0810, Japan

E-mail: [email protected]

E-mail: [email protected]

2010 Mathematical Subject Classification: 13A35, 14A22, 14B05, 14B07, 14B15, 14C17, 14D06, 14E15, 14E18, 14F99, 14H25, 14J17, 14M25, 14N99, 18F30, 19K10, 32A27, 32C37, 32F75, 32G15, 32SXX, 35Q75, 52C35, 53C05, 55N35, 55R40, 57N05, 57R18, 57R20, 58A30, 58K10, 58K40, 58K60, 60D05, 83C57 Key words: singularity theory, singularities, characteristic classes, Milnor fiber, jet schemes, equisingularity, intersection homology, knot theory, Hodge theory, Fulton–MacPherson bivariant theory, mixed weighted homogeneous, nearby cycles, vanishing cycles, affine toric variety, (versal) deformation of surface singularities, noncommutative resolution, cyclic quotient surface singularity, splice quotient singularity, F-regular singularities, semiquasihomogeneous isolated singularities, general relativity, statistical learning theory, singular distributions, localization of characteristic classes, Frobenius morphism, b-function, motivic Grothendieck group, motivic Hirzebruch class, monodromy covering, algebraic local cohomology, Riemann–Roch theorem for embeddings, birational invariant, Riemann surface, stable reduction, Teichmüller space, moduli space, orbifold

ISBN 978-3-03719-118-7 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2012 European Mathematical Society Contact address: European Mathematical Society Publishing House ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: M. Zunino, Stuttgart Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Preface In August 2009 we organized the fifth Franco–Japanese Symposium on Singularities at the Department of Mathematics of Strasbourg University. This symposium followed the fourth one held in Toyama, Japan, two years before. The first day we scheduled a JSPS Forum on Singularities and Applications, and some applications of singularity theory in physics, medicine and statistics were presented. The following days we had a conference; there were advanced talks in topology, algebraic geometry and complex geometry, and recent results on singularities were discussed. In this volume we collected some research papers from participants of the conference and surveys of some talks in the JSPS Forum. Moreover we add two lecture notes of T. Suwa and S. Yokura. All papers in this volume have been refereed and are in final form. We hope that this book will give an opportunity to readers to get a deeper understanding of the marvelous field of Singularities. On behalf of the editors of this proceedings, we would like to express our thanks to Strasbourg University, JSPS, CNRS, Region Alsace and CEEJA, for their support, and to all contributors for the proceedings and the participants of the symposium. Vincent Blanlœil, Strasbourg Toru Ohmoto, Sapporo

The participants of the Conference in front of the Opéra de Strasbourg

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Alain Joets Optical caustics and their modelling as singularities

(JSPS Forum) . . . . . . . . . . . . . 1

Helmut A. Hamm On local equisingularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Shihoko Ishii, Akiyoshi Sannai, and Kei-ichi Watanabe Jet schemes of homogeneous hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Tatsuhiko Koike Singularities in relativity (JSPS Forum) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Yukio Matsumoto On the universal degenerating family of Riemann surfaces . . . . . . . . . . . . . . . . . . . . . 71 Yayoi Nakamura and Shinichi Tajima Algebraic local cohomologies and local b-functions attached to semiquasihomogeneous singularities with L.f / D 2 . . . . . . . . . . . . . . . . . . . . . . . 103 T. Ohmoto, A note on the Chern–Schwartz–MacPherson class . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Mutsuo Oka On mixed projective curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Tomohiro Okuma Invariants of splice quotient singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Oswald Riemenschneider A note on the toric duality between An;q and An;nq . . . . . . . . . . . . . . . . . . . . . . . . . 161 Jörg Schürmann Nearby cycles and characteristic classes of singular spaces . . . . . . . . . . . . . . . . . . . . 181 Tatsuo Suwa Residues of singular holomorphic distributions

(lecture) . . . . . . . . . . . . . . . . . . . . 207

viii

Contents

Sumio Watanabe Two birational invariants in statistical learning theory

(JSPS Forum) . . . . . . . . . . 249

Takehiko Yasuda Frobenius morphisms of noncommutative blowups . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Shoji Yokura Bivariant motivic Hirzebruch class and a zeta function of motivic Hirzebruch class (lecture) . . . . . . . . . . . . . . . . . . . . 285 Masahiko Yoshinaga Minimality of hyperplane arrangements and basis of local system cohomology . . 345

Optical caustics and their modelling as singularities Alain Joets Laboratoire de Physique des Solides, Bât. 510 Université Paris-Sud, 91405 Orsay cedex, France e-mail: [email protected]

Abstract. Optical caustics are bright patterns, formed by the local focalization of light rays. They are caused, for instance, by the reflection or the refraction of the sun rays through a wavy water surface. In the absence of an appropriate mathematical frame, their main characteristics have remained unrecognized for a long time and the caustics appeared in the literature under different names: evolutes, envelopes, focals, etc. The creation of the singularity theory in the middle of the XX century radically changed the situation. Caustics are now understood as physical realizations of Lagrangian singularities. In this modelling, one predicts their local classification into five stable types (R. Thom, V. Arnold): folds, cusps, swallowtails, elliptic umbilics and hyperbolic umbilics. This local classification is indeed observed in experiments. However the global properties of the caustics are only partially taken into account by the Lagrangian model. In fact, it has been proved by Yu. Chekanov that the special form of the eikonal equation governing the propagation of the optical wave fronts imposes the existence of a topological constraint on the singular set (representing the caustic in the phase space) and restricts the number of possible bifurcations. Our experiments on caustics produced by bi-periodic structures in liquid crystals confirm the existence of the topological constraint, and validate the modelling of the caustics as special types of Lagrangian singularities.

Caustics constitute a phenomenon of light focalization, usually studied in the frame of geometrical optics or of wave optics. It is remarkable that they now constitute also a purely mathematical object, expressed in terms of singularities. These two notions are not uncorrelated. The mathematical notion is the final outcome of a long process of successive modellings of the physical phenomenon, that we will call hereafter optical caustics. The aim of this paper is to show how the singularity theory drastically changed our viewpoint about the optical caustics. We will show that some problems, for instance the local classification of caustic points, may be solved only with the help of the singularity theory, and that, conversely, the singularity theory is at the origin of new problems and new experiments on optical caustics.

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Alain Joets

1 Physical aspects 1.1 The physical phenomenon of optical caustics There are different ways to present caustics, according to whether one considers light as composed of rays, or of scalar waves or of electromagnetic waves. However, in view of our purpose, we shall mainly consider the geometrical description in which light is composed of rays, or equivalently of wave fronts. In other words, the wavelength of the light will be supposed to be 0, or very small with respect to the characteristic dimensions of the system. Given a set of rays (a congruence of rays), a caustic point is a point (of our physical space) where the rays are locally focusing, i.e. where two finitely close rays intersect (see Figure 1). At a non caustic point, that is to say at a regular point of the congruence, the rays form a local beam (or the superposition of a finite number of local beams). In contrast, the light beam is shrunk at a caustic point and the energy density becomes infinite (at least in the frame considered here). This is the reason for the name “caustic”, that comes from the Greek root “kausticos” meaning “burning”. From the geometrical viewpoint, the caustic is the envelope of the ray congruence. This means that the rays are tangent to the caustic (at the corresponding caustic point). In the usual case of straight rays in our physical 3D-space, each ray contributes to 2 caustic points and the caustic is composed of 2 sheets. A very simple example of caustics is provided by the bright moving lines one sees on the bottom of a swimming pool. Another example is provided by a perfect focus. However, this example is a somewhat misleading, since a focus is a fully unstable caustic point disappearing under any small perturbation of the congruence. Such an unstable situation must be excluded from the general study of caustics. In the plane, the caustic points constitute curves (Figure 1). In the physical 3D-space, they constitute surfaces. These geometrical objects are generally not regular. They may possess special points: regression points for the caustic curves, and regression edges for the caustic surfaces. The regression edges themselves may possess more particular points. In brief, caustics are structured objects and an important problem is to understand their structure into different types of points. There is no special condition for producing optical caustics. Every congruence of rays generates a caustic, more or less intricate. Even in the case of a beam of parallel rays, one may consider that a caustic point is generated at infinity. The caustics then constitute an optical phenomenon of great generality.

1.2 Observation of caustics As (singular) surfaces in our physical space R3, the caustics cannot be directly observed, since they are not material surfaces. However they are easily visualized by interposing some screen transversely to the rays. In other words, one sees only 2D-sections of a caustic surface and the whole caustic itself necessitates a (tedious)

Optical caustics and their modelling as singularities

3

W

caustic

Figure 1. In the plane, a congruence of rays (represented here by arrows) has an envelope curve: its caustic. The wave font W propagates (normally to the rays in the case of a homogeneous and isotropic medium) and its regression point glides along the caustic.

work of reconstruction section by section (see Figure 2). Inside the screen, the trace of the caustic forms a set of bright curves called folds (symbol A2 ). These curves may be not regular at some points forming there a tip called cusp (symbol A3 ). Figure 2 shows pairs of cusps forming “lips” (section 1-left) and a “beak-to-beak” (section 3-right). One has to remember that these curves and points are in fact the traces of fold surfaces A2 and of cusp lines A3 . In addition to self-intersections, there is generically no other type of caustic points in a 2D-section. However, for special positions of the screen, one may observe other types of bright points. They are associated with three types of caustic points: the swallowtails (symbol A4 ), the elliptic umbilics (symbol D4 ) and the hyperbolic umbilics (symbol D4C ). Examples of these three types may be found in Figure 5 (simulation) and Figure 8 (photo). The five types A2 , A3 , A4 , D4 , and D4C constitute the complete list of the generic caustic points of the physical space. The description of a caustic given here, in terms of different types of caustic points, corresponds to a modern presentation, using the results, the names and symbols coming from the singularity theory. However, the usual presentation in textbooks on optics is much more elementary, very often limited to a formal definition of a caustic point and to some elementary properties. A reason for that is perhaps the high mathematical level of the singularity theory. In fact, a more fundamental reason is that the traditional aim of optics is the lens design forcing light beams to be concentrated at well defined focal points. For that reason, optical systems have special symmetries. The caustics they produce are not generic and very special techniques have been developed to their study, for instance the geometrical theory of optical aberrations [1]. In a sense, instrumental optics are interested by degenerate caustics, not by the generic ones. On the other hand, the light focusing by natural systems always produces generic caustics [2]. Interesting examples include the optics of the eye [3], the electronic optics [4] and [5], the gravitational lensing, [6], the visualization techniques called shadowgraph methods [7] and [8].

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Alain Joets

3) 2)

3) A3

A3 2)

A2 1)

1)

3)

3)

2)

2)

1)

1)

Figure 2. Two examples of the reconstruction of a cusp line A3 and its twofold surfaces A2 : lips (left) and beak-to-beak (right).

2 Modelling caustics The history of the mathematical notion of caustic is long and intricate. Our aim is not to present a detailed picture of it, but rather to give some elements which allow one to estimate the role of the notion of singularity in the recent revival of the subject.

2.1 First modellings Surprisingly, the first study about caustics seems to be due to a Greek mathematician of the 3rd century B.C., Apollonius of Perga, who considered the problem of finding segments of extremal length linking an arbitrary point in the plane to a given conic section [9] and [10]. The conic section plays the role of an initial wave front and the


5

extremal segments, normal to the conic, play the role of the rays. In this analogy, the locus where the number of extremals changes represents the caustic. Apollonius found a geometrical construction for determining this locus. In the modern terminology, he studied the generic Lagrangian singularities of the plane (A2 and A3 ). One observes no substantial progress until the introduction of the name “caustica” by Tschirnhausen, who studied the reflection of sun rays in a circular mirror [11]. He observes that the concentration of the rays occurs along an “entire curved line, which is produced by the intersections of reflected rays” (see Figure 3-top left). The name itself appears in 1690, in the Tschirnhausen paper [12], in the Latin expression “caustica curva”, quickly abbreviated to “caustica”. In fact, a few years before, C. Huygens had obtained more accurate results about caustics by reflection or by refraction, including the propagation of the wave front along the caustic (see Figure 3-top right). However, his book, Traité de la lumière, appeared later, in 1690 [13]. Caustics (in the plane) appear in the L’Hospital’s book “Traité des infiniment petits” (1696), the first book on differential calculus [14]. They illustrate the power of the new Calculus. Caustics in the 3D-space appeared latter, after the introduction of the notions of curvature, lines of curvature, principal curvatures, etc. We have to recall the creation of the word “umbilic” by G. Monge (1795), a point of a surface at which the two principal curvatures are equal [15]. In 1873, A. Cayley studies the congruence formed by the normals to an ellipsoid [16]. He shows that the “centro-surface”, i.e. the caustic associated with the normals, possesses four special points, called by him “umbilicar centres” or “omphaloi” (see Figure 3-bottom). In the modern terminology, they are named “hyperbolic umbilics” and denoted by D4C . The study of the umbilics has been continued by Darboux [17], in 1896. More precisely, the author analyses the lines of curvature in the vicinity of an umbilic of a given surface, and not of a caustic surface. The link with the caustics and their umbilics exists only if one considers that the surface represents a wave front propagating in an homogeneous and isotropic medium. Darboux succeeded in classifying these umbilics into 3 types. Nevertheless, let us note that the Darbouxian classification is different from – although not unrelated to – the modern classification of umbilics of a caustic into hyperbolic and elliptic types and also from the classification according to their index (see [18] for the details). Since Darboux’s work and until the singularity theory, the subject of caustics seems to have been neglected, and at best considered as a source of academic exercises for students. To sum up, all these partial results show that the caustics have been recognized from the very beginning as complex objects, presenting a rich structure. However, the usual direct approach in the frame of the Euclidean space proved to be too restricted and inadequate to obtain general results. The situation radically changed in 1955 with the creation of the singularity theory by H. Whitney [19] and R. Thom [20].

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Figure 3. In his 1682 publication [11], W. Tschirnhausen defines a caustic line as the curved line produced by the “intersections des rayons réfléchis” [top left]. In his treatise [13] (written four years before, but published only in 1690), C. Huygens obtains, for the same problem, a more accurate description including the propagation of the wave front, called by him “onde repliée” [top right]. In 1871 [16], A. Cayley shows that the caustic of an ellipsoid possesses umbilics points, that is to say meeting points of the two sheets of the caustic (here in the plane x-z) [bottom].

2.2 Caustics as singularities of maps The modelling has made an important progress thanks to the singularity theory. This progress is based on the distinction between two spaces where the rays are represented: • Our physical space R3 D fx1 ; x2 ; x3 g, in which lie the rays and the caustic. In this space the rays may intersect. • An abstract “ray space” R D fr1 ; r2 ; r3 g above the physical space R3 , where each ray is represented by some curve (also called “ray”). R is only composed of rays. It is a smooth 3D-manifold and it is constructed in such a way as the “rays” cannot intersect. A simple way for constructing R is the following. One considers a surface W R3 transverse to the rays, for instance an initial wavefront. Each ray is thus parametrized by the two parameters of W, say r1 and r2 . In order to specify the position of the current point along a ray .r1 ; r2 /, one needs a third coordinate r3 , for example its distance to W along the ray. The space R is then parametrized by these three coordinates r1 , r2 , and r3 . It is clear that the intersections of rays cannot occur in this space R, since different rays have different values for .r1 ; r2 /.


7

We now recover the observed congruence by associating with each point of R its position .x1 ; x2 ; x3 / in the physical space. This defines a mapping p W R ! R3 by .x1 ; x2 ; x3 / D p.r1 ; r2 ; r3 /. The mapping p is called projection. Let us note that the source space R and the target space R3 have the same dimension, equal to 3. In this frame, the local ray focalization at a caustic point is expressed by saying that the rank of the derivative dp of p is less than its maximal possible value 3. Such a point in R is called singular or critical. The set † R of the critical points is called the singular set. Finally, the caustic C is the projection of the singular set: C D p.†/. In practice, the equation for † is obtained by cancelling the Jacobian determinant associated with p, det @.x1 ; x2 ; x3 /[email protected] ; r2 ; r3 / D 0. By solving this equation, one obtains one of the ri ’s as a function of the others, say r3 D r3 .r1 ; r2 /. Thus the caustic is found in a parametric form: x1 D x1 .r1 ; r2 ; r3 .r1 ; r2 //, etc. At this stage, we merely have a mathematical definition for the physical notion of caustic point. The singularity theory allows us to go farther and to find the nature of the caustic point. More precisely, let us recall that one defines the Thom–Boardman set †i of p as the set of points of R where dp has a kernel of dimension i [20]. Then ones defines inductively the set †i;:::;j;k as the set †k of the restriction of p to †i;:::;j . Thus, †0 represents the regular points of the congruence, †1;0 the fold-surface, †1;1;0 the cusp-lines, †1;1;1;0 the swallowtails, and †2 the umbilics (hyperbolic or elliptic). By definition, each Thom–Boardman set is obtained by cancelling some functional determinants associated with p or with the restriction of p to some other Thom– Boardman set. Therefore the effective calculation of the sets †I can always be performed at least numerically. However, this classification “by the rank” is not totally satisfactory. First, it does not distinguish between the hyperbolic umbilics D4C and the elliptic umbilics D4 . Worse, as singularities of a (general) map, the umbilics, having a codimension (4) higher than the dimension of the space (3), are not stable. The fact that they are experimentally observed shows that the modelling of caustics as singularities of a map is incomplete. In fact, it ignores an important element, namely the Fermat principle or, in mathematical terms, the symplectic structure of the problem.

2.3 Caustics as Lagrangian singularities The wave propagation along a ray is described by the wave vector, or “momentum”, pE [1]. The local ray direction is along p. E One has the fundamental relation

where S is the optical length eikonal equation:

R

pE D rS;

(2.1)

nds and n the (local) refractive index. S follows the .rS/2 D n2 :

(2.2)

Relation (2.1) shows that the “function” S must be considered as a multi-valued function, since several local beams may be passing through a given point. This suggests a new representation of the rays in a bigger space including at once the spatial coor-

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dinates xi and the vectorial coordinates pi . More precisely, one considers the phase space is characterized by its symplectic structure, space T R3 D fpi ; xi g. The phase P that is, the differential 2-form ! D dpi ^ dxi , which is nondegenerate and closed (d! D 0). One sees immediately that ! cancels at the points for which pE D rS. One is thus led to keep the cancellation of ! as the characteristic property of a congruence of rays represented in the big space T R3 . One says that the submanifold L T R3 of dimension 3 (half of the dimension of the phase space) is a Lagrangian submanifold if !jL D 0. The base space fx1 ; x2 ; x3 g is called the configuration space. Every congruence of rays is described by a Lagrangian submanifold. In this frame, the role of the projection p is played by the natural projection into the configuration space, .p; x/ D x, or more precisely by its restriction to L, p D jL (see Figure 4). As in the previous section, one defines the singular set † L as the set of points where p has a non trivial kernel. The caustic C is .†/. By reference to the name for L, the singular points are called Lagrangian singularities. The advantage of the new definition comes from the properties attached to the Lagrangian submanifolds. Indeed these submanifolds are constructed by starting from functions or from families of functions, rather than from maps. There are two important formulations [21]. The first formulation is the generalization of the relation pE D rS, valid also for the singular points of L. It takes the (local) form p˛ D

@S ; @x˛

xˇ D

@S : @pˇ

(2.3)

The generating function S is no more defined on the configuration space, but rather on the Lagrangian submanifold L (locally parametrized by the coordinates x˛ and pˇ of (2.3)). In the second formulation, the Lagrangian submanifold is given by a generating family, i.e. a function F defined on the configuration space R3 D fx1 ; x2 ; x3 g and depending on some parameter s: n

@F @F o : (2.4) D 0 and p D @s @x The first equation @F=@s D 0 determines the rays passing through .x1 ; x2 ; x3 /, whereas the second one distinguishes these rays according to their wave vector p. E So the generating-family technique links the caustics to the theory of singularities of functions depending on some parameters, that is to say to catastrophe theory [22] and [23]. L D .p; x/ W there exists s such that

2.4 Caustics and wave front singularities Figure 3-middle, extracted from the pioneering work of Huygens [13] recalls that, in the case of the plane, the propagating wave front presents a singular point gliding along the caustic curve. In fact, the entire caustic curve results from the sweeping


9

Lagrangian submanifold L p1 p2 x2 x1

singular set †

Lagrangian projector initial wave front W configuration space

x2 x1 caustic C Figure 4. When represented in the phase space (here the space fp1 ; p2 ; x1 ; x2 g), the rays constitute a regular surface L called the Lagrangian submanifold. The points of the singular set † L are characterized by a vertical tangent plane to L. The caustic C is the projection of the singular set †: C D .†/.

motion of the singularities of W. This remarkable duality linking rays and wave fronts remains valid in the general case of caustics of the 3D-space. However, in this case, a typical instantaneous wave front W has more singularities: it may possess cuspidal curves and swallowtails points. During the motion of W, governed by the eikonal equation (2.2), the cuspidal curves generate surfaces and the swallowtails generate curves. The generated surfaces are exactly the fold surfaces A2 of the caustic C, whereas and the generated curves are the cusp lines of C. To obtain the other caustic types, i.e. the swallowtails A4 and the umbilics D4 , one has to consider the bifurcations of the wave front, at some times of its motion.

3 Local and global aspects of caustics 3.1 Local types In order to distinguish different types of singularities, one defines an equivalence relation between Lagrangian projections, called Lagrange equivalence. This is a diffeomorphism between the two phase spaces, preserving simultaneously the symplectic and the fiber structures and sending the first Lagrangian submanifold L1 to the second Lagrangian submanifold L2 (see [21] for the details). In fact one considers rather local

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situations, expressed in terms of germs. A Lagrangian singularity is then a Lagrange equivalence class of a germ at a critical point. The same equivalence relation allows one to define the stability of a singularity. A singularity is stable if its equivalence class constitutes a neighborhood of it. D4

A4

D4C

A2 A3

A2 A2 Figure 5. The five generic types of Lagrangian singularities: the fold type A2 constitutes surfaces; the cusp type A3 constitutes edges of regression; the three other types are point singularities: the swallowtail type A4 (at the meeting point of twe lines A3 and a self-intersection line A2 A2 ), the elliptic umbilic type D4 (at the meeting point of three cusp lines A3 ) and the hyperbolic umbilic type D4C (at the meeting point of a cusp line A3 and a self-intersection line A2 A2 ).

The fundamental result of the Lagrangian singularity theory is the local classification of Lagrangian singularities: every stable Lagrangian singularity is equivalent to one of the five following types: A2 , A3 , A4 , D4 , D4C (see Figure 5). In terms of generating families, the list is given by [21]: A2 W F D s 3 C q1 s; A3 W F D ˙s 4 C q1 s 2 C q2 s; A4 W F D s 5 C q1 s 3 C q2 s 2 C q3 s; D4˙ W F D s12 s2 ˙ s23 C q1 s22 C q2 s2 C q3 s1 : These polynomial functions are called normal forms. They constitute a local model describing the fine structure of every caustic type. The stability means that every singularity of the above list survives the action of infinitely small perturbations. Conversely, any other singularity type not pertaining to the above list is destroyed by perturbations and is replaced by singularities belonging to the list.


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It is important to note that the vector pE describes the phase behavior of the optical wave. As a consequence, the normal forms describe at once the shape of the caustic surface (via the Lagrangian projection) and the amplitude of the interference pattern around it (via the Fresnel–Kirchhoff integral [1]); see Figure 6. In addition to its normal form, one may associate some numbers with any caustic type. We have already seen that each type forms a set of some dimension d, or better of some codimension d 0 D 3 d . We have d 0 D 1 for the folds, d 0 D 2 for the cusps, and d 0 D 3 for the swallowtails and the umbilics. Another important number is the rank r of the projection p, or equivalently the corank c D 3 r. Folds, cusps and swallowtails have a corank equal to 1, and a tangent plane is defined at the caustic point, despite the fact that the caustic is not a regular surface at the points A3 and A4 . In contrast, the umbilics have a corank equal to 2, and the caustic has no tangent plane there, but rather a local direction corresponding to the direction of the ray passing through the umbilic. There is also a “singularity index” governing the asymptotic increase of the amplitude of the diffraction pattern in the limit of wavelengths tending towards 0; see [24]. The values of this index show that cusps must appear brighter than the folds, and the swallowtails and the umbilics brighter than the cusps. The local classification accounts for all of the observed (non-degenerate) caustics and for their diffraction patterns; see [2], [24], and [22]. It is the basis of a fine study of each caustic type and the agreement between theory and experiment is found to be excellent (see for example [25] for the case of the D4 ).

3.2 Bifurcation of caustics Another local classification concerns the bifurcations of caustics themselves. When the system of rays depends on some control parameter (for example a temperature or a magnetic field), the caustics produced may undergo a topological transformation for some value of the parameter. This transformation is called bifurcation. There are eleven possible bifurcations of Lagrangian singularities [21]. Some of them describe how point singularities appear or disappear by pairs. To our knowledge, these caustic bifurcations have not yet been experimentally studied in detail.

3.3 Global aspects The global properties of caustics are less understood than the local ones. However, the generalization of the notion of Maslov’s index to spaces of higher dimensions has led to the discovery of new invariants [26]. These invariants control the number of some types of singularities. For instance, in dimension n D 4, the number of butterflies A5 (taking account of sign) is equal to zero. For the 3D space, the case mainly considered here, there exists in addition a remarkable theorem due to Yu. Chekanov.

12

Alain Joets A2

A3

D4C

A4

D4

Figure 6. Diffraction patterns associated with the five generic caustics.

3.3.1 Chekanov’s formula for the singular set †. Chekanov’s formula is a relation between the Euler characteristic .†/ of the singular set † and the number ]D4 .1=2/ of umbilics of index 1=2. More precisely, one has [27]: .†/ C 2]D4 .1=2/ D 0

(3.1)

In order to understand the definition of the index, let us recall at first that the eikonal equation expresses the fact that the Lagrangian submanifold L lies on a hypersurface E T R3 . The rays correspond the (skew) orthocomplements of E, and are called characteristic l. Moreover, an umbilic point T 2 L is a singular point where the surface † L is locally a cone (in fact a double cone) with vertex at T . Then, since the corank of the projection p at T is equal to 2, a 2D plane … D kerp is defined. Finally, cusp lines A3 † pass through T . Now, the index is defined according to the relative positions of these elements. If l and A3 are separated by …, the index is equal to C1=2, and 1=2 in the other case (see Figure 7). One shows that the index of an elliptic umbilic is always equal to 1=2. The index of a hyperbolic umbilic may be equal either to 1=2, and it is denoted by D4Ct , or to C1=2, and it is denoted by D4Cd . Thus, formula (3.1) writes: .†/ C 2.]D4 C ]D4Ct / D 0

(3.2)

Optical caustics and their modelling as singularities characteristic d

D4Cd

…

characteristic d

D4Ct

†

13

A3

characteristic d

†

† …

… A3 (a)

A3 (b)

(c)

Figure 7. In the neighborhood of a hyperbolic umbilic D4C , the critical set † is a cone (D4C is at its vertex). The kernel … of the Lagrangian projection at the point D4C cuts the cone. If … separates the characteristic l from the cusp line A3 , the index is C1=2 and the umbilic is denoted by D4Cd (a). In the other case, the index is 1=2 and umbilic is denoted by D4Ct (b). Simulation of the singular set † in the neighborhood of a hyperbolic umbilic of the caustic represented in Figure 8(a). By comparison with (a) and (b), one sees that, in this case, the index is C1=2.

Chekanov’s theorem requires some assumptions. In particular, the hypersurface E is supposed to be convex with respect to the wave vector p. E This special condition is always satisfied in geometrical optics, because of the general form of the eikonal equation (2.2). For that reason, in this framework, the Lagrangian singularities are called optical singularities. It is also assumed that † is a compact surface. Since it contains elements defined in the abstract space T R3 , Chekanov’s formula cannot be directly checked by experiment. Nevertheless, it may be possible, in the best cases, to obtain experimental informations about the ray congruence sufficient to calculate numerically these elements. This reconstruction has been successfully made in the case of a biperiodic caustic produced by the deflection of a light beam through a nematic liquid crystal layer [28]. The biperiodicity in the plane of the layer makes the emerging wave front topologically equivalent to a torus T 2 . Now, through each point of this torus passes one straight ray bearing two caustic points. These two points coincide only at the umbilics. In other words, † is a topological surface obtained by gluing together two torus at the umbilics points. One deduces immediately that .†/ is related to the number of umbilics ]D4 through the relation .†/ D ]D4 . In the experiment one counts eight umbilics per cell: .†/ D 8. The remaining work is a careful simulation of the deflection of the rays inside the liquid crystal, the numerical calculation of the projection, and the determination of its Thom–Boardman sets †1 (giving the double cones at the umbilic points) and †1;1 (giving the A3 lines which pass through the vertices of the double cones). In the case under consideration, one finds that all hyperbolic umbilics (four per cell) have a positive index. Since one counts per cell four elliptic umbilics (index 1=2) and four umbilics D4Cd , one has .†/ C 2D.1=2/ D 8 C 2.4 C 0/ D 0. The Chekanov relation is verified. It is interesting to recall that M. Kazarian [29] gave an alternative characterization of the indices of the umbilics in the configuration space. This characterization is based

14

Alain Joets

(a)

(b)

Figure 8. Two particular sections of the biperiodic caustic produced in a certain experiment using a liquid crystal as a light deflector. The caustic contains hyperbolic umbilics (a) and elliptic umbilics (b).

on the behavior of the ray direction along a cusp line A3 passing through the umbilic point. At each point of a cusp line a tangent plane to the caustic surface is defined (even if the surface is a non-regular surface there) and the ray lies inside this plane. At the umbilic point, the ray becomes parallel to the cusp line. Along A3 , there are two possibilities for the ray direction. If it points inside the cuspidal edge, the line is said to be AC 3 , and A3 when it points outside the cuspidal edge. Now, the direction of the ray at the umbilic point T defines an orientation of the cusp lines passing through T. C Following this orientation, a cusp-line AC 3 (resp. A3 ) becomes A3 (resp. A3 ) at the umbilic point and the index is equal to C1=2 (resp. 1=2). To our knowledge this new characterization has not yet been exploited experimentally. Chekanov’s relation has an important consequence on the caustic bifurcations. Among the eleven possible caustic bifurcations, considered as bifurcations of general Lagrangian singularities, four of them cannot be realized as bifurcations of optical Lagrangian singularities: they are incompatible with the Chekanov relation. So Chekanov’s relation reduces the number of optical metamorphoses to seven (see Figure 9).

3.3.2 Topological formula for caustics. Chekanov’s formula describes the topology of to the singular set † but gives no information about its image C D .†/ in the configuration space. We want to give here new elements for this issue. It is known that the Euler characteristic of a regular surface A with boundary B and corners Ci is determined by the total curvature associated with the surface (gaussian curvature ), with the boundary (geodesic curvature g ) and with the corners (external

Optical caustics and their modelling as singularities 1

2

5

3

15 4

7

6

Figure 9. Chekanov’s relation implies that only seven caustic bifurcations can be realized optically. Each drawing shows the caustic before, at and after the bifurcation (after [21]).

angles ˛i ). More precisely, one has: Z Z X 2.A/ D ds C g d l C ˛i : A

B

(3.3)

i

A natural issue is then to generalize this formula to the case of caustics C. We have found that such a generalization may be made. For a caustic without boundary, the new formula writes [30]: Z Z 2.C / D ds C 2g d l C .2]A2 A2 A2 C ]A4 C 2]D4 /: (3.4) A2

A3

The first contribution is the gaussian contribution of the fold surface A2 . The second contribution is the geodesic contribution. The factor 2 means that the cusp lines

16

Alain Joets

A3 may be considered as a kind of double boundary, along which 2 sheets A2 join together. In the third contribution, that each type of Lagrangian point singularity gives a different contribution proportional to , the factor being an integer: 0 for the hyperbolic umbilics, 1 for the swallowtails, and 2 for the elliptic umbilics. This number may be interpreted as the number of Whitney umbrellas (contribution ) “contained” in the singularity [30]. There is also a contribution coming from the triple points A2 A2 A2 . In conclusion, optical caustics are complex physical objects, structured in different types. Because of this complexity, they were analyzed as the physical realization of various mathematical notions: envelopes, evolutes, focals, centers of curvature, asymptotics, etc. Their local properties are now satisfactorily understood when they are modelled as singularities, obtained by projecting the Lagrangian manifold representing the set of rays in the phase space into the physical space where the caustics are observed. However, the understanding of their global properties has necessitated a refinement of the model, taking into account the particular form of the light propagation, expressed by the eikonal equation. At present, a consequence of the new model, namely the existence of a topological invariant, has been experimentally checked. However new experiments are needed to verify the other theoretical consequences of the model.

References [1]

M. Born and E. Wolf, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light, Pergamon Press, Oxford etc. 1965. 3, 7, 11

[2]

J. F. Nye, Natural focusing and fine structure of light. Caustics and wave dislocations, Institute of Physics Publishing, Bristol 1999. 3, 11

[3]

A. Gullstrand, Einiges über optische Bilder, Naturwissenschaften 28 (1926), 653–664. 3

[4]

W. Glaser and H. Grumm, Die Kaustikfläche der Elektronenlinsen, Optik 7 (1950), 96–120. 3

[5]

S. Leisegang, Zum Astigmatismus von Elektronenlinsen, Optik 10 (1953), 5–14. 3

[6]

H. Levine, A. O. Petters and J. Wambsganss, Singularity theory and gravitational lensing, Progress in Mathematical Physics 21, Birkhäuser, Basel 2001. 3

[7]

W. Merzkirch, Flow visualization, Academic Press, Orlando 1987. 3

[8]

A. Joets, Caustics and visualization techniques, in Singularity theory – Dedicated to Jean-Paul Brasselet on his 60 th birthday. Proceedings of the 2005 Marseille Singularity School and Conference, CIRM, Marseille, France, 24 January – 25 February 2005, ed. by D. Chériot, N. Dutertre, C. Murolo, A. Pichon and D. Trotman, World Scrientific, Singapore 2007, 277–284. 3

[9]

Apollonius, Conics, Books V to VII: the Arabic translation of the lost Greek original in the version of the Banu Musa, ed. and transl. by G. J. Toomer, Springer, New York 2006, 37–41. 4

[10] A. Joets, Apollonios, premier géomètre des singularités, Quadrature 66 (2006), 37–41. 4


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[11] W. Tschirnhausen, Nouvelles découvertes proposées à Messieurs de l’Académie Royale des Sciences, Journal des Sçavans (1682), 176–179. 5, 6 [12] W. Tschirnhausen, Curva Geometrica quae se ipsam sui evolutione describit, Acta Eruditorum IX (1690), 169–172. 5 [13] C. Huygens, Traité de la lumière, Pieter van der Aa, Leiden 1690. 5, 6, 8 [14] G. F. A. Marquis de l’Hospital, Analyse des infiniment petits pour l’intelligence des lignes courbes, Imprimerie Royale, Paris 1696, reprint ACL-éditions, Paris 1988. 5 [15] G. Monge, Feuilles d’analyse appliquée à la géométrie à l’usage de l’Ecole Polytechnique, Baudolin, Paris 1795, reprint Editions Jacques Gabay, Sceaux 2008. 5 [16] A. Cayley, On the centro-surface of an ellipsoid, Transactions of the Cambridge Philosophical Society XII (1873), 319–365. 5, 6 [17] G. Darboux, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, tome IV, Gauthier-Villars, Paris 1896, reprint Editions Jacques Gabay, Sceaux 1993, 448–465. 5 [18] I. R. Porteous, Geometric differentiation for the intelligence of curves and surfaces, Cambridge University Press, Cambridge 1994. 5 [19] H. Whitney, On singularities of mappings of Euclidean spaces I. Mappings of the plane into the plane, Ann. of Math. (2) 62 (1955), 374–410. 5 [20] R. Thom, Les singularités d’applications différentiables, Ann. Inst. Fourier, Grenoble 6 (1956), 43–87. 5, 7 [21] V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of differentiable maps. The classification of critical points, caustics and wave fronts, Vol. I, translated from the Russian by I. Porteous and M. Reynolds, Monographs in Mathematics 82, Birkhäuser, Boston 1985. 8, 9, 10, 11, 15 [22] R. Thom, Topological models in biology, Topology 8 (1969), 313–335. 8, 11 [23] R. Thom, Stabilité structurelle et morphogenèse, Interéditions, Paris 1977. 8 [24] M. V. Berry and C. Upstill, Catastrophe optics: morphologies of caustics and their diffraction patterns, Progress in Optics XVIII (1980), 257–346. 11 [25] M. V. Berry, J. F. Nye, and F. J. Wright, The elliptic umbilic diffraction catastrophe, Trans. R. Soc. Lond. A 291 (1079), 453–484. 11 [26] V. A. Vassilyev, Lagrange and Legendre characteristic classes, Advanced Studies in Contemporary Mathematics 3, Gordon and Breach Science Publishers, New York 1988. 11 [27] Yu. V. Chekanov, Caustics in geometrical optics, Funct. Anal. Appl. 20 (1986), 223–226. 12 [28] A. Joets and R. Ribotta, Experimental determination of a topological invariant in a pattern of optical singularities, Physical Review Letters 77 (1996), 1755–1758. 13 [29] M. E. Kazarian, Umbilical characteristic number of Lagrangian mappings of 3- dimensional pseudooptical manifolds, in Singularities and differential equations, Singularities and differential equations. Proceedings of a symposium, ed. by S. Janeczko, W. M. Zajaczkowski, and B. Ziemian, Bogdan, Banach Center Publications 33, Polish Academy of Sciences, Inst. of Mathematics, Warsaw 1996, 161–170. 13 [30] A. Joets, Gauss–Bonnet formula for caustics, in preparation. 15, 16

On local equisingularity Helmut A. Hamm Mathematisches Institut, Westf. Wilhelms-Universität Einsteinstr. 62, 48149 Münster, Germany e-mail: [email protected]

Abstract. We will prove some generalization of the theorems of Lê and Ramanujam resp. of Timourian for the case where the ambient space is no longer Cm . Furthermore we will derive some weaker result in the case of a family of non-isolated singularities.

1 Introduction Essentially, a family of singularities is called “equisingular” if the topological type of the singularities is constant. In order to be precise we will stick to the notions of topological type and local topological triviality. Let X and X 0 be topological spaces. Then X and X 0 are said to have the same topological type if they are homeomorphic. Suppose that we have a family of topological spaces F t ; t 2 T , given by a continuous mapping g W X ! T between topological spaces such that F t D g 1 .ftg/: Then we know that all F t have the same topological type as soon as T is connected and g is a locally trivial fibration, which means that for each t 2 T there is a neighborhood T 0 of t in T , a topological space F and a homeomorphism h W g 1 .T 0 / ! F T 0 such that the following diagram is commutative: g 1 .T G0 / GG GG G g GG G# Work

h

T0

/ F T0 ; x xx xx x x pr2 {xx

partially supported by Deutsche Forschungsgemeinschaft.

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Helmut A. Hamm

where pr2 is the projection onto the second factor. This is obvious because F t is homeomorphic to F for t 2 T . In fact we are only interested in the complex-analytic context. Let g W X ! T be a holomorphic mapping between complex spaces. We call g a topologically locally trivial fibration if the underlying continuous map is a locally trivial fibration. Lemma 1.1 (Thom’s first isotopy lemma). Suppose that g is proper, T smooth and that there is a Whitney regular stratification of X such that g is a stratified submersion, i.e. the restriction of g to each stratum of X defines a submersion. Then g is a topologically locally trivial fibration. This is proved using integration of suitable stratified vector fields, see [3] I 1.5. Now let us pass to the local situation. Let .X; x/ be a germ of a complex space. We can assume that it is a subgerm of .Cm ; x/; after translation we can assume x D 0. Let k:::k be the Euclidean norm in Cm , X a representative of .X; 0/ in Cm and B D fz 2 Cm j kzk g: If > 0 is sufficiently small the topological type of X \ B is independent of , we call it the topological type of .X; 0/. (In fact, the embedding plays no role too.) If two germs .X; 0/ and .X 0 ; 0/ of complex spaces have the same topological type they are homeomorphic but it is doubtful whether the inverse implication holds. We may extend the notion of topological type to pairs of spaces or to mappings. If .X; 0/ and .X 0 ; 0/ are embedded into .Cm ; 0/ it is not clear whether a given homeomorphism X \ B ! X 0 \ B extends to a homeomorphism .B ; X \ B / ! .B ; X 0 \ B /: For this question it is useful to look at germs of pairs of complex spaces. Now .X; 0/ can be viewed as the zero locus of a holomorphic map germ f W .Cm ; 0/ ! .Ck ; 0/: This motivates the transition from spaces to functions. We restrict to the case k D 1 (and change the notation, taking .X; 0/ to be the domain of f ). Let f W .X; 0/ ! .C; 0/ and f 0 W .X 0 ; 0/ ! .C; 0/

On local equisingularity

21

be holomorphic map germs. Then f and f 0 have the same topological type if for 0 0 is small enough we know that S intersects S transversally at p. According to [5] Lemma 3.1 we have that fg D 0g is transversal to L if L D lim Ln where Ln is the tangent space to .X nY /\ff D f .pn /g at pn and pn ! p. Because of Thom’s af -condition we have Tp S L. On the other hand, we have Tp S fg D 0g, of course. So Tp S L \ fg D 0g. Now S is transverse to S, so S intersects L \ fg D 0g transversally at p. Altogether S , L and fg D 0g are transversal. If s and t are sufficiently small compared with , s ¤ 0, we obtain that S , X \ ff D sg and fg D tg are transversal too. This implies the lemma. Now we can prove the following which is to a large extent a generalization of [4] Theorem 1.1. Theorem 3.3. Suppose that ˆg j CXnY is acyclic outside 0. Then the following conditions are equivalent: a) .ˆg j CXnY /0 is acyclic; b) for 0 < max.˛; ˇ/ 1, .f; g/ W B \ X \ f 1 .DP ˛ / \ g 1 .Dˇ / ! DP ˛ Dˇ defines a C 1 locally trivial fibration; c) for 0 < jtj 1, .B \ .X n Y / \ fg D tg/ D 0I d) for 0 < jtj 1, .B \ X \ fg D tg/ D .B \ Y \ fg D tg/I e) for 0 < jsj jtj 1, .B \ X \ ff D sg/ D .B \ X \ fg D t; f D sg/I


27

f) .f; g/jX n Y has no critical points with jf j ˛; jgj ˇ, and 0 < max.˛; ˇ/ 1. Proof. a) () c) Let us fix a Whitney regular stratification of .X; Y /. By a result of Sullivan [11] we know that .B \ .X n Y // D .S \ .X n Y // D 0 because the strata of S \ X are odd-dimensional. Therefore c) is equivalent to .B \ .X n Y /; B \ .X n Y / \ fg D tg/ D 0; i.e. ..ˆg j CXnY /0 / D 0; but CXnY Œn is perverse (with respect to the middle perversity, see [12]), because X nY is smooth of dimension n, so j CXnY Œn and hence ˆg j CXnY Œn too, see [12]. Since ˆg j CXnY is acyclic outside 0 by assumption we have that .ˆkg j CXnY /0 D 0 for k ¤ n, so ..ˆg j CXnY /0 / D 0 () .ˆg j CXnY /0 is acyclic. This implies our assertion. b) H) a) Obvious. a) H) b) By Lemma 3.2 we have that .f; g/jS \ .X n Y / is submersive above DP ˛ \Dˇ . On the other hand, Lemma 3.1 of [5] implies that .f; g/jX nY is submersive in B \ f 1 .DP ˛ / \ g 1 .Dˇ /. So we can construct vector fields which lead to a local trivialization. b) H) f) Obvious. f) H) b) This follows from Lemma 3.2 and the assumption f), see proof of a) H) b). c) () d) We look at jf j W B \ .X n Y / \ fg D tg. We know that .B \ X \ fjf j D ˛; g D tg/ D .B \ X \ fjf j D ˛; g D 0g/ D .B \ X \ f0 < jf j ˛; g D 0g/ D .B \ .X n Y / \ fg D 0g/ D .S \ .X n Y / \ fg D 0g/ D0 by [11] (see above), 0 < jtj ˛ 1. Because of Lemma 3.2, we obtain: c) () .B \ X \ f 1 .DP ˛ / \ fg D tg/ D 0 () .B \ X \ f 1 .DP ˛ / \ fg D tg; B \ X \ f 1 .@D˛ / \ fg D tg/ D 0 () f j.X n Y / \ fg D tg has no critical point in B \ f 1 .D˛ / () .B \ X \ f 1 .D˛ / \ fg D tg; B \ Y \ fg D tg/ D 0 () d), since .B \ X \ f 1 .D˛ / \ fg D tg/ D .B \ X \ fg D tg/.

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Helmut A. Hamm

f) H) e) We look at gjB \ X \ ff D sg. Because of f) we have no critical points in the interior above Dˇ , because of Lemma 3.2 there are no critical points for the restriction to the boundary (above Dˇ ). This implies our assertion, because .B \ X \ ff D sg/ D .B \ X \ ff D sg \ g 1 .Dˇ //: e) H) f) Assume that f) is wrong. Let us then look at the critical locus C of .f; g/j.X n Y / \ BV . We have dim C 1; because of Lemma 3.2, .f; g/jC \ f 1 .DP ˛ / \ g 1 .Dˇ / ! DP ˛ Dˇ is proper, so the image is analytic of dimension 1, because of Sard’s theorem: D 1. The image curve must intersect s D const, where 0 < jsj ˛. So we have that gjB \ X \ ff D sg has critical points. These cause a difference between the Euler characteristics considered in e), contradiction. Proof of Theorem 2.3. This follows from the preceding theorem: a) H) b). Now the question arises how to verify the hypothesis of Theorem 3.3 or condition a). Here the following proposition is useful. Proposition 3.4. Assume that ˆg j CXnY is acyclic outside some analytic set of dimension k. Then the following conditions are equivalent: a) ˆg j CXnY is acyclic outside some analytic subset of dimension k 1; b) for all 1 j1 < < jk m there is a subset V of Ck whose complement has Lebesgue measure 0 such that for all z 2 g 1 .f0g/ \ Y with .zj1 ; : : : ; zjk / 2 V the following holds: .B .z /\.X nY /\fzj1 D zj1 ; : : : ; zjk D zjk ; g D tg/ D 0;

0 < jtj 1:

Here B .z / D fz 2 CmC1 j kz z k g. Proof. We take a Whitney regular stratification of g 1 .f0g/\X adapted to ˆg j CXnY . If S is a stratum the set of points where S ! Ck : z 7! .zj1 ; : : : ; zjk / has rank < k is mapped onto a set of measure 0. a) H) b) Let 1 j1 < < jk m. Take z 2 g 1 .f0g/ \ X such that lies outside a suitable set of measure 0. Then z is contained in some stratum S of dimension k and S ! Ck W z 7! .zj1 ; : : : ; zjk / has rank k at z , so we have transversality of fzj1 D zj1 ; : : : ; zjk D zjk g to S at z . Therefore .zj1 ; : : : ; zjk /

.B .z /\.X nY /\fzj1 D zj1 ; : : : ; zjk D zjk ; g D tg/ D ..ˆg j CXnY /z / D 0: b) ) a): Look at a stratum S of dimension k. Choose j1 ; : : : ; jk such that S ! Ck : z 7! .zj1 ; : : : ; zjk / is of rank k somewhere. If z 2 S is such that


29

.zj1 ; : : : ; zjk / lies outside a suitable set of measure 0 we have that ..ˆg j CXnY /z / D .B .z / \ .X n Y / \ fzj1 D zj1 ; : : : ; zjk D zjk ; g D tg/ D 0; so ˆg j CXnY jS D 0.

4 An auxiliary result The situation of Section 3 is not sufficient in order to discuss equisingularity: we should look at a one-parameter family of space or map germs. The family should be given by the function g, the germ should be taken at .0; t/. So it is reasonable to suppose that Z D .f0g C/ \ U is contained in Y , see Section 2. First we will give equivalent conditions which are necessary in order to have the statement of Theorem 2.1. Recall the notion of rectified homological depth (with complex coefficients): rHd.X; C/ D n means that CX Œn is perverse, cf. [6] Corollary 1.10. This condition holds in particular if X is locally a complete intersection of dimension n or, more generally, if rhd.X / D n. Let B , S be defined as in the last section. Theorem 4.1. Suppose that ˆg j CXnY is acyclic outside 0, Z Y , rHd.X; C/ D n, ˆf CX\fgD0g is acyclic outside 0, and gjsuppˆfk CX is finite for every k. Then the following conditions are equivalent: a) .ˆg CX /0 is acyclic, and t D dim.ˆfn1 CX\fgDtg /.0;t/ is independent of t, b) ˆf CX is acyclic outside Z, and .ˆg j CXnY /0 is acyclic. Note that the condition that .ˆg CX /0 is acyclic means that .B \ X \ fg D tg/ D 1 for 0 < jtj 1. Proof. Let t ¤ 0. We have .ˆf CX\fgDtg /z D .ˆf CX Œ1/z , see [12]. Now CX\fgDtg is perverse, because CX is perverse, so ˆf CX\fgDtg too. By hypothesis, ˆf CX\fgDtg is acyclic outside a finite set, so ˆfk CX\fgDtg D 0 for k ¤ n 1. Similarly for ˆf s CX\fgDtg , where s ¤ 0, because here we are looking at the vanishing cycles at critical points of f j.X n Y / \ fg D tg. These are isolated, see proof of Theorem 3.3 e) H) f).

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Choose s with jsj < ˛ general enough so that ˆf s CX\fgDtg is acyclic. We obtain .B \ X \ fg D tg/ D .B \ X \ fjf j ˛; g D tg/ D .B \ X \ ff D s; g D tg/ X C .1/n1 dim.ˆfn1 f .z/ CX\fgDtg /z : z W jf .z/j˛

Note that .B \ X \ ff D s; g D tg/ D .B \ X \ ff D s; g D 0g/ D 1 C .1/n 0 ; and .B \ X \ fg D tg/ D 1 C .1/n1 dim.ˆng CX /0 : Altogether, we obtain the following equation: X dim.ˆfn1 dim.ˆng CX /0 D 0 C t C f .z/ CX\fgDtg /z :

(*)

z¤.0;t/

a) H) b) Equation (*) yields X dim .ˆfn1 0D f .z/ CX\fgDtg /z : z¤.0;t/

So ˆfn1 CX\fgDtg is acyclic outside .0; t/, hence ˆf CX jfg D tg too. Furthermore, by Theorem 3.3 f) H) a) we get that .ˆg j CXnY /0 is acyclic. b) H) a) By the second assumption we obtain from Theorem 2.3, a) H) d): .B \ X \ fg D tg/ D .B \ Y \ fg D tg/;

0 < jtj 1;

which means 1 C .1/n1 dim H n1 .B \ X \ fg D tgI C/ D 1 C .1/n2 dim H n2 .B \ Y \ fg D tgI C/: Here we use the fact that rHdC Y D n1. Therefore H n1 .B \X \fg D tgI C/ D 0, which implies .ˆg CX /0 D 0. So (*) yields X dim .ˆfn1 0 D 0 C t C f .z/ CX\fgDtg /z ; z¤.0;t/

and the sum on the right vanishes by the assumption. Indeed note that ˆf CX is acyclic outside Z, so f .z/ ¤ 0 if .ˆfn1 C / f .z/ X\fgDtg z ¤ 0, so z is a critical point of .f; g/jX n Y , which contradicts Theorem 3.3, a) H) f). So 0 D t for t ¤ 0.


31

5 Proof of Theorem 2.1 and Theorem 2.2 Now we want to follow the arguments of Lê and Ramanujam [8]. Here, we need stronger hypotheses. In particular we will use the notion of rectified homotopical depth rhd introduced by A. Grothendieck, see [6]. For example, rhd.X / D n as soon as X is locally a complete intersection of dimension n. Assume that X n Z and Y n Z are smooth and that g 1 .f0g/ intersects these spaces transversally. Then ˆg j CXnY is concentrated upon Z \ g 1 .f0g/ D f0g. Assume furthermore that rhd.X / D n, n ¤ 4. Let t be defined as in Theorem 4.1. First, we have the following lemma (where we could use rHd.X; C/ instead of rhd.X /). Lemma 5.1. The following conditions are equivalent: a) .ˆg CX /0 is acyclic, and t is constant; b) .f; g/ W X n Z ! C2 is submersive above D˛ Dˇ . Proof. By hypothesis, ˆg j CXnY and ˆf CX\fgD0g are acyclic outside 0, and the sheaves ˆf CX are acyclic outside Z, in particular gjsupp ˆfk CX is finite. Because of Theorem 4.1, a) () .ˆkg j CXnY /0 D 0 for all k. The equivalence of Theorem 3.3 a) () f) implies that the last condition is equivalent to the condition that .f; g/ W X n Y has no critical points above D˛ Dˇ . The last condition can be rewritten as follows: .f; g/ W X n Z has no critical points above D˛ Dˇ , because gjY n Z is submersive. Altogether we obtain the assertion. Now we have the following result which is related to [8]; here Dˇ .t0 / D ft 2 C j jt t0 j ˇg: Theorem 5.2. Assume that t is constant. Then there is a homeomorphism h such that the following diagram is commutative: h / Xı;˛ \ g 1 .Dˇ .t0 // ; X;˛ \ g 1 .Dˇ .t0 // RRR l RRR lll RRR lll l R l ll .f;g/ RRRR ) ulll .f;g/ D˛ Dˇ .t0 /

where 0 < max.˛; ˇ/ ı jt0 j 1. Proof. We use the definition of B and S introduced before Theorem 3.1! Let t 2 Dˇ .t0 /, s ¤ 0, n 5. First of all, we observe that S \ Y \ fg D 0g is .n 4/-connected, hence simply connected, because of the local Lefschetz theorem for X , see [6] Theorem 2.9: We have rhd.X / D n. Therefore we have that S \ X is

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Helmut A. Hamm

.n2/-connected and has the homotopy type of a space obtained from S \Y \fg D 0g by attaching cells of dimension n 2. The same holds for S \ Y \ fg D tg because this space is homeomorphic to the space S \ Y \ fg D 0g before. Similarly, Sı \ Y \ fg D tg is simply connected too. By Morse theory (use z 7! kzk2 ), .B n BV ı / \ Y \ fg D tg has the homotopy type of a space obtained from S \ Y \ fg D tg by attaching cells of dimension n 2, so it is simply connected too. (**) Furthermore, B \ X \ fg D 0g is contractible, and by stratified Morse theory, we have that B \ X \ fg D 0g has the homotopy type of a space obtained from B \X \fg D 0; f D sg by attaching 0 .n1/-spheres, so B \X \fg D 0; f D sg has the homotopy type of a bouquet of 0 .n 2/-spheres. The same holds for the homeomorphic space B \ X \ fg D t; f D sg. Similarly, Bı \ X \ fg D t; f D sg has the homotopy type of a bouquet of t .n 2/-spheres. Using z 7! kzk2 as a Morse function we see that B \X \fg D t; f D sg has the homotopy type of a space obtained from Bı \ X \ fg D t; f D sg by attaching cells of dimension n 2. So H k .B \ X \ fg D t; f D sg; Bı \ X \ fg D t; f D sgI Z/ D 0 for k > n 2. Since 0 D t by assumption we get that H k .B \ X \ fg D t; f D sgI Z/ ' H k .Bı \ X \ fg D t; f D sgI Z/; i.e. H k ..B n BV ı / \ X \ fg D t; f D sg; Sı \ X \ fg D t; f D sgI Z/ ' H k .B \ X \ fg D t; f D sg; Bı \ X \ fg D t; f D sgI Z/ D0 for all k. The same holds for homology instead of cohomology. Also, by duality, we can deduce that Hk ..B n BV ı / \ X \ fg D t; f D sg; S \ X \ fg D t; f D sgI Z/ D 0 for all k. (***) 1 V Now .f; g/j.B n Bı / \ X defines a C fibre bundle over D˛ Dˇ .t0 / because gj.Y n f0g/ C is a submersion. The fibre is .B n BV ı / \ X \ fg D t; f D sg, it is simply connected, the same holds for its boundary components: this follows from (**) because we may pass to the case s D 0. The inclusion of each boundary component into .B n BV ı / \ X \ fg D t; f D sg defines a homotopy equivalence, by Whitehead’s theorem, see [13], and (***). By the h-cobordism theorem (cf. [9]) we conclude that .B n BV ı /\X \fg D t; f D sg is diffeomorphic to .S \X \fg D t; f D sg/Œ0; 1. So there is a diffeomorphism of .B n BV ı / \ X \ f 1 .D˛ / \ g 1 .Dˇ .t0 // onto .S \ X \ f 1 .D˛ / \ g 1 .Dˇ .t0 /// Œ0; 1 which is compatible with .f; g/. This implies our assertion.


33

The case n 3 is easier: Assume n D 3. Put A D .B n BV ı / \ X \ fg D t; f D sg; B D Sı \ X \ fg D t; f D sg; B 0 D S \ X \ fg D t; f D sg: We have that H0 .BI Z/ ' H0 .AI Z/, so the irreducible components Aj of A and Bj of B correspond to each other. Now we have Hk .Bj I Z/ ' Hk .Aj I Z/. Furthermore Bj must be homeomorphic to S1 . Similarly for B 0 , Bj0 instead of B, Bj . So Aj has the homotopy type the complement of two points in a compact Riemann surface. Let g be its genus, then .Aj / D 2g. On the other hand, .Aj / D .Bj / D 0. Therefore g D 0, and Aj is diffeomorphic to the complement of two disjoint disks in the Riemann sphere. So Aj ' S1 Œ0; 1, as expected. In the case n 2 we must have A D B D B 0 D ;. Proof of Theorem 2.2. This follows from Lemma 5.1 and Theorems 5.2 and 5.3 below. In fact, Y nZ is smooth and g 1 .f0g/ intersects the spaces X nZ and Y nZ transversally because .f; g/ W X n Z ! C2 is a submersion. Now we want to prove 2.1. Using Lemma 5.1 we reformulate it as follows. Theorem 5.3. Under the equivalent hypotheses of Lemma 5.1, we have that, for 0 < max.˛; ˇ/ 1 the family of mappings f W g 1 .ftg/\X;˛ ! D˛ , jtj ˇ, is locally trivial, i.e. there is a homeomorphism h W X;˛ \ g 1 .Dˇ / ! F Dˇ and a continuous mapping f0 W F ! D˛ such that the following diagram is commutative: h / F Dˇ : X;˛ \ g 1 .Dˇ / PPP r r PPP rr PPP r r r .f;g/ PPP ' yrrr .f;id/ D˛ Dˇ

Example. We modify the example of Briançon and Speder [2]. Let X D f.z1 ; z2 ; z3 ; t/ 2 C4 j z35 C tz26 z3 C z27 z1 C z115 D 0g; f .z1 ; z2 ; z3 ; t/ D z2 ; g.z1 ; z2 ; z3 ; t/ D t: Then we may apply Theorem 5.3: in particular, t is constant because of the weighted homogeneous situation; furthermore, .ˆg CX /0 is acyclic. By [2], the pair .X n Z; Z/ does not satisfy Whitney’s regularity condition, so the local triviality is not merely a consequence of stratification theory.

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Helmut A. Hamm

Remark. The hypothesis that .ˆg CX /0 is acyclic is fulfilled in particular when X D CmC1 (with n D m C 1, of course). Then it is sufficient to suppose that t is constant, see Lemma 5.1. It is possible that we may apply Theorem 5.2 but not Theorem 5.3. Example. Let X D f.x; y; t/ 2 C3 j y 2 D x 2 .x t/g; f .x; y; t/ D x: Then Y D f0g C, .ˆf CX\fgDtg /.0;t/ has dimension 1, so t is constant. However, the conclusion of 5.3 does not hold. On the other hand, Theorem 5.2 is applicable. Proof of Theorem 5.3. Put .z/ D k.z1 ; : : : ; zn ; 0/k: Let † be the critical set of .; Re g/jY \ fIm g D 0g n Z. After shrinking U if necessary, each branch of the closure of † is parametrized by a real analytic curve 7! . /, with . / D 0, and we may choose the parametrisation in such a way that . . // D ˙ . In this way we see that there is k > 0 such that along † we have the inequality jRe gjk < < jRe gj1=k . We may assume ˇ < 1 and 2ˇ 1=k < . On .Y n Z/ \ fIm g D 0g, in a neighborhood of fjRe gj ˇ; 1=2jRe gjk g we can find a vector field v with d.v/ 0; dg.v/ 1. Similarly in a neighborhood of fjRe gj ˇ; 2jRe gj1=k g. Note that Re gj.Y n Z/ \ fIm g D 0; jRe gjk jRe gj1=k ; 0 < jRe gj ˇg defines a fibre bundle over Œˇ; ˇ n f0g which is trivial over Œˇ; 0Œ, resp. 0; ˇ; by the h-cobordism theorem (for n 5) or a simple direct argument (for n 3), see proof of Theorem 5.2, the fibre is diffeomorphic to F Œ0; 1, resp. FC Œ0; 1, where fjRe gjk D g corresponds to F˙ f0g and fjRe gj1=k D g to F˙ f1g. The projection onto Œ0; 1 induces a mapping W .Y n Z/ \ fIm g D 0; jRe gj ˇ; jRe gjk jRe gj1=k g ! R: Along jRe gjk D and jRe gj1=k D we have no z with dz .jfg D constg/ D dz . jfg D constg/, with 0. Therefore there is a vector field v on a neighborhood of fjRe gj ˇ; 1=2jRe gjk jRe gjk g in .Y n Z/ \ fIm g D 0g such that g d.v/ 0, dg.v/ D 1, and d.v/ D 0 along f1=2jRe gjk D g, d .v/ D 1=g along fjRe gjk D g. On a neighborhood of fjRe gj ˇ; jRe gjk jRe gj1=k g in .Y n Z/ \ fIm g D 0g we can find a vector field v such that dg.v/ D 1, d .v/ D 1=g, and gd.v/ 0 along f D jRe gjk g or f D jRe gj1=k g. Finally, on a neighborhood of fjRe gj ˇ; jRe gj1=k 2jRe gj1=k g in .Y n Z/ \ fIm g D 0g


35

we can find a vector field v such that g d.v/ 0, dg.v/ D 1, and d.v/ D 0 along f2jRe gj1=k D g, d .v/ D 1=g along fjRe gj1=k D g. We may arrange that the vector fields match together. The resulting vector field can be extended on .Y nZ/\fjgj ˇ; g such that it is controlled on a neighborhood of Z \ fIm g ¤ 0g, dg.v/ 1, and along f D g, d.v/ 0. Finally we may extend to .X n Z/ \ fjf j ˛; jgj ˇ; g such that dg.v/ 1; df .v/ 0 and, along D , d.v/ 0. Note here that .f; g/jX n Y and .f; g/jX \ f D g are without critical points in this set. Now fix ˇ > 0 sufficiently small. Then the flow ˆ corresponding to v is defined on f.z; t/ j jRe g.z/j ˇ; t 2 Œˇ Re g.z/; ˇ Re g.z/g. Assume that the lower bound of the interval is not correct. Then there is a t0 > 0 and an integral curve c such that c.t/ is defined for t0 < t 0 and .0; t1 / is an accumulation point of c.t/ for t ! t0 . Then necessarily t1 D 0 and g.c.t// D t Ct0 . We must have that c is a curve in Y \ fIm g D 0g. If .c.t// < .Re g.c.t///k for all these t we have that .c.t// is monotonously decreasing, contradiction. So there is a t with .Re g.c.t ///k .c.t //. Assume that .Re g.c.t///k .c.t// .Re g.c.t///1=k for t0 < t t ; then 1 1 D d .c.t// P D g.c.t// t C t0 for these t, so Zt .c.t //

.c.t// D

d .c.t//dt P t

Zt

D t

1 dt t C t0

D ln.t C t0 / ln.t C t0 / ! 1 for t ! t0 , in contradiction to the fact that .c.t // and .c.t// are contained in Œ0; 1. So we must have a t with .c.t // .Re g.c.t ///1=k ; then for t t we have that .c.t// is monotonously decreasing too, which gives again a contradiction. Similarly if the upper bound is not correct. y defined by Now ˆ can be extended continuously to ˆ y ˆ..0; t/; / D .0:t C / on X \ fjf j ˛; jgj ˇ; g. Let .pl / be a sequence in the complement of Z which converges to p 2 Z. Then ˆ.pl ; t/ cannot accumulate to a point in the complement of Z. Otherwise we get a contradiction using the continuity of the opposite flow. Similarly we can proceed interchanging the role of Re g and Im g: we find a suitable vector field w on B \ X n Z such that dg.w/ i; df .w/ 0. In this way we obtain the desired trivialization.

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As a consequence of Theorem 2.2 and Theorem 2.1, we have that the topological type of f jX \ g 1 .ftg/ at .0; t/ does not depend on t, as we have seen in Section 2. In fact we can say more. Corollary 5.4. Under the hypothesis of Theorem 5.3, the topological type of .f; g t/ at .0; t/ does not depend on t, where jtj < ˇ 1. Proof. Assume 0 < max.˛; ˇ 0 / jtj < ˇ 1. Then the topological type of .f; g/ at 0 is represented by the topological type of the mapping .f; g/ W X \ f g \ f 1 .D˛ / \ g 1 .Dˇ0 / ! D˛ Dˇ0 : This mapping has, by Theorem 5.3, the same topological type as .f; g t/ W X \ f g \ f 1 .D˛ / \ g 1 .Dˇ0 .t// ! D˛ Dˇ0 ; which represents, by Theorem 5.2, the topological type of .f; g t/ at .0; t/. Of course, the condition that t is constant is essential in Lemma 5.1 and Theorem 5.2 and Theorem 5.3: Example. Let X D C2 ; f .x; t/ D x.x t/; g.x; t/ D t: Then .ˆkg CX /0 D 0 for all k, 0 D 1; t D 0, for t ¤ 0.

References [1]

J. Briançon, P. Maisonobe, and M. Merle, Localisation des systèmes différentiels, stratifications de Whitney et condtion de Thom, Invent. Math. 117 (1994), 531–550. 26

[2]

J. Briançon and J.-P. Speder, La trivialité topologique n’implique pas les conditions de Whitney, C. R. Acad. Sci. Paris Sér. A 280 (1975), 365–367. 33

[3]

M. Goresky and R. MacPherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Springer, Berlin 1988. 20

[4]

H. A. Hamm, Complements of hypersurfaces and equisingularity, in Singularity theory – Dedicated to Jean-Paul Brasselet on his 60 th birthday. Proceedings of the 2005 Marseille Singularity School and Conference, CIRM, Marseille, France, 24 January – 25 February 2005, ed. by D. Chériot, N. Dutertre, C. Murolo, A. Pichon and D. Trotman, World Scrientific, Singapore 2007, 625–649. 25, 26


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[5]

H. A. Hamm, Euler characteristics and atypical values, in Real and complex singularities. Selected papers from the 10 th Workshop held at São Paulo University, São Carlos, July 27– August 2, 2008, ed. by M. Manoel, M. C. Romero Fuster, and C. T. C. Wall, London Math. Soc. Lecture Notes 380, Cambridge University Press, Cambridge 2010, 167–184. 25, 26, 27

[6]

H. A. Hamm and D. T. Lê, Rectified homotopical depth and Grothendieck conjectures, in The Grothendieck Festschrift. A collection of articles written in honor of the 60 th birthday of Alexander Grothendieck, vol. II, ed. by P. Cartier, L. Illusie, N. M. Katz, G. Laumon, Y. Manin, and K. A. Ribet, Birkhäuser, Boston 1990, 311–351. 23, 29, 31

[7]

H. C. King, Topological type in families of germs, Invent. Math. 62 (1980/81), 1–13. 23

[8]

D. T. Lê and C. P. Ramanujam, The invariance of Milnor’s number implies the invariance of the topological type, Amer. J. of Math. 98 (1970), 67–78. 23, 31

[9]

J. Milnor, Lectures on the h-cobordism theorem. Princeton University Press, Princeton 1965. 32

[10] A. Parusiński, A note on singularities at infinity of complex polynomials, in Symplectic singularities and geometry of gauge fields. Papers from the Banach Center Symposium held in Warsaw, 1995, ed. by R. Budzyński, S. Janeczko, W. Kondracki, and A. F. Künzl, Banach Center Publications 39, Polish Academy of Sciences, Inst. of Mathematics, Warsaw 1997, 131–141. 25 [11] D. Sullivan, Combinatorial invariants of analytic spaces, in Proceedings Liverpool Singularities Symposium I (1969/70), ed. by C. T. C. Wall, Lecture Notes in Mathematics 192, Springer, Berlin 1971, 165–168. 27 [12] J. Schürmann, Topology of singular spaces and constructible sheaves, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne 63, Birkhäuser, Basel 2003. 27, 29 [13] E. H. Spanier, Algebraic topology, McGraw-Hill, New York etc. 1966. 32 [14] J. G. Timourian, The invariance of Milnor’s number implies topological triviality, Amer. J. Math. 99 (1977), 437–446. 23

Jet schemes of homogeneous hypersurfaces Shihoko Ishii,Akiyoshi Sannai, and Kei-ichi Watanabe Graduate School of Mathematical Science, University of Tokyo Komaba, Meguro, Tokyo, 153-8914, Japan e-mail: [email protected] Graduate School of Mathematics, Nagoya University Furocho, Chikusaku, Nagoya, 464-8602, Japan e-mail: [email protected] Department of Mathematics, College of Human and Science, Nihon University Setagaya, Tokyo, 156-0045, Japan e-mail: [email protected]

Abstract. This paper studies the singularities of jet schemes of homogeneous hypersurfaces of general type. We obtain the condition of the degree and the dimension for the singularities of the jet schemes to be of dense F -regular type. This provides us with examples of singular varieties whose m-jet schemes have rational singularities for every m.

1 Introduction The concept jet schemes over an algebraic variety was introduced by Nash in his preprint in 1968 which is later published as [10]. These spaces represent the nature of the singularities of the base space. In fact, papers [1], [2], [8], and [9] by Musta¸taˇ , Ein, and Yasuda show that geometric properties of the jet schemes determine properties of the singularities of the base space. To summarize, their results among others are as follows. Let X be a variety of locally a complete intersection over an algebraically closed field of characteristic zero. Then Xm is of pure dimension (resp. irreducible, normal) for all m 1 if and only if X has log-canonical (resp. canonical, terminal) singularities. According to this form, it is natural to formulate the question. Problem 1.1. Does the following hold? X is non-singular if and only if Xm has at worst certain “mild” singularities for every m 1. Partially

supported by JSPS grant in aid (B) 22340004. supported by JSPS research fellow 08J08285. Partially supported by JSPS grant in aid (C) 20540050. Partially

40

Shihoko Ishii, Akiyoshi Sannai, and Kei-ichi Watanabe

Does the bound of a certain invariant of the singularities on Xm characterize the smoothness of X ? The easiest candidate for “certain mild singularities” is a rational singularity. In this paper, we show that a rationality is not appropriate for the required statement in the problem. This is proved by providing with counter examples. We study the singularities of the jet schemes of homogeneous hypersurface of general type and obtain the condition of the degree and the dimension for the singularities of the jet schemes to be dense F -regular type. over a field k of characteristic 0 defined Theorem 1.2. Let X be a hypersurface in AN k by a polynomial of general type of degree d , i.e., X f .x1 ; : : : ; xN / D i1 ;i2 ;:::;id xi1 xi2 xid ; i1 i2 ;id

where fi1 ;i2 ;:::;id g are algebraically independent over Q. If d 2 N , then the jet scheme Xm has at worst rational singularities for every m 2 N. A rational singularity is defined by using a resolution of the singularities. Since it is almost impossible to construct a resolution of the singularities of the jet scheme even for the simplest singularities on the base variety because of too many variables on the jet scheme, we use the positive characteristic method. The theorem shows examples of singular X whose jet schemes Xm for all m have at worst rational singularities. We also show that X is non-singular if and only if the F -pure threshold does not change between Xm ’s for different m. Theorem 1.3. Let X be a variety of locally a complete intersection at 0 over a field of characteristic p > 0. For m < m0 , assume also Xm ; XmC1 ; : : : ; Xm0 are complete intersections at the trivial jets 0m ; : : : ; 0m0 . Then, the following are equivalent: (i) .X; 0/ is non-singular; (ii) fpt.Xm ; 0m ; 0m / D fpt.Xm0 ; the truncation morphism.

m0 m

1

.0m /; 0m0 /, where

m0 m

W Xm0 ! Xm is

Throughout this paper the base field k is an algebraically closed field.

2 Preliminaries on jet schemes and positive characteristic methods 2.1. For a scheme X of finite type over an algebraically closed field k, we can associate the space of m-jet (or the m-jet scheme) Xm for every m 2 N. The exact definition of

Jet schemes of homogeneous hypersurfaces

41

the m-jet scheme and the basic properties can be seen in [6]. We use the notation and the terminologies in [6]. The canonical projection Xm ! X is denoted by m . defined by an equation f D 0, then the m-jet If X is a closed subscheme of AN k scheme Xm is defined in A.mC1/N D Spec kŒxi.j / j 1 i N; j D 0; 1; : : : m k by the equations fF .j / D 0gj D0;:::;m . Here, the F .j / 2 kŒxi.j / j 1 i N; j D 0; 1; : : : m is defined as follows: X X .j / X f x1.j / t j ; : : : ; xN t j D F .j / t j : j

j

j

.j / /. For a point For the simplicity of the notation, we write x.j / D .x1.j / ; : : : ; xN P 2 X , let Pm 2 Xm be the trivial m-jet at P . In particular if P is the origin 0 2 X , then 0m is defined by the maximal ideal .x.0/ ; : : : ; x.m/ / kŒx.0/ ; : : : ; x.m/ in AN k

. A.mC1/N k 2.2. The Frobenius map of rings of positive characteristic has been important tool to study the singularities of positive characteristic. The concepts F -pure, strongly F regular, weakly F -regular and F -rational appear in this stream. These notions have close relations with rationality and log-canonicity: A singularity is of dense F -rational type (i.e. it is F -rational by the reduction to characteristic p for infinitely many prime number p) if and only if it is rational by Smith [11], Hara [4], and Mehta and Srinivas [7]. If a normal Q-Gorenstein singularity is of dense F -pure type (i.e., it is F -pure by the reduction to characteristic p for infinitely many prime number p), then it is logcanonical by Hara and Watanabe [5]. In the Gorenstein case, the three notions strongly F -regular, weakly F -regular and F -rational coincide. When we restrict ourselves in the case of a complete intersection, we call it just F -regular. The definitions of F -pure and F -regular can be found in the papers above and we do not repeat them here. Lemma 2.3. The m-jet scheme Xm is F -pure (resp. strongly F -regular, rational) 1 .P / if and only if Xm is F -pure (resp. strongly F -regular, rational) along the fiber m at Pm . Proof. Note that these conditions, F -pure, strongly F -regular, rational, are open conditions. Therefore, if Xm has one of these conditions at Pm , then Xm has that on an open neighborhood U Xm of Pm . Remember that the multiplicative algebraic 1 .P / group A1k n f0g acts on Xm and the closure of the orbit of every point y in m contains Pm (see, for example, [6]). This shows that on Xm there is an isomorphism which sends y into U . Hence, Xm has the condition at y. Lemma 2.4 ([12, Lemma 3.9]). Let .R; m/ be a local ring at a closed point of a non-singular variety over an algebraically closed field of characteristic p and I R an ideal. Fix any ideal a R and any real number t 0. Write S D R=I .

42


(i) The pair .S; .aS/t / is F-pure if and only if for all large q D p e 0, we have abt.q1/c .I Œq W I / 6 mŒq . (ii) The pair .S; .aS/t / is strongly F-regular if and only if for every element g 2 RnI , there exists q D p e > 0 such that gadtqe .I Œq W I / 6 mŒq . In the case of a complete intersection, we can regard the following criteria of Fedder type as the definition of F -pure and F -regular. Corollary 2.5. If .S; m/ is aQ regular local ring of characteristic p > 0, f1 ; f2 ; : : : ; fr is an S-sequence and f D riD1 fi , then the following are equivalent: (i) S=.f1 ; : : : ; fr / is F -pure (resp. F -regular) (ii) f p1 62 mŒp (resp. for any non-zero g 2 S, there is q D p e > 0 such that gf q1 62 mŒq ). Proof. The statement on F -purity is in [3]. If I D .f1 ; : : : ; fr / is generated by S-regular sequence, then I Œq W I D I Œq C f q1 S and our assertion on F -regularity follows from Lemma 2.4. To apply the criteria, we need to show that our jet schemes are complete intersections. The following is a characteristic free statement and is a refinement of a special case of the statement obtained by Musta¸taˇ [8] for characteristic zero. defined by a homogeneous polynomial Lemma 2.6. Let X be a hypersurface of AN k f of degree d . Assume X has an isolated singularity at the origin 0 2 X . Then, it follows: (i) if d N , then Xm is not irreducible for every m N 1; (ii) if d N 1, then Xm is irreducible, therefore a complete intersection, for every m 2 N. Proof. First of all, we note that for a hypersurface X with the isolated singularity at 0, the jet scheme Xm is irreducible if and only if 1 .0/ < .m C 1/.N 1/: dim m

(2.1)

Indeed, as n f0g/ D .m C 1/.N 1/, “only if” part is trivial. For the “if” part, note that Xm is defined by m C 1 equations in A.mC1/N . Therefore, every k irreducible component of Xm has dimension greater than or equal to .m C 1/.N 1/. 1 If we assume the inequality (2.1), then m .0/ does not provide with an irreducible component of Xm . 1 .0/ is defined by For the proof of (i), assume d N . The fiber m 1 dim m .X

F .0/ .0/; F .1/ .0; x.1/ /; : : : ; F .m/ .0; x.1/ ; : : : ; x.m/ / D Spec kŒx.1/ ; : : : ; x.m/ and the first d polynomials are trivial because on AmN k .j / F is homogeneous of degree d and of weight j, therefore every monomial in F .j /

Jet schemes of homogeneous hypersurfaces .0/ (j < d ) has P the factor xi for some i, where the weight of a monomial defined as j. Therefore, for m N 1

43

Q

xi.j / is

1 .0/ mN maxf0; .m C 1/ d g .m C 1/.N 1/: dim m

Hence, Xm is not irreducible for m N 1. For the proof of (ii), assume d N 1. For m such that m d 1, as we see in 1 .0/ D AmN . As m N 2, the inequality (2.1) holds, the previous argument, m k therefore Xm is irreducible. For m such that m d , we will show it by induction on m. Assume that X0 D X; : : : ; Xm1 are irreducible. We note that for j d F .j / .0; x.1/ ; : : : ; x.j / / D F .j d / .x.1/ ; : : : ; x.j d C1/ /; 1 because f is homogeneous of degree d. Since m .0/ is defined by F .j / .0; x.1/ ; : : : ; .j / .1/ .m/ x / (j D d; : : : ; m) in Spec kŒx ; : : : ; x , we obtain 1 m .0/ D Spec kŒx.1/ ; : : : ; x.m/ =.F .j d / .x.1/ ; : : : ; x.j d C1/ //j d D0;::;md 1/N ' Xmd A.d : k 1 By this we have dim m .0/ D .m d C 1/.N 1/ C .d 1/N and it follows the inequality (2.1). Now we obtain the irreducibility of Xm and in this case we have the codimension of Xm equal to the number of the defining equation in A.mC1/N . k

3 Singularities of the jet schemes Definition 3.1. Under the notation in 2.1, let k be a field of characteristic zero and p a prime number. Let m be the maximal ideal .x.0/ ; x.1/ ; : : : ; x.m/ / kŒx.0/ ; x.1/ ; : : : ; x.m/ . Take a polynomial F in the ring. A monomial x 2 kŒx.0/ ; x.1/ ; : : : ; x.m/ is called a good monomial for .F; p/ if x 62 mŒp and x 2 F p1 by modulo p reduction. Here “x 2 F p1 ” means x appears in F p1 with non-zero coefficient. over a field k of characteristic 0 defined Theorem 3.2. Let X be a hypersurface in AN k by a polynomial of general type of degree d , i.e. X f .x1 ; : : : ; xN / D i1 ;i2 ;:::;id xi1 xi2 xid ; i1 i2 id

where fi1 ;i2 ;:::;id g are algebraically independent over Q. If d 2 N , then the jet scheme Xm is dense F -regular type for every m 2 N. Proof. Fix m 2 N. Let p be a prime number satisfying p > m.d 1/ C d . By Lemma 2.6, we may assume that Xm is a complete intersection. For the polynomial .j / 2 kŒx.0/ ; x.1/ ; : : : ; x.j / be as in 2.1 and put F D f 1 ; : : : ; xN , let F Qm2 kŒx .j / . Let g be any polynomial in kŒx.0/ ; x.1/ ; : : : ; x.m/ . We will show that j D0 F

44


there exist e > 0 and a monomial M 2 F p any e, we can decompose

e 1

e

with gM 62 mŒp by modulo p. For

p e 1 D .p e p e1 / C .p e1 p e2 / C .p e2 1/: p e1 p e2 ; c D p e2 1. Define three monomials Let a D p e p e1 Q;mb D .j L1 ; L2 ; L3 2 F D j D0 F / . First pick up the term L1 .j / from F .j / as follows: L1 .0/ D 1;2;:::;d x1.0/ x2.0/ xd.0/ ; .0/ L1 .1/ D d C1;d C2;:::;2d xd.0/ xd.0/ x2d x .1/ ; C1 C2 1 2d .0/ .0/ .0/ x2d x3d x .1/ x .1/ ; L1 .2/ D 2d C1;2d C2;:::;3d x2d C1 C2 2 3d 1 3d

:: : .1/ .1/ .1/ L1 .d 1/ D d 2 d C1;d 2 d C2;:::;d 2 xd.0/ 2 d C1 xd 2 d C2 xd 2 1 xd 2 ;

L1 .d / D 1;2;:::;d x1.1/ x2.1/ xd.1/ x .1/ ; 1 d .1/ L1 .d C 1/ D d C1;d C2;:::;2d xd.1/ xd.1/ x2d x .2/ ; C1 C2 1 2d

:: : .2/ x .2/ ; L1 .2d / D 1;2;:::;d x1.2/ x2.2/ x2d 1 2d

:: : Q Define L1 D jmD0 L1 .j /=(coefficients). Then, note that every variable xi.j / appears in L1 at most once. We can see that La1 2 F a by modulo p, by noting that the coefficients i1 ;:::;id are algebraically independent over Q. Next, pick up the term L2 .j / from F .j / as follows .0/ d / ; L2 .0/ D 2d;:::;2d .x2d .0/ d 1 .1/ L2 .1/ D d 2d;:::;2d .x2d / x2d ; .0/ d 1 .2/ L2 .2/ D d 2d;:::;2d .x2d / x2d ;

:: : .0/ d 1 .m/ L2 .m/ D d 2d;:::;2d .x2d / x2d :


Define L2 D

Qm j D0

45

L2 .j /=(coefficients). Then, note that a variable with positive

.j / x2d

.0/ weight (i.e., .j > 0/) appears in L2 at most once and a variable x2d appears b b m.d 1/ C d times in L2 . We can see that L2 2 F by modulo p. Finally, pick up the term L3 .j / from F .j / as follows:

L3 .0/ D d;:::;d .xd.0/ /d L3 .1/ D d d;:::;d .xd.0/ /d 1 xd.1/ L3 .2/ D d d;:::;d .xd.0/ /d 1 xd.2/ :: :

Define L3 D

Qm j D0

L3 .m/ D d d;:::;d .xd.0/ /d 1 xd.m/ : L3 .j /=(coefficients). Then, note that a variable with positive

xd.j /

weight (i.e., .j > 0/) appears in L3 at most once and a variable xd.0/ appears m.d 1/ C d times in L3 . We can see that Lc3 2 F c by modulo p. P i Define M D La1 Lb2 Lc3 . Noting that .p s 1/Š has exactly . s1 iD1 .p 1//-powers s s1 s1 of p as a factor and .p p /Š has exactly .p 1/-powers of p as a factor for every positive integer s, we obtain that .p e 1/Š=.aŠbŠcŠ/ does not have p as a factor. e Hence, it follows that M 2 F p 1 by modulo p. Every variable of weight 0 appears in M at most maxfb.md m C d /; a C c.md m C d /g times and p e maxfb.md m C d /; a C c.md m C d /g ! 1 .e ! 1/: On the other hand, every variable of positive weight appears in M at most a C b times (here, we used the fact that a C c a C b). We can also see that p e .a C b/ ! 1 .e ! 1/: e

Therefore, for any polynomial g 2 kŒx.0/ ; x.1/ ; : : : ; x.m/ we obtain gM 62 mŒp for sufficiently large e. Corollary 3.3 (Theorem 1.2). Let k be a field of characteristic zero. Let X be a hypersurface in AN defined by a homogeneous polynomial of general type of degree k 2 d . If d N , then the jet scheme Xm has at worst rational singularities for every m 2 N. Remark 3.4. It is expected that Theorem 3.2 and Corollary 3.3 also hold for the hypersurface X of Fermat type of degree d such that d 2 N. defined by a Theorem 3.5. Assume char k D p > 0. Let X be a hypersurface in AN k homogeneous polynomial f 2 kŒx1 ; : : : ; xN of degree d . If the jet scheme Xm is a complete intersection and F -pure for every m 2 N, then d 2 N.

46


Proof. As Xm is a complete intersection and F -pure, there exists a good monomial x for .F; p/. Fix an expression of x into a product of monomials of .F .j / /p1 ’s. Write Qm x D j D0 x.j /, where x.j / is the contribution from .F .j / /p1 . Let aij k be the power P of xk.i/ in x.j / and let aij D . N kD1 aij k /=.p 1/. Then the matrix A D .aij /0i;j m satisfies the following conditions: (i) A is an upper triangular matrix, Pm (ii) iD0 aij D d; .0 j m/, Pm (iii) iD0 iaij D j; .0 j m/ and Pm (iv) j D0 aij N; .0 i m/. Under these conditions we will prove that Pif m is sufficiently large, for any real number s < d 2 , there exists i such that ˛i D jmD0 aij > s, which shows d 2 N by (4). Let C be the matrix as follows: 0 1 d d 1 d 2 1 B C B C 1 2 d 1 d d 1 1 B C B C B C 1 d 1 d d 1 C DB C: B C B C 1 B C @ A :: : In other words, C D .cij /0i;j m be defined as cij D d u (if j D d i ˙ u) for every u D 0; 1; : : :P ; d 1 and cij D 0 (otherwise). Then, C also has the properties (1) (3). Let i D jmD0 cij , and assume m D d l for an integer l > 0, then 8 d.d C 1/ ˆ ˆ ; if i D 0, ˆ ˆ ˆ 2 ˆ ˆ ˆ ˆ 0. Put D D il1 ıi and P d.d 1/ 0 0 e for i D 0 and ıi e D D il ıi , so that D C D D 0. Since ıi 2 d.d 1/ 0 for 1 i l 1, we have D le C 2 and D le d.d21/ . If we put P ıi D e i for 1 i l 1 and D D le C d.d21/ , then l1 iD1 i and 2


D 0 D le

d.d 1/ 2 l1 X

47

C . By this we have iıi D e

iD0

l1 X

i

iD0

l1 X

ii e

iD0

l.l 1/ .l 1/: 2

On the other hand, noting that ıi 0 for i l C 1, we have m X

iıi lD 0 :

iDl

Thus, we conclude that m X iD0

iıi e

l1 X iD1

i C l le

d.d 1/ 1 D .el 2 C .e d 2 C d //l/: 2 2

But if l isPsufficiently large, then the latter will be positive and this contradicts to the fact that m iD0 iıi D 0. 3.6. Takagi and Watanabe [13] introduced the invariant F -pure threshold (denoted by fpt.X; Z; P /) for a scheme X over a field of positive characteristic and the closed subscheme Z X at a point P 2 X . It is closely related to the log-canonical threshold for characteristic zero. Here, we refer the formula for a complete intersection case. defined Let k be a field of characteristic p > 0. Let X be a subscheme of AN k Qr by polynomials f1 ; : : : ; fr where dim X D N r. Let f D iD1 fi . Let a closed subscheme Z be defined by an ideal I kŒx1 ; : : : ; xN and m be the maximal ideal of a point P 2 X . Let q D p e . Then, by Lemma 2.4 maxfr j I r f q1 6 mŒq g : q!1 q

fpt.X; Z; P / D lim

As we think of only local a complete intersection case, we can regard this formula as the definition of F -pure threshold. Theorem 3.7. Let X be a variety of locally a complete intersection at 0 over a field of characteristic p > 0. For m < m0 , assume also Xm ; XmC1 ; : : : ; Xm0 are complete intersections at the trivial jets 0m ; : : : ; 0m0 . Then, the following are equivalent: (i) .X; 0/ is non-singular; (ii) fpt.Xm ; 0m ; 0m / D fpt.Xm0 ; m0 m 1 .0m /; 0m0 /, where m0 m W Xm0 ! Xm is the truncation morphism. Proof. Assume (i), then Xi is non-singular for every i 2 N and the truncation morphism m0 m W Xm0 ! Xm is smooth. In this case, Xm0 ; Xm ; m0 m 1 .0m / and f0m g are all non-singular. Therefore by the formula in 3.6, we have fpt.Xm ; 0m ; 0m / D codim.f0m g; Xm / D codim. m0 m 1 .0m /; Xm0 / D fpt.Xm0 ; m0 m 1 .0m /; 0m0 /. For the proof of (ii) H) (i), we first show fpt.Xm ; 0m ; 0m / > fpt.XmC1 ;

mC1;m

1

.0m /; 0mC1 /;

(3.1)

48


if .X; 0/ is singular and Xm and XmC1 are complete intersections at the trivial jets. Let m and m0 be the maximal ideals of OXm ;0m and of OXmC1 ;0mC1 , respectively. Let f1 ; : : : ; fr define X in AN , where r D codim.X; AN /. Then, under the notation in 2.1, Xi is defined by Fl.j / .l D 1; : : : ; r; j i/ in A.iC1/N D .AN / . Let k i k Q Q Q .j / r m mC1 .j / .j / .j / 0 e G D lD1 Fl and G D j D0 G , G D j D0 G . For q D p , let rq D maxfs j ms G q1 6 mŒq g

rq0 D maxfs j ms G 0

q1

6 m0

Œq

(3.2)

g:

(3.3) 0

Let x be a monomial in G 0 q1 and c be an element of mrq such that cx 62 m0 Œq . Then x is factored as x D x0 x00 , where x0 and x00 are contributions from G q1 and from .G .mC1/ /q1 , respectively. As Fl.mC1/ is of weight m C 1, each monomial of Fl.mC1/ has at most one variable xi.mC1/ of weight m C 1. Then, if we factorize x00 D zz0 with z 2 kŒx.0/ ; : : : ; x.m/ and z0 2 kŒx.mC1/ , we have deg z .q 1/

r X

.dj 1/;

j D1

where dj D ord fj . Here, we note that dj 1 for all j D 1; : : : ; r and dj 2 for some j , since .X; 0/ is singular. ThePcondition cx 62 m0 Œq gives .cz/x0 62 mŒq . Noting r 0 that x0 2 G q1 and cz 2 mrq C.q1/ j D1 .dj 1/ , we obtain rq0 C .q 1/

r X

.dj 1/ rq ;

j D1

which yields fpt.XmC1 ;

mC1;m

1

.0m /; 0mC1 / C

r X

.dj 1/ fpt.Xm ; 0m ; 0m /

j D1

as required in (3.1). Now we can see the following in a similar and easier way as in the above discussions: fpt.Xm ; I; 0m / fpt.XmC1 ; I OXmC1 ; 0mC1 /

(3.4)

for an ideal I OXm ;0m . (This follows by just replacing ms by I s in (3.2) and (3.3).) By (3.1) and (3.4), we obtain that if .X; 0/ is singular, then fpt.Xm ; 0m ; 0m / > fpt.Xm0 ; 1 .0m /; 0m0 /. Therefore we conclude (ii) H) (i). m0 m


49

References [1]

L. Ein, M. Musta¸taˇ , and T. Yasuda, Jet schemes, log discrepancies and inversion of adjunction, Invent. Math. 153 (2003), 519–535. 39

[2]

L. Ein and M. Musta¸taˇ , Inversion of Adjunction for local complete intersection varieties, Amer. J. Math. 126 (2004), 1355–1365. 39

[3]

R. Fedder, F -purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461–480. 42

[4]

N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981–996. 41

[5]

N. Hara and K.-i. Watanabe, F -regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geom. 11 (2002), 363–392. 41

[6]

S. Ishii, Jet schemes, arc spaces and the Nash map, C. R. Math. Rep. Acad. Sci. Canada 29 (2007), 1–21. 41

[7]

V. B. Mehta and V. Srinivas, A characterization of rational singularities, Asian J. Math. 1 (1997), 249–271. 41

[8]

M. Musta¸taˇ , Jet schemes of locally complete intersection canonical singularities, with an appendix by D. Eisenbud and E. Frenkel, Invent. Math. 145 (2001), 397–424. 39, 42

[9]

M. Musta¸taˇ , Singularities of pairs via jet schemes, J. Amer. Math. Soc. 15 (2002), 599–615 (electronic). 39

[10] J. F. Nash, Arc structure of singularities, Duke Math. J. 81 (1995), 31–38. 39 [11] K. Smith, F -rational rings have rational singularities, Amer. J. Math. 119 (1997), 159–180. 41 [12] S. Takagi, F -singularities of pairs and Inversion of Adjunction of arbitrary codimension, Invent. Math. 157 (2004), 123–146. 41 [13] S. Takagi and K.-i. Watanabe, On F -pure thresholds, J. Algebra 282 (2004), 278–297. 47

Singularities in relativity Tatsuhiko Koike Department of Physics, Keio University Yokohama, 223-8522 Japan e-mail: [email protected]

Abstract. Many phenomena of importance in general relativity theory are related to singularities in mathematics. For a simple example, the spacetime regions of extreme gravitational field such as the beginning of the Universe and the fate of a massive star are described by singularities in the differential-geometric sense, i.e., curvature singularities of pseudo-Riemannian manifolds. This type of singularity is one of the main objects of interest in general relativity. A less trivial example is that the formation of a black hole horizon can be described as a blow-up solution of some partial differential equations in a certain coordinate system, which is a singularity in the analytic sense. Another is that the “shape” of the black hole horizon is fully characterised by the set of its nondifferential points which are singularities in the sense of singularity theory. I will explain these connections between singularity and relativity with some comments on my related works.

1 Introduction: A very brief review of general relativity Space and time are not absolute contents of the nature but constitute a spacetime which can bend and can be deformed, ant that is what we feel as gravitational phenomena. The spacetime is described by a Lorentzian 4-manifold .M; g/, i.e., a 4-manifold M endowed with a pseudo-Riemannian metric g of signature . C C C / (Figure 1). Table 1 shows the brief correspondence of physical and mathematical objects, where a timelike or null curve refers to a curve whose tangent vector has negative or zero squared norm, i.e., g.V; V / < 0 or g.V; V / D 0, respectively. The matter field bends the spacetime, and the matter’s equation of motion must be consistent with how the spacetime bends. This fact is described by Einstein’s equation 1 8G Rab Rgab D 4 Tab ; 2 c where Rab D Racb c is the Ricci tensor and R D Rab g ab is the scalar curvature constructed by the Riemann curvature tensor Rabc d of the metric g, and Tab is the energy-momemtum tensor of the matter field which depends on the type of matter in consideration. The constant c is the speed of light, and G is Newton’s constant. We shall take the units c D 1 and G D 1 below. The Bianchi identity rŒa Rbcd e D 0 (where

52

Tatsuhiko Koike

.M; g/

Figure 1. The physical world is represented by a curved Lorentzian manifold. The vertical direction corresponds to time and the horizontal directions to space.

Œ denotes the anti-symmetrisation) implies r a Tab D 0. This must be consistent with the matter field’s own equation of motion. Table 1. Correspondence of physical and mathematical objects in general relativity.

Physics

Mathematics

Spacetime

Lorentzian manifold .M; g/

Gravitational field strength Spacetime with no gravitational field Motion of massive particles Motion of photons

Riemann curvature Rabc d Minkowski space .R4 ; dt 2 C jd xj2 / Timelike geodesics Null geodesics

In this article, I present singularities in a wider sense appearing in general relativity. I briefly discuss geodesic incompleteness in Section 2, and curvature singularity in Section 3. These are the most widely discussed singularities in general relativity. They are the singularities of the spacetime itself and physically represent extreme gravitational situations. One can consider them as “singularities in the geometric sense.” In the later sections, I discuss other classes of objects that can be considered as “singularities”. They are in general not treated as singularities in physics and they have nothing to do with the above-mentioned geometric singularities. One such object is the endpoint set of the event horizon (Section 4) which characterise the qualitative physical feature of a black hole. We see that this set is closely related to the singularities in mathematical singularity theory. Another is the apparent horizon (Section 5) which physically characterises formation of a black hole. We see that this is closely related to singular behaviour of the solutions of a partial differential equation. One can consider these two as “singularities in the analytic sense.” To demonstrate the usefulness of this viewpoint that the apparent horizon is a singularity, I discuss critical behaviour

Singularities in relativity

53

singularity

.M; g/

Figure 2. Existence of singularity implies that there is some observer whose history suddenly ends in finite time.

in gravitational collapse in Section 6. I explain what one can call “singularities in the topological sense” in Section 7. Section 8 is a summary.

2 Geodesic incompleteness Geodesic incompleteness is what is usually called singularity in general relativity. It is one of the most important object because of its intimate relation with black holes and with the initiation of the universe. Geodesic incompleteness is defined as follows. An affinely parametrised geodesic

W 7! M is a curve satisfying rP P D 0; where the dot denotes the differentiation by and r is the Riemannian connection with respect to g. An affinely-parametrised geodesic W D ! M is complete if D D R. A point p is a future (respectively, past) endpoint a of a causal (i.e., timelike or null at all points) curve W D ! M if for each neighbourhood U of p, there is t such that .fs 2 D j s > tg/ U (respectively, .fs 2 D j s 6 tg/ U ). An incomplete geodesic without endpoints is said to have a singularity. Physically, existence of a singularity along a causal geodesic represents that there is some observer whose history suddenly ends in a finite time. Since the gravitational force is attractive, positive mass attracts matter and the light. Using this fact, Penrose established that occurrence of the singularity is inevitable in common situations. The following version states that existence of the cosmological initial singularity in usual situations. Theorem 2.1 (Penrose [1]). Spacetime .M; g/ is null geodesically incomplete if (i) Rab K a K b > 0 for all null vectors K a ; (ii) there is a noncompact Cauchy surface † in M; and (iii) there is a closed trapped surface T in the future of †.

54

Tatsuhiko Koike

The first condition is essentially the requirement of positivity of mass. In the second condition, a Cauchy surface is defined as a 3-hypersurface † such that all causal curves without endpoints intersect †. In the third, a closed trapped surface is a compact 2surface T without boundary such that the area form is converging to both null directions orthogonal to T (Figure 3).

T

†

Figure 3. The conditions in Theorem 2.1. A trapped surface lies in the future of a Cauchy surface.

Existence of the singularity is neither necessary nor sufficient for the existence of black holes, but they are closely related. The domain of future outer communication DOCC .M/ of M is the set of points p such that there is a future-directed causal curve from p which can escape to infinity. To be precise, the curve above must reach to future null infinity I C . The definition of the future and past null infinity I ˙ requires a notion of conformal completion of M (see e.g. [2] for details). The black hole region B of M is the complement of DOC.M/ in M. The event horizon H is the boundary of B. The Schwarzschild spacetime is the simplest spacetime containing a black hole region. In the most common coordinates, the metric has the form g D f .r/dt 2 C

dr 2 C r 2 hS 2 ; f .r/

f .r/ D 1

2MBH ; r

(2.1)

where hS 2 is the metric of a unit 2-sphere and MBH is called the black hole mass. Figure 4 shows the causal structure of the Schwarzschild spacetime, where two angular directions are suppressed. In fact, at r D 2MBH , called the Schwarzschild radius, the metric components in (2.1) become singular, and the coordinate system covers any one of the four regions in Figure 4 (two diamonds and two triangles). There is a coordinate system which covers the whole spacetime. The event horizon H is a null surface whose spatial section is a two-sphere with the radius being the Schwarzschild radius. There are singularities at r D 0; geodesics toward them are incomplete.


55

r D0 JC H B J r D1 Figure 4. Causal structure of the Schwarzschild black hole. The white part is the domain of future outer communication DOCC .M/. whose complement B and boundary H are the black hole region and the event horizon, respectively.

3 Curvature singularities Curvature singularity is most commonly encountered “singularity” in general relativity. A scalar singularity is said to exist when some scalar combination of the Riemann tensor such as Ricci scalar R, Ricci tensor squared Rab Rab , Riemann tensor squared Rabcd Rabcd , etc., diverge along a curve in the spacetime manifold .M; g/. Physically, it represents the existence of infinitely strong tidal force where everything approaching there would be destroyed. In most common situations in relativity, a singularity (geodesic incompleteness) is a curvature singularity. A simple example is the one at r D 0 in the Schwarzschild spacetime represented by (2.1). If a singularity is not a curvature singularity, the observer’s life ends in a finite time (geodesic incompleteness) without a catastrophe (curvature singularity). Physically, this is a somewhat puzzling and unwanted situation. In those cases, the notions of singularities weaker than curvature singularity are defined and discussed. Since curvature singularities are most common and discussed in enormous number of works, which the reader should be able to access easily, we do not go into them here (see e.g. [2] for further reading on basic concepts).

4 The endpoint set of the event horizon Mathematical singularity theory gives a good characterisation of an event horizon in relativity. An event horizon H is generated by null geodesics. A future event horizon cannot have future endpoints but can have past endpoints. The endpoint set E of a horizon H is an arc-wise connected acausal set, where an acausal set is a set such that no two points thereof can be connected by a causal curve in the spacetime M (Figure 5).

56

Tatsuhiko Koike JC H E Figure 5. The endpoints set E of the event horizon H .

In relativity, there is no preferred time coordinate. For example, if E is a twodimensional surface, the spacetime allows the following interpretations, among many others, according to different choices of time slices (Figure 6); see [3]: (i) an S 2 black hole forms, and grows; (ii) two black holes form, and collide; (iii) a torus black hole forms, and the handle pinches to make an S 2 black hole. Note that if E consists of a point, e.g. in the case a spherically symmetric black hole, the first interpretation above applies for any choice of time slices. Thus, the endpoint set determines the qualitative feature of the black hole. JC

t D const

JC

E

E

Figure 6. Physical interpretation depends on the choice of time slices: formation of a single black hole, collision of two formed black holes, …, etc.

Points u 2 E are classified by the multiplicity m.u/ of u, the number of the null geodesic generators emanating from u: E D C t D;

C D fu 2 E j m.u/ > 1g;

D D fu 2 E j m.u/ D 1g:

(4.1)

The set C is called the crease set of the horizon. The crease set contains the interior of the endpoint set, i.e., the closure of C contains E. The crease set C coincides with the set of points of E on which the horizon is not differentiable, i.e., the horizon is differentiable at u 2 E if and only if u 2 D [4]. The sets C and D can be naturally understood in the context of singularity theory in mathematics. Assume M is globally hyperbolic with a Cauchy surface †. Then there is a global time function t, and M is a direct product of the time R and the space †: W R † 3 .t; q/ 7! .t; q/ 2 M:


57

We assume that in the sufficiently late times a t=constant section H of the event horizon H is stable. We fix a compact surface S in † so that .t; S/ D H with that t. The generic structure of the endpoint set E can be classified by the singularity theory [5]. Let us define the Fermat potential F for .x; q/ 2 S † by F .x; q/ D supft 2 R j there is a future-directed causal curve from .t; q/ to xg: (4.2) Let BMaxwell .F / be the Maxwell set of F , i.e., BMaxwell .F / D fq 2 † j F . ; q/ has two or more global minimum pointsg: Let B.F / be the bifurcation set of F, i.e., the set of points where the minimum bifurcates. Then we have C Š BMaxwell .F /;

DŠ

.BMaxwell .F / \ B.F //:

This correspondence can be used for classifying the stable structure of E. The Maxwell set of a generic l 6 6-parameter family of functions is locally P stably diffeomorphic to one of the types Am1 Amk , where the mi are odd and mi 6 l C 1. The concrete types other than Ak1 are shown in Table 2 (see e.g. [6]). For example, one has A21 D .x12 C y12 ; x22 C y22 C q1 /; A3 D x 4 C q2 x 2 C q1 x C y 2 : In particular, the Maxwell set Ak1 is merely the intersection of walls separating k domains. In the case of four-dimensional spacetime M, we have l 6 dim † D 3 and the possible locally stable structure of C , hence of E, can be summarised by the following diagram: A3 A?1 ? ??? ?? ?? ?? A31 _______/ A21 o_______ A3 , O _?? ?? ?? ?? ?? ? A41 where an arrow means that the structure at the origin of the arrow has the structures at the target of the arrow in the vicinity and the box means that the structure appears at the boundary of the Maxwell set and is not contained therein. See Figure 7 for an example. By making use of Table 2, one can easily construct such diagrams for the cases dim M D 5; 6; 7.

58

Tatsuhiko Koike

A3 A1 A1 A1 A1 A1 A1 A1 A1 A3 A1

Figure 7. An example of a generic endpoint set.

Table 2. Locally stable Maxwell sets.

l

2

3

4

5

6

Type

A3

A1 A3

A21 A3 ; A5

A23 ; A31 A3 ; A1 A5

A1 A23 ; A41 A3 ; A21 A5 ; A7

5 Apparent horizons The event horizon is an important object in relativity but it may not be determined easily because it depends on the causal structure of the whole spacetime. i.e., it depends heavily on the behaviour at infinitely late times. For the same reason, it is hard to relate the event horizon with the dynamics of the gravitational field. Thus one often discusses apparent horizons which can be defined (quasi-)locally. An apparent horizon usually emerges near the event horizon, and, if the cosmic censorship holds, inside thereof (see e.g. [2]). In particular, if the event horizon exists and if the spacetime is static (i.e. if it has a one-parameter family of isometries with image curve of each point being timelike outside the black hole), the event horizon coincides with the apparent


59

horizon. The apparent horizon is not considered as a singularity in general relativity, but the viewpoint that it is so is possible and useful. An apparent horizon is a closed 2-surface (or 3-hypersurface foliated thereby) such that the surface area form is converging in one null direction and is stationary in the other null direction (see Figure 8). In terms of coordinate systems consisting of a “time” coordinate and “space” coordinates, the apparent horizon is characterised by its singularity.

ordinary surface

apparent horizon

Figure 8. An ordinary surface and an apparent horizon.

a.r/ 1

r O

rAH

Figure 9. An apparent horizon formation can be described by a blow-up of a certain component of the metric.

A simplest example is a spherically symmetric spacetime. In a commonly used coordinate system, the metric reads g D ˛.t; r/2 dt 2 C a.t; r/2 dr 2 C r 2 hS 2 : The product of expansion rates in null directions is C D

lC .r 2 / l .r 2 / 2 D 2 ; 2 2 r r r a.t; r/2

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Tatsuhiko Koike

where ˙ are the expansion rates of the area form in null directions 1 @t @r : ˙ l˙ D p a 2 ˛ Thus, from the viewpoint of partial differential equation, the blow-up a.t; r/ % 1 of the solution corresponds to the emergence of an apparent horizon. Thus, when and where a.t; r/ blows up have the information on the apparent horizon (see Figure 9). The mass MBH of the black hole, i.e. of the event horizon is usually well estimated by the mass MAH of the apparent horizon. In spherically symmetric spacetimes, there is a natural definition of the mass (or energy) M.t; r/ contained in a given sphere specified by .t; r/. It is simply M.t; r/ D r=2. Therefore, if we find an apparent horizon appearing at r D rAH , i.e., if lim t!tAH a.t; rAH / D 1 with some tAH , then the mass contained in the apparent horizon is given by MAH D M.tAH ; rAH / D rAH =2. An application of this fact is presented in the following section.

6 Critical behaviour in gravitational collapse In this section, we shall present an application of the occurrence of the apparent horizon as a “singularity” discussed in the previous section. Formation of black holes is important in astrophysics as well as in general relativity. By the quasistationary analysis of realistic stars, it is suggested that relatively heavy stars (those more than several times heavier than the sun) become a black hole, while the minimum mass of resulting black hole is around twice that of the sun. In general relativity, one is also interested in more dynamical situations of black hole formation called gravitational collapse, and in fundamental theoretical problems such as the conditions for formation of singularities, event horizons and Cauchy horizons.1 However, it is difficult in general since Einstein’s equation is highly nonlinear. Critical behaviour in gravitational collapse presented here is a universal and characteristic phenomenon appearing in the limiting situation that the initial matter distribution evolves into a black hole with infinitesimal mass. It was found by numerical simulation [7] and allowed an interpretation similar to critical phenomena in statistical physics. The mechanism of this peculiar phenomenon was not known for a while, but was later revealed that a certain structure of the phase space as a dynamical system [8] is responsible. The understanding has natural connection to that of critical phenomena in statistical physics and also gives a method for quantitative analysis. Though the study of the subject had been driven by purely theoretical interests, it now has direct cosmological applications, related to the nature of black holes in an early stage of the Universe [9]. 1A Cauchy horizon is a hypersurface in the spacetime beyond which the physics is not predictable, i.e. it is the boundary of globally hyperbolic region of the spacetime.


61

6.1 The phenomena Let us consider some matter field and its gravitational collapse, i.e., the formation of a black hole. If the initial configuration of the matter is sufficiently “weak”, then the matter will disperse and the black hole will not form. For a sufficiently “strong” one, the black hole will form (for examples of rigorous analysis, see Christodoulou’s works [10]). Then the questions arise. What happens if one gradually changes the initial configuration? What happens at the threshold of the formation of the black hole (Figure 10)? matter

threshold?

gone!

black hole

Figure 10. What is the behaviour of the initial data near the threshold between the ones which will evolve into a black hole and those which will not?

By numerical simulations, Choptuik [7] found critical behaviour in gravitational collapse which resembles critical behaviour in statistical physics, characterised by scaling and universality, in a spherically symmetric system of gravitational and real massless scalar fields. The energy-momentum tensor of the matter is given by Tab D 1 r rb , where W M ! R is the real massless scalar field, and the spacetime 2 a metric is given by g D ˛.t; r/2 dt 2 C a.t; r/2 dr 2 C r 2 hS 2 . The behaviour is as follows (the description here relies on [11]) Let I.x/ be a generic 1-parameter family of initial data such that a black hole will form in the future for sufficiently large x and it will not form for sufficiently small x. Then the following properties hold (Figure 11). (E) There is xc such that for x > xc a black hole forms and for x < xc no black hole forms. (S1) The x xc solutions once approach a discretely self-similar solution and then either forms a black hole or approaches the flat spacetime. > xc , the mass MBH of the formed black hole satisfies a scaling law (S2) For x MBH / .x xc /ˇBH .

62

Tatsuhiko Koike

(U1) The self-similar spacetime in (S1) is unique for all x xc . (U2) The critical exponent ˇBH , which was approximately 0:37, is universal in the sense that it does not depend on the choice of the one-parameter family I.x/ in the space of initial data.

t 0

r MBH

MBH / .x xc /ˇBH

xc

x

Figure 11. Critical behaviour in gravitational collapse. Up: any near-critical spacetime approaches a self-similar one before it evolves into a black hole or diverges. Down: the mass of the formed black hole in the super-critical spacetimes satisfies a universal power law.

Similar behaviour was found in the axial symmetric system of pure gravity (i.e. with no matter field) and in the spherically symmetric system of gravity and a radiation fluid. The former system showed discrete self-similarity and the critical exponent was ˇBH 0:37; see [12]. In the latter, the energy-momentum tensor was Tab D ua ub C p.ua ub C gab /, where p D =3 is the pressure, > 0 is the density, and ua is the velocity vector of the fluid. The system showed continuous self-similarity instead of discrete self-similarity in (S1) and the value of ˇBH was approximately 0:36; see [13]. The mechanism of this interesting phenomenon was not understood.


63

6.2 Renormalisation group method The mechanism of the critical behaviour in gravitational collapse is revealed by Hara, Adachi and the present author [8] by using the method of renormalisation group (RG). The method of RG for partial differential equation was used for the analysis of intermediate asymptoticswhere self-simlar solutions play an important role. In the gravitational system, as is discussed in the previous section, the formation of an apparent horizon can be considered as blow-up of the solution. We apply the RG method to the analysis of blow-up solutions. Consider a partial differential equation (PDE) @u @u

L

;

@t @r

; u; t; r D 0;

u.1; r/ D U.r/;

(6.1)

where L is a function from R5 to Rn . We use the notation as if n D 1 for simplicity; the recovery to general n is simple. We assume that the system is invariant under the scaling transformation s , s u.t; r/ D e ˛s u.e s t; r ˇs /;

˛, ˇ, fixed reals:

Namely, if u is a solution to the PDE, so is s u. Let us call D fU g (the space of initial data) the phase space. The renormalisation group transformation (RGT) Rs on is defined by Rs U.r/ D s u.1; r/ D e ˛s u.e s ; r ˇs /; where u is a solution to the PDE. The RGT depends on the real parameters ˛ and ˇ but we omit them in the notation Rs . The RGT Rs can be described as “the time evolution from t D 1 to t D e s , followed by a spatial scaling transformation”.2 The family of RGTs has semi-group property Rs1 Cs2 D Rs2 ı Rs1 . For simplicity, we explain the case of continuous self-similarity in the following. The generalisation to the case of discrete self-similarity is straightforward. The generator of the RGT Rs is defined by R1 ; RP D lim s!0 s P The generator RP defines a vector and thereby the RGT is expressed as Rs D exp.s R/. field on , which we call the RG flow. A fixed point U of the flow RP satisfying P D U corresponds to a self-similar solution uss D s uss of (6.1) by RU uss .t; r/ D .t/˛ U

r : .t/ˇ

2 We use negative t because we consider a “shrinking” spatial scaling (and we want t to increase toward future).

64

Tatsuhiko Koike

The tangent map of Rs around the fixed point U is defined by Rs .U C "F / U : "!0 " The tangent map sends the tangent space at U to that at Rs U . Its generator is given by Ts F D lim

Ts 1 TP D TPU D lim ; s!0 s so that Ts D exp.s TP /; An eigenmode of TP is defined by TP F D F: The eigenmode F is called relevant if Re > 0, irrelevant if Re < 0, and marginal if Re D 0.

6.3 The mechanism The mechanism and all aspects of the critical behaviour discussed in Section 6.1 can be understood as the structure of the RG flow. We describe them as assumptions. Assumption 6.1 (Spectrum of T at U ). The spectrum of T is .T / D fg [ 0 ; where Re > 0 and 0 f 0 2 C j Re 0 6 g with some < 0. Assumption 6.2 (Global information). The two directions of W u .U / are the flat and the black hole spacetimes. If U is sufficiently far from U , it becomes a black hole in a finite “time” s. The stable manifold of U is defined by W s .U / D fU 2 j lim Rs .U / D U g: s!1

The unstable manifold of U is defined by W u .U / D fU 2 j lim Rs .U / D U g: s!1

The properties (E), (S1), and (U1) are direct consequences of the assumptions. Proposition 6.3 (E, S1, U1: global structure of the flow). We have dim W u .U / D 1


65

and codim W s .U / D 1: The two directions of W u .U / are the flat and the black hole spacetimes (Figure 12). Uc .x D xc / Uinit I x > xc

x < xc

Q

Flat spacetime

Black hole

U

Wu

Ws Figure 12. Schematic diagram of the structure of the RG flow on the phase space.

We can also show (S2) and (U2) from the same assumptions. Proposition 6.4 (S2, U2: mass of the black hole). The mass of the black hole forms is given by MBH ' K.x xc /ˇBH ;

ˇBH D

ˇ ;

when x ! xc . We have also assumed that the mass MBH of the black hole is well estimated by the mass MAH of the apparent horizon, as discussed in the previous section. Let U be a small neighbourhood of U . Let U D Uc C "F be a near-critical initial data. Because U is the attractor of the critical surface W u .U /, Rs U enters in U at some s D s0 (Figure 13): Rs0 U D Rs0 .Uc C "F / Rs0 Uc C "F 0 : Near U , we can use the linearised equation: 0 : Rs0 Cs1 U D Rs1 .Rs0 Uc C "F 0 / ' Rs0 Cs1 Uc C "Ts1 F 0 ' U C "e s1 Frel

66

Tatsuhiko Koike

Choose s1 as the time spent in U. Then we have "e s1 1 (i.e. independent from "): From Assumption 6.2, Rs U will blow up at s D s0 C s1 C s2 with finite s2 . The blow-up occurs at position 1 (independent from "). However, its physical length scale is given by rAH expfˇ.s0 C s1 C s2 /g/ "ˇ= because of the scaling transformation contained in the RGT Rs . Thus rAH "ˇ= . Since MBH D rAH =2, MBH ' rAH e ˇs1 "ˇ= : We have shown the scaling relation and have obtained the formula for the critical exponent ˇBH D ˇ . This is obviously independent of the choice of the 1-parameter family I.x/ of initial data. Initial data s0 C s1 C s2 s0

s0 C s1 Black hole

Minkowski spacetime

U

Figure 13. Illustration of how the structure of the RG flow explains all characteristics of the critical behaviour. Note that the flow spends essentially the whole “time” near the fixed point when x ! xc .

For a radiation fluid, we found, with ˛ D 1 and ˇ D 1, a unique relevant mode with D 2:81055255 by numerically solving the eigenmode equation of T, which is a twopoint boundary problem of an ordinary differential equation [8]. We further established the uniqueness of the relevant mode by Liapunov analysis (numerical scheme that selectively finds large Re 0 modes) [11]. These show that our interpretation of the critical behaviour is correct and give the critical exponent ˇBH D 0:35580192 (recall that the simulation value was ˇBH 0:36).

7 Topological singularities due to quotienting Making the quotient of a manifold by a group action may yield a space with “topological” singularities, and this may have some physical implications.


67

Many cosmological observations suggest that the universe underwent an inflation. This means that the spacetime was close to the de Sitter spacetime, i.e., the spacetime of positive constant curvature. Observations also suggest that the present universe has negative spatial curvature so that the spatial sections can be approximated by hyperbolic 3-manifolds. The de Sitter spacetime does not have a curvature singularity and can be extended smoothly to the past. On the other hand, spatial compactness of the universe is an appealing notion, especially in the context of the canonical treatments of the universe or quantum gravity. Compactness provides a finite value of the action integral and gives the natural boundary conditions for the matter and gravitational fields in the universe. It has been shown [15] that these three conditions, inflation, spatial hyperbolicity and spatial compactness cannot hold simultaneously. Namely, though the universal covering space of the universe can be extended analytically, beyond the so-called past Cauchy horizon, the extended region has densely many points which correspond to singularities of the compact universe. This is done by carefully analysing the group action for quotienting around the Cauchy horizon, and the proof relies on the ergodicity of the geodesic flow on a compact negatively curved manifold (Figure 14). X0 H2

B1

X2

A1 De Sitter 2

O C1

X1

Figure 14. Topological singularities appear in a spatially compact, spatially hyperbolic, de Sitterlike spacetime.

68

Tatsuhiko Koike

8 Summary I have reviewed singularities in a wider sense which appear in relativity. Geodesic incompleteness, curvature singularity, the endpoint set of the event horizon, apparent horizons with an application of the critical behaviour in gravitational collapse, and topological singularity. Since relativity is formulated as differential geometry, the ideas and the techniques of the latter have been applied to the former. I hope that singularity theory and other field of mathematics will have more and more interactions with relativity and lead to new discoveries.

References [1]

R. Penrose, Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14 (1965), 57–59. 53

[2]

S. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Monographs on Mathematical Physics 1, Cambridge University Press, London and New York 1973. 54, 55, 58

[3]

M. Siino, Topology of event horizons, Phs. Rev. D 58 (1998), 104016. 56

[4]

J. K. Beem and A. Królak, Cauchy horizon end points and differentiability, J. Math. Phys. 39 (1998), 6001–6010 56

[5]

M. Siino and T. Koike, Topological classification of black hole: Generic Maxwell set and crease set of horizon, Internat. J. Mod. Phys. D 20 (2011), 1095–1122. 57

[6]

V. I. Arnold (ed.), Dynamical systems VIII: Singularity Theory II, Classification and Aplications, Encyclopedia of Mathematical Science 39, Springer-Verlag,Berlin etc., 1991. 57

[7]

M. Choptuik, Universality and scaling in gravitational collapse of a massless scalar field, Phys. Rev. Lett. 70 (1993), 9–12. 60, 61

[8]

T. Koike, T. Hara, and S. Adachi, Critical behavior in gravitational collapse of radiation fluid: A renormalization group (linear perturbation) analysis, Phys. Rev. Lett. 74 (1995), 5170–5173. 60, 63, 66

[9]

J. C. Niemeyer and K. Jedamzik, Near-critical gravitational collapse and the initial mass function of primordial black holes Phys. Rev. Lett. 80 (1998), 5481. 60

[10] D. Christodoulou, A mathematical theory of gravitational collapse, Comm. Math. Phys. 109 (1987), 613–647, and references therein. 61 [11] T. Koike, T. Hara, and S. Adachi, Critical behavior in gravitational collapse of a perfect fluid, Phys. Rev. D 59, 104008; see T. Hara and T. Koike, in Blowup and aggregation, ed. by M. Mimura, University of Tokyo Press, Tokyo 2006. 61, 66 [12] A. M. Abrahams and C. R. Evans, Critical behavior and scaling in vacuum axisymmetric gravitational collapse, Phys. Rev. Lett. 70 (1993), 2980–2983. 62 [13] C. R. Evans and J. S. Coleman, Critical phenomena and self-similarity in the gravitational collapse of radiation fluid, Phys. Rev. Lett. 72 (1994), 1782–1785. 62


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[14] J. Bricmont, A. Kupiainen, and G. Lin, Renormalization group and asymptotics of solutions of nonlinear parabolic equations, Comm. Pure Appl. Math. 47 (1994), 893–922. [15] A. Ishibashi, T. Koike, M. Siino, and S. Kojima, Compact hyperbolic universe and singularities, Phys. Rev. D 54 (1996), 7303–7310. 67

On the universal degenerating family of Riemann surfaces Yukio Matsumoto Department of Mathematics, Gakushuin University Mejiro, Toshima-ku, Tokyo 171-8588, Japan e-mail: [email protected]

> 2/. Over the TeichAbstract. Let †g be a closed oriented (topological) surface of genus g .D müller space T .†g / of †g , Bers constructed a universal family V .†g / of curves of genus g, which would be well called “the tautological family of Riemann surfaces”. The mapping class group g of †g acts on V .†g / ! T .†g / in a fibration preserving manner. Dividing the fiber space by this action, we obtain an “orbifold fiber space” Y.†g / ! M.†g /, where Y.†g / and M.†g / denote V .†g /=g and T .†g /=g , respectively. The latter quotient M.†g / is called the moduli space of †g . The fiber space Y.†g / ! M.†g / can be naturally compactified to another orbifold fiber space Y.†g / ! M.†g /. The base space M.†g / is called the Deligne– Mumford compactification. Since this compactification is constructed by adding “stable curves” at infinity, it is usually accepted that the compactified moduli space M.†g / is the coarse moduli space of stable curves of genus g. In this paper, we will sketch our argument which leads to a conclusion, somewhat contradictory to the above general acceptance, that the compactified family Y.†g / ! M.†g / is the universal degenerating family of Riemann surfaces, i.e. it virtually parametrizes not only stable curves but also all types of degenerate and non-degenerate curves. Our argument is a combination of the Bers–Kra theory and Ashikaga’s precise stable reduction theorem.

Contents 1. 2. 3. 4. 5. 6. 7. 8.

Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Complex orbifolds and fiber spaces over orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Types of mapping classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Fenchel–Nielsen coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Compactification process of M.†g / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Bers’ deformation spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Subdeformation spaces D" .C/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 The universal degenerating family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Supported

by the Grant-in-Aid for Scientific Research (B) 20340014.

72

Yukio Matsumoto

1 Main theorem First we will recall the notation and some basic results. Let †g be a closed oriented > 2/. Let g D .†g / be the mapping class group topological surface of genus g .D of †g , which is defined as follows: g D ff W †g ! †g j orientation preserving homeomorphismg=isotopy: Let Tg D T .†g / denote the Teichmüller space of †g . The definition of T .†g / begins with pairs .S; w/ consisting of a Riemann surface S and an orientation preserving homeomorphism w W S ! †g : Two such pairs .S1 ; w1 / and .S2 ; w2 / are said to be equivalent and denoted by .S1 ; w1 / .S2 ; w2 /, if there exists a biholomorphic map t W S1 ! S2 such that the following diagram is homotopically commutative1 : S1

w1

t

s2

/ †g id

w2

/ †g :

The Teichmüller space of †g is defined2 by T .†g / D f.S; w/g= : The equivalence class to which .S; w/ belongs will be denoted by ŒS; w. By the works of Ahlfors [3], and Weil [40], [41], the Teichmüller space T .†g / is a complex manifold of complex dimension 3g 3. Bers [6] embedded T .†g / in C3g3 as a bounded domain. The mapping class group g acts on T .†g / through Œf .ŒS; w/ D ŒS; f ı w: Here Œf is an element of g and ŒS; w is a point of T .†g /. It can be proved that this action is properly discontinuous and holomorphic [7]. See also [30], Chapter 2, and [18], Chapter 6. Consequently, the moduli space M.†g / D T .†g /=g is a normal complex analytic space. See [12]. Bers [7] constructed a family of Riemann surfaces V .†g / ! T .†g / and showed that g acts on the total space V .†g / and on the base space T .†g / simultaneously in 1 In

our situation, this condition is equivalent to saying that the diagram is isotopically commutative. usual definition of Teichmüller spaces starts with pairs .S; w/ in which w is an orientation preserving homeomorphism of †g onto S, being in the opposite direction to our homeomorphism which is from S onto †g . Cf. [18]. The author thinks that our definition is more natural so far as the action of g on T .†g / is concerned 2 The

On the universal degenerating family of Riemann surfaces

73

such a way that the projection is equivariant with respect to these actions3 (in short, the action of g preserves the fibered structure of V .†g / ! T .†g /). His construction uses quasiconformal machinery and is quite involved, but as a final product, his fiber space is “tautological” in the sense that over the point ŒS; w 2 T .†g / the identical surface S is situated. The action of g on the fiber space is also tautological: if Œf .ŒS; w/ D ŒS 0 ; w 0 , we have a homotopically commutative diagram in which t is a biholomorphic mapping S

w

f

t

S0

/ †g

w0

(1.1)

/ †g

and the action of Œf on the total space V .†g / takes the fiber S onto the fiber S 0 by the mapping t. Let us recall Bers’ theorem. Theorem 1.1 ([7]). There exists an orbifold fiber space Y.†g / ! M.†g / such that over each point p D ŒS 2 M.†g /, one has a fiber which is isomorphic to the quotient S=Aut.S/. Note that Aut.S/ ¤ f1g if and only if p D ŒS; w.2 T .†g // is fixed under the action of a finite subgroup of g , isomorphic to Aut.S/. Any finite subgroup of g has a non-empty fixed point set in T .†g /; see [21]. The fiber space of Theorem 1.1 is the quotient of V .†g / ! T .†g / by the action of g , i.e. Y.†g / D V .†g /=g and M.†g / D T .†g /=g . The moduli space M.†g / is compactified by adding “stable curves” at infinity. This compactification is called the Deligne–Mumford compactification (see [14]) and the compactified moduli space will be denoted by M.†g /. Because of this process of compactification, it is usually accepted that M.†g / is the coarse moduli space of stable curves. Somewhat contradictory (at first sight) to this consensus, our main result states the following. 3As a matter of fact, Bers did not mention this fiber space explicitly in his paper [7]. He constructed instead a fiber space F .G/ ! T .G/, where G is a Fuchsian group isomorphic to the fundamental group 1 .†g /, and T .G/ is canonically identified with T .†g /. Each fiber of F .G/ is a (Jordan) domain of discontinuity of a quasi-Fuchsian group isomorphic to G. The group G acts on F .G/ so that it preserves the fibering structure. To get the fiber space V .†g / ! T .†g /, we have only to take the quotient F .G/=G ! T .G/ and put V .†g / D F .G/=G. Cf. [22] §4.6.

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Yukio Matsumoto

Theorem 1.2. (1) The orbifold fiber space Y.†g / ! M.†g / can be compactified to an orbifold fiber space Y .†g / ! M.†g /: The total and the base spaces of this fiber space are compact complex normal analytic spaces (hence Hausdorff ). (2) The compactified fiber space is the universal degenerating family, or more precisely, the universal orbifold fiber space in the following sense: given a degenerat> 3) over a (either compact ing family of Riemann surfaces ' W M ! B of genus g( D or non-compact) Riemann surface B, one can canonically associate with it an orbifold fiber space '# W M# ! B; and if ' W M ! B is almost asymmetric, then there exists an orbifold pull-back diagram orbifold pull-back

M#

/ Y .†g /

'#

B

b#

/ M.†g / :

Note that the statement of the main theorem belongs to “biregular geometry”, not to “birational geometry”. In birational sense, the main theorem (especially as stated in the first version of this paper4 ) is almost straightforward. See Remark 8.3. A degenerating family of Riemann surfaces of genus g over a base Riemann surface B is, by definition, a proper surjective holomorphic map ' W M ! B of a 2dimensional complex manifold M to B, whose general fiber is a Riemann surface homeomorphic to †g . It may admit isolated degenerate fibers. A Riemann surface S 4 In

the first version of this paper, the main theorem was stated somewhat vaguely as follows.

Theorem (1) The orbifold fiber space Y .†g / ! M.†g / can be compactified, in a fiber preserving manner, to an orbifold fiber space Y .†g / ! M.†g /: The total and the base spaces of this fiber space are compact complex normal analytic spaces. (2) The compactified fiber space is the universal degenerating family, in the sense that for any degenerating > 3/ over a (either compact or noncompact) Riemann surface B, family of Riemann surfaces of genus g. D ' W M ! B, which is almost asymmetric, there exists a pull-back/blowing up diagram M

blowing up

/ M#

pull-back

/ Y .†g /

'#

'

B

id

/B

/ M.†g / :


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is said to be asymmetric, if Aut.S/ D f1g: We will say that a degenerating family of Riemann surfaces over B is almost asymmetric, if there exists a set of isolated points Symm in B such that for each p 2 B Symm, the fiber over p is asymmetric. Our asymmetry condition5 rules out several exceptional cases. For example, the < 2 is excluded, because in this case every Riemann surface is symmetric case where g D with respect to the hyperelliptic involution. It is well known that by blowing down .1/-spheres in M , we get relatively minimal degenerating family containing no .1/-spheres in the fibers. In what follows, we will assume that ' W M ! B is a relatively minimal degenerating family. Also by blowing up M at non-normal crossing singular points of fibers, one can make a degenerating family ' W M ! B a normally minimal family ' 0 W M 0 ! B, in which the reduced scheme F red of each fiber F is a nodal surface and every .1/-sphere in F red has at least three nodes. The normally minimal family ' 0 W M 0 ! B is uniquely determined by ' W M ! B, and is called the normally minimal model of ' W M ! B. By Ashikaga’s theorem (Theorem 8.1), we obtain an orbifold fiber space '# W M# ! B from the normally minimal model ' 0 W M 0 ! B by contracting certain (explicitly known) linear and/or Y-shaped configurations of rational curves in fibers of M 0 . For their exact shapes, see Figure 1, [32], and [33]. The length r and the self-intersection numbers (a1 ; a2 ; : : : ; ar or b3 ; : : : ; br ) are read off from the topological monodromy around the singular fiber on which the singular point in question is situated; see [4], [27], and [28]. The total space M# may have two types of singularities: cyclic quotient singularities which are the contraction images of linear configurations, and dihedral quotient singularities which are the contraction images of Y-shaped configurations. The orbifold fiber space '# W M# ! B is uniquely and explicitly determined from ' W M ! B. We will call it the orbifold model of ' W M ! B. a1

a2

a3

b3

b4

:::

ar

2

:::

br

2 Figure 1. A linear configuration and a Y-shaped configuration. 5 This condition was inspired by Ashikaga and Yoshikawa’s A-generality condition imposed on fibered surfaces, [5].

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Since the normally minimal model ' 0 W M 0 ! B is obtained from '# W M# ! B by minimal resolution (just converse to the contraction) of cyclic and/or dihedral singularities in M# ([32] and [33]), the following theorem is an immediate corollary of Theorem 1.2. Theorem 1.3. Let ' W M ! B be a degenerating family of Riemann surfaces which is relatively minimal and almost asymmetric. Then there exists the following diagram (the arrows are holomorphic maps): M o

blow down

'

Bo

id

M0

minimal resolution

/ M#

'0

'#

B

/B

id

orbifold pull-back

/ Y .†g /

/ M.†g /:

2 Complex orbifolds and fiber spaces over orbifolds Orbifolds were introduced by Satake [34] and [35] (under the name of V-manifolds) and independently by Thurston [38] (who created the terminology of orbifolds). In this section, we will recall orbifolds, especially in the category of complex analytic spaces, and will introduce “fiber spaces over orbifolds”.

2.1 Complex orbifolds There are several ways of defining orbifolds; see [34], [35], [38], [19], [11], and [26]. We will follow Satake’s definition. A complex m-dimensional orbifold (briefly a complex m-orbifold) is a -compact Hausdorff space X which is covered by an atlas of folding charts A D f.Uzi ; Gi ; i ; Ui /gi2I , each chart consisting of a connected open set Uzi of Cm , a finite group Gi acting on Uzi holomorphically and effectively, an open set Ui of X and a folding map i W Uzi ! Ui which induces a natural homeomorphism i =Gi W Gi nUzi ! Ui . The atlas A must satisfy the following conditions. (1) If a point p of X is contained in the intersection Ui \ Uj , where Ui D i .Uzi / and Uj D j .Uzj /, then there exists a folding chart .Uzk ; Gk ; k ; Uk / such that p 2 Uk Ui \ Uj . (2) If the open set Ui D i .Uzi / is contained in the open set Uj D j .Uzj /, then there exists a holomorphic embedding j i W Uzi ! Uzj (called an injection) of Uzi onto an open subset of Uzj such that i D j ı j i . It is proved that if j i and j0 i W Uzi ! Uzj are two injections, there exists a uniquely determined element g 0 of Gj such that j0 i D g 0 j i (see Lemma 1 in [35]). In particular,


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if g is an element of Gi , then there exists a uniquely determined element g 0 of Gj such that j i g D g 0 j i . The correspondence g 7! g 0 is an isomorphism of Gi onto a subgroup of Gj (see [35], p. 466), which we will denote by j[ i W Gi ! Gj : The injection j i W Uzi ! Uzj becomes an equivariant map with respect to j[ i . Two atlases A and B of folding charts are said to be equivalent, if the union A [ B is an atlas of folding charts satisfying the above two conditions. An orbifold structure on the space X is an equivalence class of atlases of folding charts. By Cartan [12], a complex orbifold X is a normal complex analytic space. A typical example of a complex m-orbifold is given by an m-dimensional complex manifold M on which a discrete group G is acting holomorphically and properly discontinuously: the quotient space GnM has a structure of a complex m-orbifold.

2.2 Orbifold maps Let X and Y be two (complex) orbifolds possibly of different dimensions.A continuous map h W X ! Y is said to be a (holomorphic) orbifold map if it satisfies the following conditions (cf. [35], p. 469). For each point p 2 X , there exist a folding chart .Uzi ; Gi ; i ; Ui / of X containing p (which means that Ui contains p) and a folding chart .Vzk ; Hk ; k ; Vk / of Y containing h.p/ with the following properties. (I) h.Ui / Vk , and there exists a lifted holomorphic map hki W Uzi ! Vzk such that the diagram Uzi

hki

i

Ui

/ Vzk k

hjUi

/ Vk

commutes. (II) Suppose that folding charts .Uzi ; Gi ; i ; Ui /; .Uzj ; Gj ; j ; Uj / of X and folding charts .Vzk ; Hk ; k ; Vk /; .Vzl ; Hl ; l ; Vl / of Y satisfy Ui Uj , Vk Vl , h.Ui / Vk , and h.Uj / Vl , and that there exist lifted holomorphic maps hki W Uzi ! Vzk and hlj W Uzj ! Vzl . Then for any injection j i W Uzi ! Uzj , there

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exists an injection 0lk W Vzk ! Vzl such that 0lk ı hki D hlj ı j i : Uzj O

hlj

0lk

j i

Uzi

/ Vzl O

hki

/ Vzk :

Let p be a point of a complex m-orbifold X , .Uzi ; Gi ; i ; Ui / a folding chart containing p. Let pQ be a point (2 Uzi ) of i1 .p/. The isomorphism class of the isotropy subgroup .Gi /pQ is proved to depend only on p and to be independent of the choice of the folding chart containing p and of the choice of pQ (cf. [26], Appendix A). If .Gi /pQ ¤ f1g, we call p an isotropic point, and the totality †.X / of isotropic points is called the isotropic set. From the definition of orbifolds, it follows that the isotropic < m 1. set †.X / is an analytic subset of complex dimension D Definition 2.1. A (holomorphic) orbifold map h W X ! Y is said to be generic, if h.Ui / \ .Vk †.Y // ¤ ;: for each pair of folding charts .Uzi ; Gi ; i ; Ui / of X and .Vzk ; Hk ; k ; Vk / of Y such that h.Ui / Vk . Lemma 2.2. A generic orbifold map h W X ! Y satisfies the following equivariance condition (III). (III) (Equivariance condition). For each pair of open sets, Ui D i .Uzi /; Vk D k .Vzk /; such that h.Ui / Vk , a (not necessarily injective) group homomorphism h[ki W Gi ! Hk is associated with each lifted holomorphic map hki , with respect to which hki W Uzi ! Vzk is an equivariant map, that is, for all .g; u/ 2 Gi Uzi we have hki .gu/ D h[ki .g/hki .u/: Proof. The argument is similar to the proof of Lemma 1 in [35]. By the assumption, we have i1 .Ui h1 .†.Y /// ¤ ;. Choose a point pQ in it. Let g be any element of Q D hi .g p/ Q D hi .p/ Q D k .hki .p//, Q there exists an element Gi . Since k .hki .g p// g 0 2 Hk such that hki .g p/ Q D g 0 hki .p/: Q (2.1) The element g 0 is uniquely determined by g and (2.1) holds independently of the choice of p, Q which follows from the connectedness of i1 .Ui h1 .†.Y ///. The correspondence g ! g 0 is our homomorphism h[ki W Gi ! Hk .


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2.3 Fiber spaces over orbifolds We will introduce the notion of fiber spaces over orbifolds. In this subsection, the spaces denoted by E will not in general be manifolds, but will always be (not necessarily normal) complex analytic spaces. Let E be a complex analytic space. Suppose X is a complex m-manifold. Then a map ' W E ! X is said to be a fiber space if ' is a surjective, proper and flat holomorphic map. We would like to extend this notion of a fiber space over a manifold X to the case where X is an orbifold. Let X be a complex m-orbifold. A map ' W E ! X is said to be a fiber space over the orbifold X , if it is a surjective, proper holomorphic map and additionally it satisfies the following two conditions (i) and (ii). (i) For each p 2 X , there exist a folding chart .Uzi ; Gi ; i ; Ui / of X containing p, a fiber space 'Qi W Ezi ! Uzi over an m-manifold Uzi , a holomorphic action of Gi on Ezi , and a fibration preserving holomorphic map Qi W Ezi ! ' 1 .Ui / (called the fibered folding map) which induces an isomorphism Qi =Gi W Gi nEzi ! ' 1 .Ui /. We assume that the actions of Gi on Ezi and on Uzi preserve the fibered structure of 'Qi W Ezi ! Uzi and that the following diagram commutes Gi nEzi

Qi =Gi .Š/

'j' 1 .Ui /

induced projection

Gi nUzi

/ ' 1 .Ui /

i =Gi .Š/

/ Ui :

We will call .'Qi W Ezi ! Uzi ; Gi ; Qi ; i ; Ui / a fibered folding chart of ' W E ! X. (ii) Let .'Qi W Ezi ! Uzi ; Gi ; Qi ; i ; Ui / and .'Qj W Ezj ! Uzj ; Gj ; Qj ; j ; Uj / be two fibered folding charts of ' W E ! X with Ui Uj . Then there exits a fibration preserving holomorphic embedding Q j i W Ezi ! Ezj (called a fibered injection) which covers j i W Uzi ! Uzj , and satisfies Qi D Qj ı Q j i . A fiber space over an orbifold ' W E ! X is called an orbifold fiber space if E is an orbifold and ' is an orbifold map.6

2.4 Orbifold pull-back diagram Let ' W E ! X be a fiber space over a complex m-orbifold X. Let X 0 be another complex orbifold whose dimension may be different from m. Let h W X 0 ! X be a generic orbifold map. Then we can canonically construct a new fiber space ' 0 W E 0 ! X 0, and 6 The reason that we do not confine ourselves to the orbifold fiber spaces is that a pull-back of an orbifold fiber space is not necessarily an orbifold fiber space.

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a commutative diagram E0

hQ

'0

X0

/E '

h

/ X;

which we will call an orbifold pull-back diagram. zk ; Jk ; k ; Wk / a The construction is as follows. Let p 2 X 0 be a point and let .W 0 folding chart of X containing p. If we take Wk small enough, then there exists a fibered folding chart .'Qi W Ezi ! Uzi ; Gi ; Qi ; i ; Ui / of ' W E ! X such that h.Wk / Ui . Let zk ! Uzi be a lifted holomorphic map. Since h W X 0 ! X is generic, we may hik W W assume by Lemma 2.2 that the following equivariance condition is satisfied: (*) there exists a group homomorphism h[ik W Jk ! Gi with respect to which zk ! Uzi is an equivariant map. hik W W zk be the fiber space zk ). Let 'Q 0 W Ez 0 ! W Step 1 (Construction of a fiber space over W k k zk pulled back from the fiber space 'Qi W Ezi ! Uzi by the lifted holomorphic map over W zk ! Uzi , that is, Ez 0 is defined by hik W W k

zk Ezi j hik .w/ D 'Qi .e/g; Ezk0 D f.w; e/ 2 W zk is the projection: 'Q 0 .w; e/ D w. Also hQ ik W Ez 0 ! Ezi is the and 'Qk0 W Ezk0 ! W k k projection: hQ ik .w; e/ D e. The map 'Qk0 is proper, surjective and flat holomorphic map. See the diagram below: Ezk0

hQ i k

0 'Qk

zk W

/E zi 'Qk

hi k

/ Uzi :

zk ). By the definition of Step 2 (Group action of Jk on the fiber space 'Qk0 W Ezk0 ! W zk holomorphically and effectively. We let Jk act on orbifolds, the group Jk acts on W 0 z Ek as follows: g.w; e/ D .gw; h[ik .g/e/;

zk Ezi : g 2 Jk ; .w; e/ 2 Ezk0 W

(2.2)

If .w; e/ 2 Ezk0 , then g.w; e/ 2 Ezk0 . In fact, assuming hik .w/ D 'Qi .e/, we have hik .gw/ D h[ik .g/hik .w/ D h[ik .g/'Qi .e/ D 'Qi .h[ik .g/e/: zk , The action of Jk is holomorphic and preserves the fibered structure of 'Qk0 W Ezk0 ! W (i.e. 'Qk0 .g.w; e// D gw).


81

Step 3 (Constructions of the pulled back fiber space ' 0 W E 0 ! X 0 ). Let Ek0 be the quotient space Jk nEzk0 , and let Qk W Ezk0 ! Ek0 be the quotient map. We define 'k0 W Ek0 ! Wk to be the map Ek0

.Q k =Jk /

D

zk Jk nEzk0 ! Jk nW

.k =Jk /

D

Wk ;

zk . The fiber space which is naturally induced from the projection 'Qk0 W Ezk0 ! W 0 0 0 ' W E ! X is constructed by pasting together all the fiber spaces 'k0 W Ek0 ! Wk zk ; Jk ; k ; Wk /. The quotient map Qk W Ez 0 ! E 0 can be conover all folding charts .W k k sidered the fibered folding map Qk W Ezk0 ! .' 0 /1 .Wk /. Note that we have constructed fibered folding charts zk ; Jk ; Qk ; k ; Wk / .'Qk0 W Ezk0 ! W of ' 0 W E 0 ! X 0 . We have to check the compatibility condition (ii) for them. zk ; Jk ; Qk ; k ; Wk / and .'Q 0 W Ez 0 ! W zl ; Jl ; Ql ; l ; Wl / be the Let .'Qk0 W Ezk0 ! W l l two fibered folding charts constructed as above. We assume Wk Wl , and let zk ! W zl be an injection. Choose fibered folding charts .'Qi W Ezi ! Uzi ; Gi ; Qi ; 0lk W W i ; Ui / and .'Qj W Ezj ! Uzj ; Gj ; Qj ; j ; Uj / of ' W E ! X such that Ui Uj ; h.Wk / Ui and h.Wl / Uj . Let j i W Ui ! Uj (or Q j i W Ezi ! Ezj ) be a corresponding injection (or a fibered injection). Then, since zk Ezi j hik .w/ D 'Qi .e/g; Ezk0 D f.w; e/ 2 W zl Ezj j hj l .w 0 / D 'Qj .e 0 /g; Ezl0 D f.w 0 ; e 0 / 2 W the map Q 0lk W Ezk0 ! Ezl0 defined by Q 0lk .w; e/ D .0lk .w/; Q j i .e// is a fibration preserving holomorphic embedding onto an open subset, which covers zk ! W zl . 0lk W W We have the actions of Jk on Ezk0 and of Jl on Ezl0 , defined by (2.2). The embedding Q 0lk W Ezk0 ! Ezl0 is equivariant with respect to these actions together with the (injective) group homomorphism .0lk /[ W Jk ! Jl . The embedding Q 0 W Ez 0 ! Ez 0 descends to the quotient fiber spaces to identify lk

k

l

the fiber space 'k0 W Ek0 ! Wk with a fibered subspace of 'l0 W El0 ! Wl , and the compatibility condition (ii) is verified. The construction of the pulled back fiber space ' 0 W E 0 ! X 0 is now completed.

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3 Types of mapping classes Let us recall the Thurston-Bers classification of mapping classes of †g . We will follow Bers’ terminology: Theorem 3.1 ([39], [10], and [16]). Let f W †g ! †g be an orientation preserving homeomorphism. Then the mapping class Œf is one of the four types. (1) periodic (elliptic): there exists a positive integer n such that Œf n D 1 2 g . (2) parabolic: Œf is reduced by a system of disjoint simple closed curves C D C1 [ C2 [ [ Cr on †g and its component maps are periodic, in other words, f preserves C and its restriction to the complement of C is isotopic to a periodic self-homeomorphism of †g C . (3) hyperbolic: f is a pseudo-Anosov homeomorphism in Thurston’s sense. (4) pseudo-hyperbolic: “reducible” by a system of disjoint simple closed curves, but not parabolic. The dynamical aspects of the actions of g on T .†g / and of the Fuchsian group on the upper half plane are much alike: a periodic mapping class Œf has fixed points on T .†g /. The other types of mapping classes have fixed points on the Thurston boundary of T .†g /; see [39]. But for our purpose, Bers’ “extremum” formulation is more suitable; see [10]. Let p1 D ŒS1 ; w1 and p2 D ŒS2 ; w2 be two points of the Teichmüller space T .†g /. Then the distance (Teichmüller distance) between the two points is defined as d.p1 ; p2 / D

1 log inf K.g/ 2

where g W S1 ! S2 is a homeomorphism isotopic to w21 ı w1 , and K.g/ is the dilatation7 of g. For Œf 2 g , let a.Œf / denote the infimum of d.Œf .p/; p/ for p 2 T .†g /. Bers’ classification of mapping classes is the following; see [10], p. 80: Œf is elliptic if it has a fixed point in T .†g / (cf. [21]), parabolic if there is no fixed point but a.Œf / D 0, hyperbolic if a.Œf / > 0 and there is a p 2 T .†g / with d.Œf p; p/ D a.Œf /, pseudo-hyperbolic if a.Œf / > 0 and d.Œf .p/; p/ > a.Œf / for all p 2 T .†g /. The property of being elliptic, parabolic, hyperbolic, or pseudo-hyperbolic is preserved by inner automorphisms of the mapping class group g . Definition 3.2. An orientation preserving homeomorphism f W †g ! †g or its mapping class Œf is called pseudo-periodic, if it is periodic or parabolic. 7 For the definition of the dilatation K.g/, see [10]. K.g/ takes real values between 1 and C1, with K.g/ D 1 meaning that g is conformal, and K.g/ D C1 meaning that g is not quasi-conformal.


83

Given a degenerating family ' W M ! of Riemann surfaces of genus g over a unit disk D fz j jzj < 1g, the topological monodromy f W †g ! †g is known to be pseudo-periodic of negative twist.8 (We are assuming that there is a possible singular fiber only over the origin z D 0.) For a pictorial explanation of the topological monodromy, see Figure 2.

†g M

f

'

Figure 2. Degenerating family and its monodromy f.

The pseudo-periodic nature of the homological and topological monodromy and its negativity have been clarified by several mathematicians: in Milnor fiberings, these facts were discovered by Lê [23], A’Campo [2], Lê, Michel, and Weber [24], and Michel and Weber [29], and in families of Riemann surfaces, by Imayoshi [17], Shiga and Tanigawa [36], and Earle and Sipe [15]. The converse is also true. Two degenerating families of Riemann surfaces of genus g, ' W M ! and ' 0 W M 0 ! , are said to be topologically equivalent and will be Top

denoted by .M; '; / Š .M 0 ; ' 0 ; /, if there exist orientation preserving homeomor8 A pseudo-periodic homeomorphism f W † ! † is of negative twist, if it is periodic or, in the parabolic g g case, if all of its screw numbers (see [31] and [28]) about the reducing curves Ci are negative: s.Ci / < 0.

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Yukio Matsumoto

phisms H W M ! M 0 and h W ! (with h.0/ D 0) such that the diagram M

H

'0

'

/ M0

h

/

commutes. Theorem 3.3 ([27], [28], and [25]). The set g of all topological equivalence classes > 2) of (relatively minimal) degenerating families of Riemann surfaces of genus g ( D over is in a bijective correspondence with the set Pg of all conjugacy classes (in g ) of pseudo-periodic mapping classes of negative twists. The correspondence is given by the topological monodromy g 3 ŒM; '; 7! Œf 2 Pg : A right-handed full Dehn twist about an essential simple closed curve C on †g is the simplest example of a pseudo-periodic homeomorphism of negative twist (we consider that such a Dehn twist gives a .1/-twist about C.) To this Dehn twist corresponds, under Theorem 3.3, a degenerating family over whose central fiber is a stable curve with one node, obtained by pinching the curve C to a point.

4 Fenchel–Nielsen coordinates The closed surface †g has a system of disjoint simple closed curves L D fL1 ; L2 ; : : : ; L3g3 g such that the closure of each connected component of †g L is a pair of pants (i.e. a compact surface homeomorphism to a 2-sphere with three disjoint open disks deleted); see [1] and [13]. Let p D ŒS; w be a point of T .†g /. Since the Riemann surface S has a natural hyperbolic metric (descending from the Poincaré metric of the upper half plane H D SQ ), each simple closed curve w 1 .Li / on S is isotopic to a simple closed y i . Moreover, these simple closed geodesics Ly D fL y 1; L y 2; : : : ; L y 3g3 g are geodesic L y i / depends on p 2 T .†g / real disjoint; see [13]. The hyperbolic length li D l.L y we obtain 2g 2 analytically. Decomposing S along the simple closed geodesics L, pairs of pants with geodesic boundaries. This process is called the pants decomposition of S by the system L of simple closed curves. To recover the Riemann surface S, one has to glue these pants together along the y i using certain twisting parameters i ; i D 1; 2; : : : ; 3g 3. The mapping geodesics L ‰.p/ D .l1 ; : : : ; l3g3 ; 1 ; : : : ; 3g3 /


85

gives a diffeomorphism of T .†g / onto R3g3 R3g3 . These coordinates ‰ are called C the Fenchel–Nielsen coordinates based on the system L of simple closed curves. Cf. [1] and [18], Chapter 3. For the proofs of the following two basic lemmas, see [20] and [1], p. 95. Lemma 4.1. There exists a universal positive constant M0 such that distinct simple closed geodesics C1 and C2 on a compact Riemann surface of genus g do not intersect if their hyperbolic lengths are shorter than M0 , i.e. l.C1 /; l.C2 / < M0 H) C1 \ C2 D ;: Figure 3 explains the intuitive meaning of Lemma 4.1. longer

longer

shorter

shorter

Figure 3. When closed geodesics become shorter, transverse curves become longer.

Lemma 4.2. There exists a universal positive constant M1 such that every compact Riemann surface of genus g has a pants decomposition by a system of simple closed y 3g3 g each member of which has the length shorter than y 1; : : : ; L geodesics Ly D fL M1 , i.e. y i / < M1 ; i D 1; : : : ; 3g 3: l.L

5 Compactification process of M.†g / First note that, up to the action of g , there are at most a finite number of topological ways to decompose a surface of genus g into 2g 2 pairs of pants. Given an infinite sequence of points fpn g1 nD1 T .†g /, Lemma 4.2 tells us that we may assume, under the action of g and in particular under the action of finite products of certain Dehn twists to pn , that there is a certain infinite subsequence, denoted by the same notation fpn g1 nD1 again, which is contained in .0; M1 .0; M1 Œ0; 2 Œ0; 2 „ ƒ‚ … „ ƒ‚ … 3g3

3g3

with respect to the Fenchel–Nielsen coordinates based on a certain system L of simple closed curves.

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Thus either (1) there exists a subsequence fpn.j / gj1D1 converging to a point of T .†g / or (2) there exist a subsequence fpn.j / gj1D1 and a subset fLi.1/ ; : : : ; Li.k/ g of L such y i.k/ / of the corresponding geodesics y i.1/ /; : : : ; l.L that the hyperbolic lengths l.L (for pn.j / ) converge to 0 as j ! 1. Therefore, considering .0; M1 Œ0; 2 to be the polar coordinates of a punctured disk D.M1 / f0g of radius M1 , the disk D.M1 / compactifies .0; M1 Œ0; 2 by adding the origin, and the product D.M1 / D.M1 / of 3g 3 copies of D.M1 / compactifies the part of M.†g / corresponding to the direct product .0; M1 .0; M1 Œ2; 2 Œ2; 2. The above product of the disks D.M1 / is real analytically isomorphic to a part near the “boundary” of Bers’ deformation space D.L/, [9], [8]; see [1], p. 104.

6 Bers’ deformation spaces Bers [9] defines Riemann surfaces with nodes. Let us quote relevant passages from [9] (with notation slightly changed). > 2/ is a “A compact Riemann surface with nodes of (arithmetic) genus g.D > 0/ points P1 ; : : : ; Pk called connected complex space S, on which there are k.D nodes, such that (i) every node Pj has a neighborhood homeomorphic to the analytic set fz1 z2 D 0; jz1 j < 1; jz2 j < 1g, with Pj corresponding to .0; 0/; > 1/ components †.1/ ; : : : ; †.r/ , called (ii) the set S fP1 ; : : : ; Pk g has r.D parts, each †.i/ is a Riemann surface of some genus gi compact except for ni > 0, and n1 C C nr D 2k; and (iii) we have punctures, with 3gi 3 C ni D g D .g1 1/ C C .gr 1/ C k C 1.” Let S and S 0 be Riemann surfaces with nodes. “A continuous surjection u W S 0 ! S is called a deformation if for every node P 2 S, u1 .P / is either a node or a Jordan curve avoiding all nodes and, for every part †.i/ of S, u1 j†.i/ is an orientation preserving homeomorphism.” Bers then proceeds to define an equivalence relation of deformations. “Two deformations, u W S 0 ! S and v W S 00 ! S are called equivalent if there are homeomorphisms ' W S 0 ! S 00 and W S ! S, homotopic to an isomorphism and to the identity, respectively, such that v ı ' D ı u.” “The deformation space D.S/ consists of all equivalence classes ŒS 0 ; u of deformations onto S.”


87

Bers associates to each node P 2 S a subspace of D.S/ called a distinguished subset consisting of all ŒS 0 ; u 2 D.S/ for which the preimage u1 .P / of P is a node of S 0 . Theorem 6.1 ([9]). D.S/ is a cell. There is a (canonical) homeomorphism of D.S/ onto C3g3 which takes each distinguished subset onto a coordinate hyperplane. Bers realized D.S/ as a bounded domain in C3g3 ; see [9]. The complex structure coming from the bounded domain is taken to be the natural complex structure of D.S/. A system of disjoint simple closed curves C D fC1 ; C2 ; : : : ; Ck g on †g is said to be admissible, if each connected component of †g C has negative Euler characteristic. < 3g 3, and if k D 3g 3, we call C a terminal system. Let Obviously, we have k D C D fC1 ; C2 ; : : : ; Ck g be an admissible system of simple closed curves. Let †g .C / denote a Riemann surface with nodes which is obtained by pinching each curve in C > 2/. For simplicity, we will to a point (i.e. a node). †g .C / has arithmetic genus g.D denote the deformation space D.†g .C // by D.C /. Let C 0 be a sub-system of simple closed curves of C , i.e. C 0 C . Then the deformation h W †g .C 0 / ! †g .C / which pinches each curve in C C 0 to a node induces a mapping h W D.C 0 / ! D.C / called allowable mapping, which takes each ŒS; u 2 D.C 0 / into ŒS; h ı u. In particular, if C 0 D ;, then D.;/ is nothing but the Teichmüller space T .†g /, and the allowable mapping h W T .†g / ! D.C / is understood as follows (cf. Kra [22]). By adding certain simple closed curves CkC1 ; : : : ; C3g3 to C D fC1 ; : : : ; Ck g, we obtain a terminal system L which gives a pants decomposition of †g . We can speak of the Fenchel–Nielsen coordinates based on L. Let .C / denote the subgroup of g generated by .1/-Dehn twists .Ci / about the simple closed curves Ci 2 C , i D 1; : : : ; k. Then .C / Š Z ˚ ˚ Z (with k summands). Two points p1 D ŒS1 ; w1 and p2 D ŒS2 ; w2 have the same image under the allowable mapping h W T .†g / ! D.C / if and only if p1 D Œf .p2 / for some Œf 2 .C /, i.e., in terms of the Fenchel–Nielsen coordinates, if and only if ‰.p1 / D .l1 ; : : : ; l3g3 ; 1 C 2m1 ; ; : : : ; k C 2mk ; kC1 ; : : : ; 3g3 /; where we assume ‰.p2 / D .l1 ; : : : ; l3g3 ; 1 ; : : : ; k ; kC1 ; : : : ; 3g3 /; and m1 ; : : : ; mk are integers. We can adopt .l1 e i1 ; : : : ; lk e ik ; lkC1 C ikC1 ; : : : ; l3g3 C i3g3 /

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as the coordinates for the image h .T .†g // in D.C /. Thus h .T .†g // is the quotient T .†g /=.C / and is real analytically isomorphic to C: .C f0g/ .C f0g/ C „ ƒ‚ … „ ƒ‚ … 3g3k

k

Let …i denote the i-th coordinate plane in C3g3 : …i D Ci1 f0g C3g3i : The deformation space D.C / is the completion of the quotient T .†g /=.C / by adding k [

…i :

iD1

(See [22], Theorem 9.4.) The coordinate plane …i corresponds to the distinguished subset parametrizing conformal structures on the Riemann surface with one node †g .fCi g/. More generally, for a subset C 0 C , the intersection \ …i Ci 2C 0

corresponds to the subset of D.C / that parametrizes conformal structures on the Riemann surface with nodes †g .C 0 /. Thus we can identify this subset with the Teichmüller space T .C 0 / of the Riemann surface with nodes †g .C 0 / 9 . It is easy to see that dimC T .C 0 / D 3g 3 #C 0 : To summarize, D.C / is the completion of the quotient T .†g /=.C / by the union of the Teichmüller spaces [ T .C 0 /: ;¤C 0 C

Similarly to the above, the general allowable mapping h W D.C 0 / ! D.C / is the projection map onto the quotient D.C 0 /=.C C 0 /, and this quotient space is identified with an open subset [ T .fCi g/: D.C / Ci 2C C 0

Kra [22] announces the following result. 9 We

think that more precisely the notation T .C 0 / should be TC .C 0 /.


89

Theorem 6.2 ([22], §3.10). There exists a family of Riemann surfaces (with nodes) C W V .C / ! D.C /: The total space V .C / is a .3g 2/-dimensional complex manifold. Over a point p 2 T .C 0 /, C 0 being a subset of C , we have a Riemann surface with nodes canonically homeomorphic to †g .C 0 /. Figure 4 illustrates this “tautological” fibration. j i

D.C / D

j

i

Figure 4. The deformation space D.C/ parametrizes Riemann surfaces with nodes.

Let N .C / denote the normalizer of .C / in the mapping class group g : N .C / D fŒf 2 g j Œf .C /Œf 1 D .C /g: Let W .C / denote the quotient group N .C /=.C /. This group is canonically identified with the mapping class group of the surface with nodes †g .C /: W .C / D ff W †g .C / ! †g .C / j orientation preserving homeomorphismg=isotopy: W .C / acts on D.C / as automorphisms. The action is defined as follows: Œf .ŒS; u/ D ŒS; f ı u;

for Œf 2 W .C /; ŒS; u 2 D.C /:

The following theorem is an extension of Theorem 1.1. Cf. [22], §4.5, §4.6, and §7.

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Theorem 6.3. The group W .C / acts on the fiber space C W V .C / ! D.C / in a fibration preserving manner. This action is properly discontinuous and holomorphic, and by taking quotients, we obtain an orbifold fiber space C0 W Y.C / ! M.C /; where Y.C / and M.C / denote V .C /=W .C / and D.C /=W .C /, respectively. Unfortunately, the relation between the spaces M.†g / and M.C / is very difficult to see. To remedy this difficulty, we will introduce “subdeformation spaces”.

7 Subdeformation spaces D" .C / Let C D fC1 ; : : : ; Ck g be an admissible system of disjoint simple closed curves on †g , and let h W T .†g / ! D.C / be the allowable mapping corresponding to the inclusion ; C . Choose positive real numbers "1 ; 1 ; "2 ; 2 ; : : : ; "3g3 ; 3g3 satisfying 0 < "1 < 1 < "2 < 2 < < "3g3 < 3g3 < M0 ;

(7.1)

and fix them (M0 being the number appearing in Lemma 4.1). We define the subdeformation space D" .C / as follows:

3

< l.u1 .ŒCi // < "k ; i D 1; : : : ; k; D" .C / D fŒS; u 2 D.C / j 0 D and other simple closed geodesics on S are longer than k g; where ŒCi denotes the node of †g .C / which is obtained by pinching the simple closed curve Ci 2 C to a point, and the suffix k of "k and of k is the number of simple closed curves in C . Thus u1 .ŒCi / is the simple closed curve or the node on S which is mapped to the point ŒCi under the deformation u W S ! †g .C /. The

3 3

notation u1 .ŒCi / denotes the simple closed geodesic which is isotopic to u1 .ŒCi /, and l.u1 .ŒCi // is its length. The equality l.u1 .ŒCi // D 0 is understood to mean

3 .ŒC / is a node in S. In particular, l.u3 .ŒC // D 0 for all i D 1; : : : ; k, if

1 that u1 i i and only if ŒS; u is in the Teichmüller space T .C /. The subdeformation space D" .C / is an open “handle” whose “core” is T .C / \ D" .C /. (See Figure 5.) The action of W .C / on D.C / preserves the subdeformation space D" .C /.

Lemma 7.1. Suppose that a mapping class Œf 2 g takes a point ŒS; w 2 T .†g / into another point ŒS 0 ; w 0 2 T .†g /, and that h .ŒS; w/ and h .ŒS 0 ; w 0 / are in D" .C /, where h W T .†g / ! D.C / is the allowable mapping. Then Œf belongs to N .C /. Proof. Note that S and S 0 are nonsingular Riemann surfaces. Since we have ŒS 0 ; w 0 D Œf .ŒS; w/, there exists a biholomorphic map t W S ! S 0 such that the following


91

Figure 5. Subdeformation space D" .C/.

diagram homotopically commutes: S

w

/ †g f

t

S0

w0

/ †g :

The map t W S ! S 0 preserves the hyperbolic (Poincaré) metric. Since h .ŒS; w/ and h .ŒS 0 ; w 0 / belong to the subdeformation space D" .C /, the closed geodesics

3

3

w 1 .Ci /; i D 1; : : : ; k; (or w 0 1 .Ci /; i D 1; : : : ; k) are only simple closed geodesics on S (or on S 0 ) that are shorter than "k . Thus for each i D 1; : : : ; k, the map t takes

3

3

the geodesic w 1 .Ci / to a geodesic w 0 1 .Cj / with some j 2 f1; : : : ; kg. Hence f W †g ! †g induces a permutation (up to isotopy) of the simple closed curves of C . This implies that Œf belongs to the normalizer N .C /. Lemma 7.1 is an analogy of Shimizu’s lemma on Fuchsian groups; see [18]. As we said above, the relation between M.†g / and M.C / is complicated. However, if we define the submoduli space M" .C / to be the quotient D" .C /=W .C /, its relation to M.†g / is clear: By Lemma 7.1, M" .C / projects homeomorphically onto an open subset of M.†g /. We will identify M" .C / with the projected image. Note that there are only finitely many subdeformation spaces D" .C / and submoduli spaces M" .C /, because, if C and C 0 are equivalent under an orientation preserving homeomorphism f W †g ! †g (i.e. C 0 D f .C /), then .D" .C /; W .C // and .D" .C 0 /; W .C 0 // are “identical”. Lemma 7.2. The “boundary” M.†g / M.†g / is covered by the submoduli spaces: [ M.†g / M.†g / M" .C /: C¤;

Proof. Let "1 ; 1 ; "2 ; 2 ; : : : ; "3g3 ; 3g3 be positive real numbers satisfying inequality (7.1). Let S be a Riemann surface or a Riemann surface with nodes. Let

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C1 ; C2 ; : : : ; Cn be the totality of simple closed geodesics on S whose lengths l.Ci /; i D < 3g 3. (Here we adopt 1; 2; : : : ; n; are shorter than M0 . By Lemma!4.1 we have n D a convention that a node is a special case of a simple closed geodesic whose length is equal to 0.) Put ci D l.Ci /, for simplicity. Permuting Ci ’s, if necessary, we may assume < c1 D < c2 D < D < cn : 0D 0; i D 1; 2; 3 where g.z1 ; z2 ; zN 1 ; zN 2 / is a convenient non-degenerate polar weighted homogeneous polynomial. Then the Milnor fibers F .fi /; i D 1; : : : ; 3 and F .fi0 /; i D 2; 3 are simply connected and they have homotopy types of bouquets of spheres S 2 _ _ S 2 . Let g be the Milnor number of g. The Euler characteristics and the Milnor numbers are given as follows: .F .f1 // D .a3 1/g C 1; .f1 / D .a3 1/g .F .f2 // D .F .f20 // D a1 a2 a3 a2 a3 C a3 ; .f2 / D .f20 / D .F .f2 // 1; .F .f3 // D a1 a2 a3 C 1; .f3 / D a1 a2 a3 ; .F .f30 // D a1 a2 a3 1 .f30 / D a1 a2 a3 2; Proof. We consider first F1 D f11 .1/ where f1 .z; zN / D g.z1 ; z2 ; zN1 ; zN 2 / C z3a3 Cb3 zN3b3 where g.z1 ; z2 ; zN1 ; zN2 / is a convenient non-degenerate polar weighted homogeneous polynomial. For two variables case, the Milnor fiber Fg of g.z1 ; z2 / has the homotopy type of a bouquet of S 1 as it is a connected open Riemann surface (see [8], Proposition 38). Let g be the Milnor number (that is the first Betti number) of Fg . Then the Milnor fiber F1 of f1 is homotopic to the join Fg a where a is the set of a-th roots of unity (see [2]). This join is obviously homotopic to a bouquet of g .a 1/ S 2 spheres. Consider F2 D f21 .1/ or F20 D f21 .1/. The Euler characteristic can be easily computed from the additivity of the Euler characteristics, applied on the toric stratification F2 D F2f1;2;3g q F2f2;3g q F2f3g ; F20 D F20f1;2;3g q F20f2;3g q F20f3g ;

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and Theorem 10 of [7], where F2I is defined by F2 \ CI and CI D fz 2 C3 j zi ¤ 0; i 2 I; zj D 0; j … I g for I f1; 2; 3g. The Euler characteristics of F3 D f31 .1/ and F30 D f30 1 .1/ can be computed in the exact same way. The assertion on the homotopy types are now obtained simultaneously as follows. First Fjf1;2;3g and Fj0 f1;2;3g are CW-complex of dimension 2 by Theorem 10 of [7]. Secondly Fj and Fj0 are simply connected by the 1-convenience assumption (see [7]). Using the above decomposition and Mayer–Vietoris exact sequences, we see that the (reduced) homology groups are non-trivial only on dimension 2 and no torsion on H2 .Fj / and H2 .Fj0 / for j D 2; 3. Thus by the Whitehead theorem (see for example [10]), we conclude that Fj and Fj0 are homotopic to bouquets of two dimensional spheres.

5.2 Projective mixed curves We consider projective curves of degree q: C D fŒz1 W z2 W z3 2 CP 2 j f .z1 ; z2 ; z3 / D 0g; where f is a strongly polar homogeneous polynomial with pdeg f D q. We have seen that the topological degree of C is q by Theorem 4.1. The genus g of C is not an invariant of q. Recall that for a differentiable curve C of genus g, embedded in CP 2 , with the topological degree q, we have the following Thom’s inequality, which was conjectured by Thom and proved by, for example, Kronheimer–Mrowka [4]: g

.q 1/.q 2/ 2

where the right side number is the genus of algebraic curves of degree q, given by the Plücker formula. Recall that for a mixed strongly polar homogeneous polynomial, the genus and the Euler characteristic of the Milnor fiber are related as follows (cf. Proposition 2.3(ii)). Proposition 5.2. We have 2 2g D 3

.F / q

(5.1)

where F D f.z1 ; z2 ; z3 / 2 C3 j f .z1 ; z2 ; z3 ; zN1 ; zN2 ; zN 3 / D 1g: Now we will see some examples which shows that .F / is not an invariant of q.

On mixed projective curves

145

I. Simplicial polynomials. We consider the following simplicial polar homogeneous polynomials of polar degree q: fs1 .z; zN / D z1qCr zN 1r C z2qCr zN2r C z3qCr zN 3r ; fs2 .z; zN / D z1qCr1 zN1r z2 C z2qCr1 zN2r z3 C z3qCr zN3r ; fs3 .z; zN / D z1qCr1 zN1r z2 C z2qCr1 zN2r z3 C z3qCr1 zN3r z1 ; fs4 .z; zN / D z1qCrC1 zN1r zN2 C z2qCrC1 zN2r zN3 C z3qCr zN3r ; fs5 .z; zN / D z1qCrC1 zN1r zN2 C z2qCrC1 zN2r zN3 C z3qCrC1 zN3r zN1 : Let Fsi be the Milnor fiber of fsi and let Csi be the corresponding projective curves for i D 1; : : : ; 5. First, the Euler characteristic of the Milnor fibers and the genera are given by Proposition 5.1 and Proposition 5.2 as follows: .q 1/.q 2/ ; i D 1; 2; 3; 2 q.q C 1/ .Fs4 / D q.q 2 C q C 1/; ; g.Cs4 / D 2 .q C 2/.q C 1/ .Fs5 / D q.q 2 C 3q C 3/; g.Cs4 / D : 2 In [9], we have shown that Cs1 and Cs2 are isomorphic to algebraic plane curves defined by the associated homogeneous polynomials of degree q: .Fsi / D q 3 3q 2 C 3q;

g.Csi / D

gs1 .z/ D z1q C z2q C z3q gs2 .z/ D z1q1 z2 C z2q1 z3 C z3q : We also expect that Cs3 is isotopic to the algebraic curve z1q1 z2 C z2q1 z3 C z3q1 z1 D 0; as the genus of Cs3 suggests it (see also [9]). II. We consider the following join type polar homogeneous polynomial: N C z3qCr zN3r ; hj .z; zN / D gj .w; w/ N D .w1qCj wN 1j C w2qCj wN 2j /.w1rj ˛w2rj /.wN 1rj ˇ wN 2rj /; gj .w; w/ where 0 j r and where ˛; ˇ 2 C are generic. The Milnor fiber Fgj of gj is connected. The Euler characteristic of .Fgj / (Fgj D Fgj \ C2 ) is given by .Fgj / D rg q where rg is the link component number of g D 0 which is q C 2.r j /. Thus .Fgj / D .Fgj / C 2q where the last terms come from .Fgj /I / with

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// I D f1g or f2g. Thus .Fgj / D q.q 2 C 2.r j // and g D .q1/.q2C2.rj . We 2 observe that the genus can take the following values by taking j D r; : : : ; 0:

.q 1/.q C 2r 2/ .q 1/.q 2/ .q 1/q ; ; :::; : 2 2 2 As we can take the positive number r arbitrary large, we have the following result. Proposition 5.3. There exist differentiable curves embedded in CP 2 with a fixed degree q 2 whose genera are given as fg0 C k.q 1/ j k D 0; 1; : : : g;

g0 D

.q 1/.q 2/ : 2

In particular, taking q D 2, we obtain the following corollary. Corollary 5.4. For any smooth surface S of genus g, there is an embedding S CP 2 such that the degree of S is 2.

References [1]

A. Borel and A. Haefliger, La classe d’homologie fondamentale d’un espace analytique, Bull. Soc. Math. France 89 (1961), 461–513. 140

[2]

J. L. Cisneros–Molina, Join theorem for polar weighted homogeneous singularities, in Singularities II. Geometric and topological aspects. Proceedings of the International School and Workshop on the Geometry and Topology of Singularities held in honor of the 60 th birthday of Lê D˜ung Tráng in Cuernavaca, January 8–26, 2007, ed. by J.-P. Brasselet, J. L. Cisneros-Molina, D. Massey, J. Seade, and B. Teissier, Contemporary Mathematics 475, American Mathematical Society, Providence, RI, 2008, 43–59 143

[3]

P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley & Sons, New York 1978; reprinted it Wiley Classics Library, John Wiley & Sons, New York 1994. 135

[4]

P. B. Kronheimer and T. S. Mrowka, The genus of embedded surfaces in the projective plane. Math. Res. Lett. 1 (1994), 797–808. 134, 144

[5]

J. Milnor, Singular points of complex hypersurfaces. Annals of Mathematics Studies 61, Princeton University Press, Princeton 1968. 137

[6]

M. Oka, On the cohomology structure of projective varieties, in Manifolds–Tokyo 1973. Proceedings of the International Conference on Manifolds and Related Topics in Topology held in Tokyo, April 10–April 17, 1973, ed. by A. Hattori, University of Tokyo Press, Tokyo 1975, 137–143. 138

[7]

M. Oka, Topology of polar weighted homogeneous hypersurfaces. Kodai Math. J. 31 (2008), 163–182. 135, 140, 144

[8]

M. Oka, Non-degenerate mixed functions. Kodai Math. J. 33 (2010), 1–62. 135, 139, 140, 142, 143

On mixed projective curves [9]

147

M. Oka, On mixed Brieskorn variety, in Topology of algebraic varieties and singularities. Papers from the Conference on Topology of Algebraic Varieties, in honor of Anatoly Libgober’s 60 th birthday, held in Jaca, June 22–26, 2009, ed. by J. I. Cogolludo-Agustín and Eriko Hironaka, Contemporary Mathematics 538. American Mathematical Society and Real Sociedad Matemática Española, Providence, R.I., and Madrid 2011, 389–399. 135, 145

[10] E. H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York etc. 1966; corrected reprint Springer, New York and Berlin, 1981. 144

Invariants of splice quotient singularities Tomohiro Okuma Department of Education, Yamagata University Yamagata 990-8560, Japan email: [email protected]

Abstract. This article is a survey of results on analytic invariants of splice quotient singularities induced by Neumann and Wahl. These singularities are natural and broad generalization of quasihomogeneous surface singularities with rational homology sphere links. The “leading terms” of the equations are constructed from the resolution graph. Some analytic invariants of splice quotients can explicitly be computed from their graph.

1 Introduction Splice quotient singularities were introduced by Neumann and Wahl ([21], [22], [23], [18], and [30]). These singularities form a rich class of complex surface singularities with rational homology sphere (QHS for short) links. This class contains rational singularities, minimally elliptic singularities and weighted homogeneous singularities with QHS links. We consider a resolution graph of a normal complex surface singularity with QHS link with certain conditions. Then from we can construct, according to Neumann–Wahl algorithm, the equations of a family of surface singularities in which each fiber has the resolution graph ; these singularities are called splice quotients. For example, if a resolution graph of a rational singularity is given, we can explicitly write down equations for a rational singularity with that graph. From the point of the definition of the splice quotient, it is very natural to expect that some fundamental analytic invariants of them can be computed from the resolution graph. The purpose of this paper is to survey known results on some invariants of splice quotients. Neumann and Wahl [19] proved the “end curve theorem”, that is, the splice quotients are characterized by the existence of “end curve functions” (cf. [27] and [3]). The results on the dimension of cohomology groups of certain invertible sheaves depend on the existence of the end curve functions. The author is grateful to the referee for pointing out mistakes and helpful suggestions.

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2 Splice quotients The system of equations for a splice type singularity is originally associated with a weighted tree called a splice diagram, which is obtained from the resolution graph of a surface singularity with QHS link satisfying the “semigroup condition” (see [22] and [23] for details). In this section, we introduce the splice type singularities and splice quotient singularities in terms of “monomial cycles” (see [26] for details). The equivalence of these two definitions is verified in [22], §13. We will see later that the notion of monomial cycles is useful for connecting the combinatorics of the resolution graph to analytic objects. Let .X; o/ be a normal complex surface singularity whose link † is a QHS. We may assume that X is homeomorphic to the cone over †. By definition, a topological invariant of .X; o/ is an invariant of †. There uniquely exists a finite morphism q W .X u ; o/ ! .X; o/ of normal surface singularities that induces an unramified Galois covering X u n fog ! X n fog with Galois group H1 .†; Z/. The morphism q W X u ! X is called the universal abelian covering of X . Let W Xz ! X be the minimal good resolution with the exceptional divisor E D 1 .o/. Let fEv gv2V denote the set of irreducible components of E and let denote the weighted dual graph of E. It is known that and † determine each other (see [17]). The assumption on † is equivalent to that every Ev is a rational curve and is a tree. Hence and the intersection matrix I.E/ D .Ev Ew / have the same information. Since I.E/ is negative-definite, for each v 2 V there exists an effective Q-cycle Ev such that Ev Ew D ıvw for every w 2 V . Let X X ZEv and L D ZEv : LD v2V

v2V

We call an element of the group L (resp. L ˝ Q) a cycle (resp. Q-cycle). We have a natural isomorphism (cf. [8], §2, and [25], §2) H D L =L ! H1 .†; Z/: Thus H1 .†; Z/ is a finite group of order jdet I.E/j. Let ıv denote the number of irreducible components of E intersecting Ev , i.e., ıv D .E Ev / Ev . A curve Ev is called an end (resp. a node) if ıv D 1 (resp. ıv 3). Let E (resp. N ) denote the set of indices of ends (resp. nodes). A connected component of E Ev is called a branch of Ev . Let Cfzg D Cfzw I w 2 Eg be the convergent power series ring in #E variables. P Definition 2.1. An element of a semigroup M D w2E Z0 Ew , where Z0 is the setPof nonnegative integers, is called a monomial cycle. For a monomial cycle Q ˛w , we associate a monomial z.D/ D w2E zw 2 Cfzg. D D w2E ˛w Ew

Invariants of splice quotient singularities

151

For every v; w 2 V , we define positive integers ev , lvw , and mvw as follows: .lvw / D jH j.I.E//1 ; ev D jH j= gcd flvw j w 2 V g; mvw D ev lvw =jH j: Q ˛w Definition P 2.2. For any v 2 V , we define the v-degree of a monomial w2E zw to be w2E ˛w mvw . The leading form of f 2 Cfzg with respect to the v-weight is called the v-leading form of f and denoted by LFv .f /. The v-degree of LFv .f / is called the v-order of f. Note that v-deg z.D/ D ev D Ev . In Section 3, we shall see the geometric nature of the degree. Definition 2.3. We say that E (or ) satisfies the monomial condition if for any node Ev and any branch C of Ev , there exists a monomial cycle D such that D Ev is an effective cycle supported on C . For such D, the monomial z.D/ is called an admissible monomial belonging to the branch C. Definition 2.4. Assume that the monomial condition is satisfied. Let Ev be an arbitrary node with branches C1 ; : : : ; Cıv , and let Mi denote an admissible monomial belonging to Ci . Let Fv be the set of polynomials f1 ; : : : ; fıv 2 defined by fi D

ıv X

cij Mj

j D1

with

0

1 0 B0 1 B .cij / D B : : : :: @ :: :: 0 0

0 0 :: :

a1 a2 :: :

1 b1 b2 C C C; A

bıv 2 S where ai ; bi 2 C and ai bj aj bi ¤ 0 (i ¤ j ). We call v2N Fv a Neumann–Wahl system associated with E. 1 aıv 2

Example 2.5. Let us consider the following graph corresponding to a QHS link: s 1 P P

6 PP P s s 2

9

7

s

s 8 sH s4 3 HHs

s3 s5

10

where the positive integers i indicate Ei and the weights 2 are omitted. This graph satisfies the monomial condition. In fact, the following polynomials form a Neumann–

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Wahl system: F6 D fz12 C z22 C z32 z52 g; F8 D fz12 C z33 C z53 ; z42 C z33 z53 g: The intersection form L L ! Q defined by I.E/ induces a pairing e

W H L ! Q=Z ! C ;

p where e.x/ D exp.2 1x/. We denote by ‚.D/ 2 HO D Hom.H; C / the character determined by . ; D/. The group H acts on the power series ring Cfzg as follows. For any .h; D/ 2 H M, we define h z.D/ 2 Cfzg by h z.D/ D .h; D/z.D/: Note that Fv consists of ‚.Ev /-eigenfunctions. Definition 2.6 (see [22], §7). Consider a set S

F D ffvjv j v 2 N ; jv D 1; : : : ; ıv 2g Cfzg:

If the set v2N fLFv .fvjv / j jv D 1; : : : ; ıv 2g is a Neumann–Wahl system associated with E, then F is called a system of splice diagram functions, and the singularity .Y; o/ .C#E ; o/ define by F is called a splice type singularity. This is an isolated complete intersection surface singularity. Furthermore if every fvjv is a ‚.Ev /-eigenfunction, then the singularity .Y =H; o/ is a normal surface singularity and called a splice quotient singularity. Definition 2.7. We say that Xz satisfies the end curve condition if for each w 2 E there exist a function uw on Xz and an irreducible curve Hw Xz , not contained in C Hw / D div.uw /. Then Hw E D Hw Ew D 1. E, such that ew .Ew w If the end curve condition is satisfied, then, taking functions sw D u1=e on X u , w we obtain an H -equivariant C-algebra homomorphism

W Cfzg ! OX u ;o ;

.zw / D sw :

Theorem 2.8 (End Curve Theorem [19]). If Xz satisfies the end curve condition, then X is a splice quotient singularity; in fact, the homomorphism is surjective, and its kernel is generated by a system of splice diagram functions with H -action. The converse is also true.

3 Filtrations We assume that .X; o/ satisfies the end curve condition. Let us recall that the universal abelian cover q W X u ! X fits into the following commutative diagram, where p is


153

finite and unramified over Xz n E, is a partial resolution (cf. [25], §3.2): Xz u

p

Xu

/ Xz

q

/ X:

Then F D p 1 .E/ is the -exceptional set on Xz u . Let v 2 V and Fv D p 1 .Ev /. For each n 2 Z0 , we denote by Inv theLideal f .f / j v-ord.f / ng OX u ;o . Let v G.v/ denote the associated graded ring n0 Inv =InC1 . Since p Ev D ev Fv by [25], §3.4, it follows from the definition of the v-degree that every function f 2 Inv satisfies divXz u .f / nFv . Proposition 3.1 ([22], §2.6 [26], §3, [24], §4). For every v 2 V , (i) Inv D . OXz u .nFv //o , for n 0, and (ii) the ring G.v/ is a reduced complete intersection ring, isomorphic to CŒz=I v , where I v is the ideal generated by fLFv .f / j f 2 F g. y , let Definition 3.2. For v 2 V and 2 H Y 1 X 1 H;v .t/ D .h/ .1 .h; Ew / t mvw /ıw 2 : jH j h2H

w2V

.t/ is the Hilbert series of the Proposition 3.3 ([26]). The rational function H;v -eigenspace G.v/ with respect to the H -action, i.e., X H;v .t/ D .dim G.v/n /t n : n0

In [12], Némethi proves the Campillo–Delgado–Gusein-Zade type identity for the Hilbert series of the multi-variable filtration associated with the exceptional divisors of the minimal good resolution. This result implies the proposition above, and furthermore that h1 of invertible sheaves associated with exceptional divisors are topological.

4 Some analytic invariants In general the fundamental invariants geometric genus and multiplicity are not topological invariants. However for splice quotients these are explicitly computed from the weighted dual graph. Assume that Xz satisfies the end curve condition. Let H1 be a subgroup of H and .X1 ; o/ D .X u =H1 ; o/. Every abelian cover of X which is unramified over X n fog is of this type.

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4.1 The geometric genus The geometric genus of .X; o/ is denoted by pg .X; o/. We define an invariant of the triple .; v; /. .t/ D p.t/ C r.t/=q.t/, p; q; r 2 CŒt, deg r < deg q. Definition 4.1. Suppose H;v Then c;v D p.1/. We omit in the notation when D 1.

Let v 2 V . Suppose that C1 ; : : : ; Cıv are the branches of Ev . Let .Xi ; xi / denote the normal surface singularity obtained by contracting Ci . Theorem 4.2 ([26]). We have the following results. (i) Each .Xi ; xi / is also a splice quotient. Pv (ii) pg .X; o/ D c;v C ıiD1 pg .Xi ; xi /. Corollary 4.3. The geometric genus of a splice quotient can be computed from . If is a star-shaped graph with node v, then pg .X; o/ D c;v . Example 4.4. Suppose that .X; o/ is a splice quotient with graph in Example 2.5. Then H;8 .t/ D t 5 C t C fractional part. Since every branch of E8 corresponds to a rational singularity, pg .X; o/ D c;v D 2. There is a formula for hi of -eigensheaves of OX u similar to Theorem 4.2(2), and thus pg .X1 ; o/ can be computed from and H1 (cf. [26]).

4.2 The Seiberg–Witten invariant In this subsection we mention results on the Seiberg–Witten invariant of the links of singularities. First let us recall Casson Invariant Conjecture (Neumann–Wahl [20]). If .V; o/ is an isolated complete intersection surface singularity with Z-homology sphere link, then D =8, where is the Casson invariant of the link and the signature of the Milnor fiber of .V; o/. In [20], Neumann and Wahl proved the conjecture for Brieskorn complete intersections (Fukuhara–Matsumoto–Sakamoto [5] independently proved it) applying the additivity properties with respect to splice decomposition and the result for Brieskorn hypersurface proved by Fintushel and Stern [4], and also proved it for suspension hypersurface singularities. This conjecture is generalized as follows. If .X; o/ satisfies the assumption of the conjecture, by Laufer–Durfee formula, we obtain D 8pg C K 2 C s, where K is


155

the canonical divisor and s the number of irreducible components of the exceptional divisor on a good resolution. Thus the equality D =8 is equivalent to K2 C s D 0: 8 Note that smoothability of the singularity is not needed in this formulation. Némethi– Nicolaescu [10] considered the Seiberg–Witten invariant sw.†/ in order to generalize the Casson Invariant Conjecture. This was a natural generalization, since sw.†/ D .†/ when H1 .†; Z/ D 0. They computed sw.†/ from the graph and formulated the following: pg C C

Seiberg–Witten Invariant Conjecture ([10], cf. [2]). Let K D KXz and s D #V . If the complex structure of .X; o/ is “nice” pg .X; o/ C sw.†/ C

K2 C s D 0; 8

The original conjecture was formulated for Q-Gorenstein singularities with QHS links, however the counterexamples are given in [7]. Thus the problem is to identify classes of singularities satisfying this identity. Némethi and Nicolaescu proved this conjecture for some classes (see [10], [11], [15], [13], and [14]), including splice quotients with star-shaped graph. This guarantees the first step of the induction for the proof of Seiberg–Witten invariant conjecture for splice quotients. For the singularity .Xi ; xi / in Subsection 4.1, we define the invariants †i , si , Ki2 in a similar way as †, s, K 2 . Theorem 4.5 (Braun–Némethi [2]). v X K 2 C si K2 C s D c;v C sw.†i / C i : 8 8

ı

sw.†/ C

iD1

This theorem and the pg -formula in Subsection 4.1 shows that the Seiberg–Witten invariant conjecture is true for splice quotients. We note that Némethi [8] formulated the conjecture in more general situation and Braun and Némethi [2] verified it for splice quotients. Remark. There exists a non splice quotient which satisfies the Seiberg–Witten invariant conjecture; in fact, there exists an equisingular deformation of a splice quotient such that general fibers are not splice quotients (see [7]).

4.3 The multiplicity We denote the multiplicity of .X; o/ by mult.X; o/. Let mX P denote the maximal ideal of OX;o . For any function f 2 mX n f0g, if divXz . f / D v2V av Ev C C , where

156

Tomohiro Okuma

P C has no component of E, then .f /E D v2V av Ev . For any set D of effective Qcycles, we denote by gcd D the maximal Q-cycle (if it exists) such that gcd D D for any D 2 D. The cycle ZXz D gcd f. f /E j f 2 mX n f0gg is called the maximal ideal cycle on Xz . Theorem 4.6 (Wagreich [29]). If OXz .ZXz / is generated by its global sections, then mult.X; o/ D Z 2z . X

Let M denote the semigroup of the monomial cycles and X1 D X u =H1 . Let MH1 denote the set of H1 -invariant monomial cycles, i.e., MH1 D fD 2 M j ‚.H1 ; D/ D f1gg: Let p W Xz1 ! Xz be as in the commutative diagram in Section 3, but Xz1 is a good P resolution of X1 . For a Q-cycle D D ai Ei , let coef Ei .D/ denote ai . The following theorems are proved in [24]. Theorem 4.7. Let Z D gcd MH1 . Assume the following: (i) for every x 2 E, there exists D 2 MH1 such that D D Z on a neighborhood of x; P (ii) for every i 2 E with Z Ei < 0, there exists F D j 2Enfig aj Ej 2 MH1 such that coef Ei .F / D coef Ei .Z/. Then p Z is the maximal ideal cycle on Xz1 , and OXz1 .p Z/ is generated by global sections. Hence mult.X1 ; o/ D jH=H1 j.Z 2 /. Theorem 4.8. There exists a modification W Xx ! Xz such that the conditions of x which can be obtained by finitely many blowing ups Theorem 4.7 are satisfied on X, at “bad points.” The graph of XN can be obtained from and H1 . Corollary 4.9. The multiplicity of a splice quotient singularity and its universal abelian cover can be computed from the weighted dual graph. In [12], Némethi has obtained a formula for the multiplicity of splice quotients. His formula does not need resolutions of the base points of OXz .ZXz /, and the contributions from the base points are expressed in terms of Newton diagrams obtained from the graph. Example 4.10. We consider a splice quotient with the weighted dual graph of Example 2.5. Let hD1 ; : : : ; Dk i denote the subgroup of H generated by the class of D1 ; : : : ; Dk 2 L . We see that H D hE1 ; E2 ; E3 i and jH j D 12. (The Hermite normal form of I.E/ shows generators of H and their relations.) We denote an element


P10 iD1

ai Ei by the sequence .a1 0 1 0 5 E1 2 B C B B B E2 C B 2 B C B C B 1 B E3 C D B B C B B BE C B @ 5A @1 E8 3

157

a10 /. Then 3 2 3 2

1

4

1

1

1

4 3 2 3

3

2

2

1

5 2

1

1

3

2

4

7 2 7 2

3

2

2

2

2

1

2 3 4 3

2

2

2

5 3 4 3

3

2

6

6

6

4

2

1

C 2C C C 4 C : 3 C C 5 C 3 A 4

Suppose H1 D f0g. Then X1 D X u and Z D 13 E8 . By Theorem 4.7, we have mult.X u ; o/ D jH j.Z 2 / D 12 .2=3/ D 8: The leading form (with respect to deg zi D 1) of the Neumann–Wahl system in Example 2.5 forms a regular sequence of degree 2; 2; 2. This also implies mult.X1 ; o/ D 8. If H1 D hE1 i, then Z D 13 E8 and mult.X1 ; o/ D jH=H1 j.Z 2 / D 6 .2=3/ D 4: Next suppose H1 D H. Then we have Z D E3 C E5 . Since 2E1 2 MH1 and coef Ei .2E1 / D coef Ei .Z/ for i D 3; 5, it follows from Theorem 4.7 that mult.X; o/ D Z 2 D 4.

4.4 The embedding dimension It is known that the embedding dimension of rational and weakly elliptic Gorenstein singularities are topological (see [1], [6], and [9]). However in general the embedding dimension e:d:.X; o/ is not topological even if .X; o/ is a weighted homogeneous singularity. Assume that .X; o/ is a weighted homogeneous singularity, Xz the minimal good resolution, and E0 the node. Let D ı0 . The complex structure of .X; o/ is determined by the weighted dual graph and the configuration of the points E0 \ .E E0 / in E0 . Hence e:d:.X; o/ is obviously topological if D 3. Theorem 4.11 ([16], §5-6). (1) If 5, then the Hilbert series HmX =m2 of the graded 2 is topological, hence so is e:d: .X; o/. artinian module mX =mX

X

(2) If e:d: .X; o/ is topological, then so is HmX =m2 . X

Example 4.12 ([16], §7.1.1). Assume that D 6, E D E0 C E1 C C E6 , and .E02 ; : : : ; E62 / D .2; 2; 2; 3; 3; 7; 7/. Then .X; o/ is defined by y34 y23 y5 C .p1 C p2 /y32 y5 C p1 p2 y52 ; y17 y23 C .p1 C p2 p3 p4 /y32 C .p1 p2 p3 p4 /y5

158

Tomohiro Okuma

where p1 ; : : : ; p4 2 C . Hence e:d: .X; o/ D

´

3 if p1 p2 p3 p4 6D 0, 4 otherwise:

In case E02 , we have an explicit expression of HmX =m2 in terms of the X Seifert invariant (see [16], §6); this is an extension of Van Dyke’s result [28] to the case E02 C 1. However we have not obtained explicit expression of HmX =m2 X for other classes.

References [1]

M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129–136. 157

[2]

G. Braun and A. Némethi, Surgery formula for Seiberg–Witten invariants of negative definite plumbed 3-manifolds, J. Reine Angew. Math. (2010), 189–208. 155

[3]

Gabor Braun, Geometry of splice-quotient singularities, preprint 2008. arXiv:0812.4403 149

[4]

R. Fintushel and R. J. Stern, Instanton homology of Seifert fibred homology three spheres, Proc. London Math. Soc. (3) 61 (1990), 109–137. 154

[5]

S. Fukuhara, Y. Matsumoto, and K. Sakamoto, Casson’s invariant of Seifert homology 3-spheres, Math. Ann. 287 (1990), 275–285. 154

[6]

H. Laufer, On minimally elliptic singularities, Amer. J. Math. 99 (1977), 1257–1295. 157

[7]

I. Luengo-Velasco, A. Melle-Hernández, and A. Némethi, Links and analytic invariants of superisolated singularities, J. Algebraic Geom. 14 (2005), 543–565. 155

[8]

A. Némethi, Line bundles associated with normal surface singularities, preprint 2003. arXiv:math/0310084 150, 155

[9]

A. Némethi, “Weakly” elliptic Gorenstein singularities of surfaces, Invent. Math. 137 (1999), 145–167. 157

[10] A. Némethi and L. I. Nicolaescu, Seiberg-Witten invariants and surface singularities, Geom. Topol. 6 (2002), 269–328. 155 [11] A. Némethi and L. I. Nicolaescu, Seiberg-Witten invariants and surface singularities II. Singularities with good C -action, J. London Math. Soc. (2) 69 (2004), 593–607. 155 [12] A. Némethi, The cohomology of line bundles of splice-quotient singularities, preprint 2008 arXiv:0810.4129 153, 156 [13] A. Némethi, On the Ozsváth–Szabó invariant of negative definite plumbed 3-manifolds, Geom. Topol. 9 (2005), 991–1042. http://www.msp.warwick.ac.uk/gt/2005/09/p023.xhtml 155 [14] A. Némethi, Graded roots and singularities, in Singularities in Geometry and Topology. Proceedings of the Trieste Singularity Summer School and Workshop, ICTP, August 15–


159

September 3, 2005. ed. by J-P. Brasselet, D. T. Lê, and M. Oka, World Scientific, Hackensack, NJ, 2007, 394–463. 155 [15] A. Némethi and L. I. Nicolaescu, Seiberg–Witten invariants and surface singularities: splicings and cyclic covers, Selecta Math. (N.S.) 11 (2005), 399–451. 155 [16] A. Némethi and T. Okuma, The embedding dimension of weighted homogeneous surface singularities, J. Tapol. 3 (2010), 643–667. 157, 158 [17] W. D. Neumann, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981), 299–344. 150 [18] W. D. Neumann, Graph 3-manifolds, splice diagrams, singularities, in Singularity theory – Dedicated to Jean-Paul Brasselet on his 60 th birthday. Proceedings of the 2005 Marseille Singularity School and Conference, CIRM, Marseille, France, 24 January – 25 February 2005, ed. by D. Chériot, N. Dutertre, C. Murolo, A. Pichon and D. Trotman, World Scrientific, Singapore 2007, 787–817. 149 [19] W. D. Neumann and J. Wahl, The end curve theorem for normal complex surface singularities, J. Eur. Math. Soc. (JEMS) 12 (2010), 471–503. 149, 152 [20] W. D. Neumann and J. Wahl, Casson invariant of links of singularities, Comment. Math. Helv. 65 (1990), 58–78. 154 [21] W. D. Neumann and J. Wahl, Universal abelian covers of surface singularities, in Trends in singularities, ed. by A. Libgober and M. Tib˘ar, Trends in Mathematics, Birkhäuser, Basel 2002, 181–190. 149 [22] W. D. Neumann and J. Wahl, Complete intersection singularities of splice type as universal abelian covers, Geom. Topol. 9 (2005), 699–755. http://www.msp.warwick.ac.uk/gt/2005/09/p017.xhtml 149, 150, 152, 153 [23] W. D. Neumann and J. Wahl, Complex surface singularities with integral homology sphere links, Geom. Topol. 9 (2005), 757–811. http://www.msp.warwick.ac.uk/gt/2005/09/p018.xhtml 149, 150 [24] T. Okuma, The multiplicity of abelian covers of splice quotient singularities, preprint 2010. arXiv:1002.2048 153, 156 [25] T. Okuma, Universal abelian covers of rational surface singularities, J. London Math. Soc. (2) 70 (2004), 307–324. 150, 153 [26] T. Okuma, The geometric genus of splice-quotient singularities, Trans. Amer. Math. Soc. 360 (2008), 6643–6659. 150, 153, 154 [27] T. Okuma,Another proof of the end curve theorem for normal surface singularities, J. Math. Soc. Japan 62 (2010), 1–11. 149 [28] F. Van Dyke, Generators and relations for finitely generated graded normal rings of dimension two, Illinois J. Math. 32 (1988), 115–150. 158 [29] P. Wagreich, Elliptic singularities of surfaces, Amer. J. Math. 92 (1970), 419–454. 156 [30] Jonathan Wahl, Topology, geometry, and equations of normal surface singularities, in Singularities and computer algebra. Selected papers of the conference, Kaiserslautern, Germany, October 18–20, 2004 on the occasion of Gert-Martin Greuel’s 60 th birthday. ed. by Ch. Lossen and G. Pfister, London Mathematical Society Lecture Note Series 324, Cambridge University Press, Cambridge 2006, 351–371. 149

A note on the toric duality between the cyclic quotient surface singularities An;q and An;nq Oswald Riemenschneider Mathematisches Seminar der Universität Hamburg Bundesstrasse 55, 20146 Hamburg, Germany email: [email protected]

Abstract. In my lecture at the Franco–Japanese Symposium on Singularities I gave an introduction to the work of Martin Hamm [3] concerning the explicit construction of the versal deformation of cyclic surface singularities. Since that part of his dissertation is already documented in a survey article (cf. [8]), I concentrate in the present note on some other aspect of [3]: the toric duality of the total spaces of the deformations over the monodromy coverings of the Artin components for the singularities An;q and An;nq which themselves are toric duals of each other. Our exhibition is based – as in Hamm’s dissertation – on the algebraic aspects, i.e. the algebras and their generators of these total spaces. We prove Hamm’s remarkable duality result in this note first in detail for the hypersurface case q D n 1 in which the interplay between algebra and geometry of the underlying polyhedral cones is rather obvious, especially when bringing also the “complementarity” of An;q and An;nq into the game. We then treat the dual case q D 1 of cones over the rational normal curves once more in order to develop the necessary ideas for transforming the generators in such a way that it becomes transparent how to compute the dual, even in the general situation (which we explain in the last section by an example).

1 Introduction The association 7! 0 D =. 1/ establishes a one-to-one correspondence for rational numbers in the open interval .1; 1/. Therefore, it gives rise to a bijection on the set of all rational cones D D f.x; y/ 2 R2 W 0 y xg R2 ;

> 1;

resp. to a bijection on the set of all (additive, finitely generated) semigroups N D N D \ .Z ˚ Z/ D \ .N ˚ N/ D f.j; k/ 2 N2 W 0 k j g N2 with > 1. Writing D n=q with 1 q < n and n; q coprime, we also set n;q D D n=q and Nn;q D N , respectively.

162

Oswald Riemenschneider

The subalgebra An;q D CŒn;q D CŒs j t k W .j; k/ 2 Nn;q CŒs; t is finitely generated and defines a two-dimensional affine toric variety Xn;q (for the general theory and these examples see e.g. [4]). It is well known that An;q is isomorphic to the invariant ring CŒu; vn;q ; where the finite cyclic group n;q is generated by the diagonal matrix diag.n ; nq / with n an n-th primitive root of unity, i.e. An;q D CŒs j t k W .j; k/ 2 Nn;q Š CŒuj v k ; j C qk 0 mod n D CŒu; vn;q and Xn;q Š C2 =n;q : The quotient space Xn;q has exactly one (normal) singularity at the (image of the) origin (under the natural projection C2 ! C2 =n;q D Xn;q ). By abuse of language, we call this the An;q -singularity resp. the singularity of type An;q or just the singularity An;q . Examples. (1) q D n 1. The semigroup Nn;n1 is generated by .1; 0/; .1; 1/ and .n 1; n/ and thus, the ring CŒn;n1 is generated by the monomials x0 D s; x1 D st; x2 D s n1 t n with the generating relation x0 x2 D x1n . Similarly, the invariant ring is generated by un ; uv; v n with the same generating relation. Hence, singularities of type An;n1 are the two-dimensional simple hypersurface singularities of type An1 . (2) q D 1. The semigroup Nn;1 is generated by .1; 0/; .1; 1/; .1; 2/; : : : .1; n/, and the ring CŒn;1 is therefore generated by the monomials x0 D s; x1 D st; x2 D st 2 ; : : : ; xn D st n . Equations for the corresponding affine variety are given in determinantal form, i.e. by the vanishing of the 2 2-minors of the matrix ! x0 x1 : : : xn2 xn1 : x1 x2 : : : xn1 xn In homogeneous coordinates of Pn , these equations define the so called rational normal curve of degree n. Hence, the singularity of type An;1 is isomorphic to the cone over this rational curve, i.e. to the closure in CnC1 of its preimage under the natural projection CnC1 n f0g ! Pn . Note that this again is in accordance to the description as an invariant ring, since CŒu; vn;1 will obviously be generated by the monomials un ; un1 v; un2 v 2 ; : : : ; uv n1 ; v n . The purpose of this note is to make the correspondence Xn;q D X 7! X0 D Xn;nq geometrically visible. In particular, we want to elucidate Martin Hamm’s result that the total spaces of the versal simultaneously resolvable deformations of Xn;q and Xn;nq are dual to each other as affine toric varieties.

A note on the toric duality

163

2 Generators for Nn;q and Hirzebruch–Jung continued fractions It is more natural to start with the complement of n;q in the half space R RC which we denote by c D f.x; y/ 2 R2 W nx qy; y 0g: n;q If we expand the ratio n=q into its Hirzebruch–Jung continued fraction n D b1 1 b2 1 br ; q then the partial fractions b1 1 b2 1 b ;

D 1; : : : ; r;

form a strongly decreasing sequence of rational numbers.1 If we write them in the form P with P ; Q relatively prime, Q it is well known that the numerators and denominators are built by the following inductive rules: P1 D

0;

P0 D 1;

P D b P1 P2 ;

Q1 D 1;

Q0 D 0;

Q D b Q1 Q2 :

In particular, successive quotients P1 =Q1 and P =Q are Farey neighbors, i.e. ! P1 P D 1: det Q1 Q From this, one can easily deduce the following lemmata. c c D n;q \ .Z N/ is minimally generated by the Lemma 2.1. The semigroup Nn;q elements .Q ; P /; D 1; 0; 1; : : : ; r.

Lemma 2.2. For each rational number P =Q between successive quotients P =Q and PC1 =QC1 the denominator Q is greater or equal to Q C QC1 . Now, the linear mapping ! x y

7!

! ! 1 1 x 0 1

i.e. D

y

D

yx y

! ;

y 7! ; x 1

1 From now on, the Greek letter denotes an index running from 1 to r rather than a rational number as in the introduction.

164


c maps obviously n;q bijectively onto n;nq and Z Z onto itself, hence gives a c onto Nn;nq . semigroup isomorphism from Nn;q Interchanging the roles of q and n q, this implies the following result.

Lemma 2.3. Expand n=.n q/ in its Hirzebruch–Jung continued fraction: n D a1 1 a2 1 am : nq Then, the semigroup Nn;q is finitely generated by the elements D 0; : : : ; m C 1;

.j ; k /; where .j0 ; k0 / D .1; 0/; .j1 ; k1 / D .1; 1/; and

.jC1 ; kC1 / D a .j ; k / .j1 ; k1 /;

xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx

n

nq

qn

xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxx

n

n

D 1; : : : ; m:

xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxxxx xx xxx x

q xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxx xx xxxxxxxxxxxxxxxxxxxxxxxx xx xx xx xx xx xx xx xx

n Figure 1. Four different manifestations of N7;4 .

q


165

3 Duality of the singularities An;q and An;nq c The dual of the cone we get by rotating n;q clockwise by 90ı equals n;q . Let us state this remark as follows.

Lemma 3.1. We have ? c n;q Š n;q :

This identity has been found and conceptually proven in more generality by Patrick c Popescu-Pampu (see [5]). Since we have Nn;q Š Nn;nq by our considerations before, this implies the following result. Lemma 3.2. The singularities An;q and An;nq are dual to each other as affine toric varieties, i.e., if An;q is defined by a lattice L and a cone L ˝ R, then An;nq is given by the dual lattice L_ and the cone ? L_ ˝ R. Corollary 3.3. The ordinary double point A1 D A2;1 is selfdual (and the unique one with that property). Lemma 3.1 suggests to leave the realm of affine toric varieties by regarding the fan we get by subdividing the closed upper half plane along the line qy nx D 0. This gives a space Xxn;q which contains a projective line P1 such that at the origin of this projective line, Xxn;q possesses a singularity of type An;q , whereas at 1, Xxn;q possesses a singularity of type An;nq (and no others). In particular, Xxn;q is a partial compactification of Xn;q .

n

q Figure 2. The fan for Xxn;q .

166


Notice that the action of the cyclic group n;q on C C can be extended to the partial compactification P1 C, i.e. to the trivial line bundle on P1 . If u; v are the coordinates of C2 , we put u; v for the coordinates at infinity, i.e. u D u1 ;

v D v:

Hence, the action of the standard generator of n;q extends uniquely at 1 to u 7! n1 u;

v 7! nq v

such that, by using the inverse of the standard generator, we see that the action at 1 is of type n;nq . We leave it as an exercise to the reader to show the following result. Lemma 3.4. We have Xxn;q Š .P1 C/=n;q :

4 Quasi-determinantal equations and the Artin component Lemma 2.3 is fundamental for finding equations of the singularities of type An;q (see [6]). We put x D s j t k ;

D 0; 1; : : : ; m; m C 1:

Then, obviously, we have a

x1 xC1 D x ;

D 1; : : : ; m:

However, these are not all equations unless m D 1. In general, one has to look at the so called quasi-matrix 1 0 x1 x2 : : : xm1 xm x0 C B am 2 C B x1a1 2 x2a2 2 ::: xm A @ x1 x2 x3 : : : xm xmC1 and to form all 2 2-quasi-determinants. By perturbing the entries of this quasi-matrix, one gets automatically deformations of the given singularity. A nice way to do this is to set x.`/ D x C t.`/ ;

` D 0; : : : ; a 1; D 1; : : : ; m;

and to form the 2 2-quasi-determinants of the quasi-matrix 0 .am 1/ x1.a1 1/ ::: xm x0 B .1/ .am 2/ B x1.1/ : : : x1.a1 2/ x2.1/ : : : x2.a2 2/ : : : xm : : : xm @ x1.0/

x2.0/

:::

xmC1

1 C C: A


167

In order to minimize the cardinality of deformation parameters, one can choose t.0/ D 0 (i. e. x.0/ D x ) or better, since more symmetrically, X

a 1

t.`/ D 0 ;

D 1; : : : ; m :

`D0

The resulting deformation has an intrinsic meaning. There exists a versal deforma.vers/ z n;q ! Tn;q of the minimal resolution Xzn;q of Xn;q which has a smooth base tion X .vers/ space Tn;q of dimension r X

.b 1/

D1

(notice that the numbers b are invariants of the minimal resolution: The selfintersection numbers of the components of the exceptional divisor). By a general result of J. Wahl and the author, this deformation can fiberwise be blown down to a deformation of the singularity Xn;q itself. It has been shown in [6] that the resulting deformation is exactly the one given explicitly above. Consequently, r X

.b 1/ D

D1

m X

.a 1/:

()

D1

(for a direct proof of this equality, see also [6] or [4], Corollary 1.23). Denote this deformation by .vers/ Yn;q ! Tn;q :

Obviously, on this deformation operates in a natural manner the group W D Wn;q which is the product of the symmetric groups Sa1 1 Sam 1 . Moreover, it acts .vers/ by reflections such that the induced deformation on Tn;q .vers/ .Art/ X.Art/ n;q D Yn;q =Wn;q ! Tn;q =Wn;q D Sn;q

has a smooth base space of the same dimension. It is well known that the (reduced) .vers/ .vers/ base space .Sn;q /red of the versal deformation X.vers/ n;q ! Sn;q of Xn;q has in general several (smooth) components. There exists exactly one such that the induced defor.Art/ mation is isomorphic to X.Art/ n;q ! Sn;q . We call this the Artin deformation, resp. the Artin component. This deformation is the versal deformation space for deformations of Xn;q which possess a simultaneous resolution after finite base change. The .vers/ is also versal for deformations of Xn;q which can be deformation Yn;q ! Tn;q resolved simultaneously without base change. We call it the (directly) resolvable Artin deformation.

168


5 The grand monodromy covering The existence of a grand monodromy covering has been conjectured by the author and was proven by S. Brohme [1] and M. Hamm [3]. In the following, the pair .n; q/ is fixed and will be suppressed as an index. Theorem 5.1. There exists a finite union Tred of linear subspaces Tk of a suitable CN z of symmetric groups which that is invariant under the action of a (larger) product W N acts canonically on C such that .vers/ z: D Tred =W Sred .vers/ The components Sj of Sred are exactly the images of the irreducible components Tk .vers/ of Tred under the canonical projection Tred ! Sred , and two irreducible components of Tred will be mapped onto the same component Sj only if they are translates of each z other under the action of W. In the induced diagram of deformations

YredaC CC CC CC CC Ca

/ Xy

Yk

T } k {{ { {{ {{ { }{ {

Tred

z =W

/ X(vers) = red {{ { {{ {{ { ={ {

/ Sj !C C CC CC CC CC ! / S (vers) red

the deformation Yred ! Tred can explicitly be constructed, and the inner squares are .vers/ . exactly the monodromy coverings of the deformations on the components of Sred Conjecture 5.2. Enlarging Tred by some embedded components with linear support to a non reduced space T , it should be possible to extend the family Yred ! Tred explicitly z -equivariant family Y ! T such that the quotient by W e is isomorphic to the to a W versal deformation X.vers/ ! S .vers/ . Remark. It had been already remarked by Christophersen [2] that the total deformation spaces Yk can be equipped with an affine toric structure. M. Hamm gives in [3] a precise description of these toric structures and a kind of toric recipe how the components intersect.


169

6 Duality of the Artin components We shortly explain the toric description of the resolvable Artin deformation via generators in a special example. Example. For n=.n q/ D 56=15 D 4 1 4 1 4 we can abbreviate the quasi-determinantal equations by the scheme 0 1 .1; 0/ .1; 1/ .3; 4/ .11; 15/ B C B C .1; 1/ .1; 1/ .3; 4/ .3; 4/ .11; 15/ .11; 15/ @ A .1; 1/ .3; 4/ .11; 15/ .41; 56/ in which the symbol .j; k/ stands either for the element .j; k/ 2 N2 or for the monomial s j t k . In order to describe the toric structure of the resolvable Artin deformation, we have to insert 3C3C3 D 9 further deformation variables. This has to be done systematically in the following way: 0 B B @

.1I 0; 0; 0I 0; 0; 0I 0; 0; 0I 0/

.1I 0; 0; 0I 1; 0; 0I 0; 0; 0I 0/

.1I 0; 1; 0I 0; 0; 0I 0; 0; 0I 0/ .1I 0; 0; 1I 0; 0; 0I 0; 0; 0I 0/ .1I 1; 0; 0I 0; 0; 0I 0; 0; 0I 0/

:::

.3I 1; 1; 1I 1; 0; 0I 0; 0; 0I 0/ .3I 1; 1; 1I 0; 0; 0I 1; 0; 0I 0/

: : : .3I 1; 1; 1I 0; 1; 0I 0; 0; 0I 0/ .3I 1; 1; 1I 0; 0; 1I 0; 0; 0I 0/

.11I 4; 4; 4I 0; 1; 1I 1; 0; 0I 0/ .11I 4; 4; 4I 0; 1; 1I 0; 0; 0I 1/ : : : .11I 4; 4; 4I 0; 1; 1I 0; 1; 0I 0/ .11I 4; 4; 4I 0; 1; 1I 0; 0; 1I 0/

1 C C: A

.41I 15; 15; 15I 0; 4; 4I 0; 1; 1I 1/ In general, this procedure defines uniquely mC2C

m X

.a 1/

j D1

vectors in Nd , where d D2C

m X

.a 1/;

j D1

which we denote by v0 ; v1.0/ ; : : : ; v1.a1 1/ ; : : : according to the second matrix at the beginning of Section 4. Writing them as column vectors into a matrix, one sees immediately that this matrix has maximal rank d such that the kernel of the corresponding

170


homogeneous linear system of equations has exactly dimension m which is 2 less than the embedding dimension of the singularity. By construction, there are indeed m linearly independent elements in the kernel, namely v0 C v2.0/ D v1.0/ C C v1.a1 1/ ;

v1.a1 1/ C v3.0/ D v2.0/ C C v2.a2 1/ ;

etc:;

from which all the other relations like v0 C v3.0/ D v1.0/ C C v1.a1 2/ C v2.1/ C C v2.a2 1/ follow. Writing the elements of the matrix symbolically as .j; k1 ; k2 ; : : :/ and projecting down each element to .j; k/ with k D k1 C k2 C , we see immediately that our construction yields a toric variety which contains the given singularity as a distinguished subspace and which, in fact, is isomorphic to the total space of the resolvable Artin deformation of the given singularity. In particular, it is automatically normal. Because of identity (), the total spaces of the resolvable Artin deformations of An;q and An;nq have the same dimension. Martin Hamm’s result says much more. Theorem 6.1 (M. Hamm). The total spaces of the resolvable Artin deformations of An;q and An;nq are dual to each other as affine toric varieties. In other words: given the generators in case An;q , which are determined by the sequence a1 ; : : : ; am , the corresponding rational cone in Rd will minimally be described (up to isomorphism) by inequalities defined by the generators in case An;nq and hence by the sequence b1 ; : : : ; br . Remark. Hamm’s proof rests on a clever matrix construction which is based on another simple manifestation of the duality in question observed by the author [6] and called “Riemenschneider’s dot diagram” by several authors (see also Section 10). In the following three sections we shall prove Hamm’s result – after treating the simplest case A1 – for the special cases q D n 1 and q D 1.

7 The case A1 Let us start with the simplest case, the singularity A1 . According to Hamm’s result, not only A1 is selfdual as a toric variety, but also its resolvable Artin deformation is. Of course, this duality can not anymore be realized by a complement in some kind of “quarter” space. However, something else happens.As one can check easily (see also the next section) the convex rational cone with generators .1; 0; 0/; .1; 1; 0/; .1; 0; 1/; .1; 1; 1/ is given by the inequalities x y1 0 ;

x y2 0 :


171

Using the “complementary” description of the algebra of the singularity A1 with generators .1; 0/, .0; 1/, .1; 2/, we have for the resolvable Artin deformation the generators .1; 0; 0/, .0; 1; 0/; .0; 0; 1/; .1; 1; 1/, and the convex rational cone generated by these points is equal to the union of the octant x 0; y1 0; y2 0 and the set f.x; x; x/ C .0; y1 ; y2 / W x; y1 ; y2 0g, in other words, it is equal to the cone satisfying the inequalities y1 x C ;

y2 x C ;

x C D max.x; 0/:

Hence this cone is “complementary” to the cone x y1 0; x y2 0 in the sense, that they intersect in the line segment .x; x; x/; x 0, only, and that their convex union is the quarter space y1 0; y2 0.

y1

y1

x

x y2

y2 Figure 3

In fact each one is (isomorphic to) the dual of the other: it is again very easy to see that the dual of the cone in the complementary description is just the cone f.x; y1 ; y2 / W y1 ; y2 0; y1 C y2 x 0g: Replacing x by x, this cone is generated by .0; 1; 0/, .0; 0; 1/, .1; 1; 0/, and .1; 0; 1/, but one has 0 1 0 10 1 1 1 1 1 0 1 1 0 0 1 1 B C B CB C @0 1 0 1A D @0 0 1A @1 0 1 0A 0 0 1 1 1 0 0 0 1 0 1 so that the dual of the cone in the complementary description is isomorphic to the original one. (For unproven statements please see the next section.) It is perhaps more suggestive to visualize the “complementarity” of both cones by drawing x–slices only.

172


y2

xC

xC

y1

Figure 4

8 The hypersurface case In order to prove Hamm’s result for q D n 1, we write down the generators for .n; n 1/ and .n; 1/. We first do this in a “small” case, say n D 5. Case .5; 4/. We have 0 1 .1I 0; 0; 0; 0I 0/ .1I 0; 0; 0; 0I 1/ B C B C .1I 0; 1; 0; 0I 0/ .1I 0; 0; 1; 0I 0/ .1I 0; 0; 0; 1I 0/ @ A .1I 1; 0; 0; 0I 0/ .4I 1; 1; 1; 1I 1/ Case .5; 1/. We have ! .1I 0I 0I 0I 0I 0/ .1I 0I 1I 0I 0I 0/ .1I 0I 1I 1I 0I 0/ .1I 0I 1I 1I 1I 0/ .1I 0I 1I 1I 1I 1/ : .1I 1I 0I 0I 0I 0/ .1I 1I 1I 0I 0I 0/ .1I 1I 1I 1I 0I 0/ .1I 1I 1I 1I 1I 0/ .1I 1I 1I 1I 1I 1/ Denoting the standard basis of RnC1 by e0 ; e1 ; : : : ; en , the “formats” of the singularities An;n1 and An;1 can thus be written in the form (An;n1 ): 0 1 e0 e0 C e n B C e0 C e 2 ; e0 C e 3 ; : : : @ A e0 C e1 .n 1/e0 C .e1 C C en / and (An;1 ): e0

e0 C e 2

e0 C .e2 C e3 /

e0 C e1 e0 C .e1 C e2 / e0 C .e1 C e2 C e3 /


e0 C .e2 C e3 C C en /

e0 C .e1 C e2 C e3 C C en /

173

! :

In order to get a somewhat simpler result, we first transform the scheme .5; 4/ in an obvious way to the “complementary” situation: 0 1 .1I 0; 0; 0; 0I 0/ .0I 0; 0; 0; 0I 1/ B C B C: .0I 0; 1; 0; 0I 0/ .0I 0; 0; 1; 0I 0/ .0I 0; 0; 0; 1I 0/ @ A .0I 1; 0; 0; 0I 0/ .1I 1; 1; 1; 1I 1/ For general n, this becomes the much nicer format .An;n1 /c : 1 0 en e0 C B e2 ; e3 ; : : : ; en1 A: @ e1 e0 C .e1 C C en / From this, it is evident that (an isomorphic copy of) the convex cone for An;n1 with arbitrary n 2 is given by the union of the “quarter space” j 0; k` 0 ;

` D 1; : : : ; n;

and the set of all points .j; j; : : : ; j / C .0; k1 ; : : : ; kn / ;

j 0; k1 ; : : : ; kn 0:

Thus, it can be described in (; 1 ; : : : ; n )-space by the inequalities ` C D max.; 0/ ;

` D 1; : : : ; n;

and therefore the situation is an obvious generalization of the picture on the right hand side of Figure 3. More precisely, these are exactly the following 2n inequalities: ` ;

` 0;

` D 1; : : : ; n:

Hence, the dual cone will be generated by the 2n vectors e` ; e` e0 ; ` D 1; : : : ; n. From the format ! e2 en e1 e1 e0

e2 e0

en e0

it follows that these generators satisfy the same relations as do the generators for the resolvable Artin deformations of the singularity An;1 . Therefore, it should be possible to transform them by a unimodular matrix. In fact, we have for n D 5 (after replacing e0 by e0 ):

174

0 B B B B B B B B B B @


0 1 0 1 0 1 0 1 0 1

1

C 1 1 0 0 0 0 0 0 0 0 C C C 0 0 1 1 0 0 0 0 0 0 C C C 0 0 0 0 1 1 0 0 0 0 C C 0 0 0 0 0 0 1 1 0 0 C A 0 0 0 0 0 0 0 0 1 1 0 B B B B B D B B B B B @

0 1

0

0

0

1 0 1

0

0

0

0

0

1 0

0 0

C 0 C C C 1 1 0 0 C C C 0 1 1 0 C C 0 0 1 1 C A

0 0

0

0 0 0 0

1

B B B B B B B B B B @

1 1 1 1 1 1 1 1 1 1

1

C 0 1 0 1 0 1 0 1 0 1 C C C 0 0 1 1 1 1 1 1 1 1 C C: C 0 0 0 0 1 1 1 1 1 1 C C 0 0 0 0 0 0 1 1 1 1 C A 0 0 0 0 0 0 0 0 1 1

It is an easy exercise to write down the unimodular transformation for arbitrary n from the “dual” set of generators e` ; e0 C e` ; ` D 1; : : : ; n, in case An;n1 to the set of generators in case of An;1 . In order to have better control we write the 2n generators in a matrix as above, replacing once more e0 by e0 . Then, we transform ek 7! e1 C e2 C C ek ; k D 1; : : : ; n, and we get a new matrix of generators ! e1 e1 C e 2 e 1 C e 2 C C en () e0 C e 1 e0 C e 1 C e 2 e0 C e 1 C e 2 C C en It remains to interchange e0 and e1 . This then finishes the proof of Martin Hamm’s duality result in the special case q D n 1.

Remark. Obviously, the unimodular transformation from the “complementary” situation to the original one in case .n; n 1/ is given by D x C y1 C C yn ;

` D y` ;

` D 1; : : : ; n:

After a few elementary considerations this implies that the original convex cone in this case has a slightly more complicated description: x

1;:::;n X

y ;

y` 0;

` D 1; : : : ; n:

¤`

For n D 3, the x-slices look as follows (x 0; for x < 0 they are, of course, empty).


175

y3 x

x y2 x y1 Figure 5

In particular, for n D 2, we have (as already used in the preceding section) x y1 0 ;

x y2 0:

In order to check the validity of this description, we compute the generators of the inequalities and the relations between them. Clearly, the generators are e1 ; : : : ; en and fj D e0 C

1;:::;n X

e` ;

j D 1; : : : ; n:

`¤j

The relations e2

en

f1 f2

fn

e1

!

are exactly those for the singularity An;1 .

9 The case An;1 It is after the results of the last section not necessary to go also in the opposite direction but quite amusing and helpful for understanding the general situation. Let us first remark that we get another interesting system of generators for An;1 by transforming the entries of the matrix () into the following one: ! e 0 C e 1 e 0 C e 1 C e 2 e 0 C e 1 C e 2 C C en : e0 e0 C e 2 e 0 C e 2 C C en

176


For n D 5, we have concretely .1I 1I 0I 0I 0I 0/ .1I 1I 1I 0I 0I 0/ .1I 1I 1I 1I 0I 0/ .1I 0I 0I 0I 0I 0/ .1I 0I 1I 0I 0I 0/ .1I 0I 1I 1I 0I 0/ .1I 1I 1I 1I 1I 0/ .1I 1I 1I 1I 1I 1/ .1I 0I 1I 1I 1I 0/ .1I 0I 1I 1I 1I 1/

! :

Defining inequalities for the corresponding cone are, as one can check: 0 y1 x ;

0 yn yn1 y2 x :

Therefore, the dual cone is generated by the vectors e0 e1 ; e1 ; e0 e2 ; e2 e3 ; : : : ; en1 en ; en : The format

0 @

e 0 e1 e0 e2

en

e2 e3 ; e3 e4 ; : : : ; en1 en

1 A

e1

shows that the dual cone in fact belongs to the case An;n1 . In particular, this new cone for An;1 is contained in the set 0 y` x; with ` D 1; : : : ; n. Recall that a cone for An;n1 is equal to y` x C D max.x; 0/, with ` D 1; : : : ; n: So, this again is a manifestation of the interplay between duality and complementarity. y2 x

x y1 x y3 Figure 6


177

Another way from An;1 to An;n1 starts with the format e 0 C e 1 e0 C e 2 e1

e2

:::

e0 C en

:::

en

!

which one obtains from former formats of An;1 by unimodular transformations (and possibly interchanging e0 and e1 ). It is not difficult at all to show that the cone generated by these vectors can be described as fxe0 C y1 e1 C C yn en W yj 0; j D 1; : : : ; n; y1 C C yn x 0g: Thus the dual cone is generated by e0 ; e1 ; : : : ; en ; e0 C e1 C C en , and this is obviously isomorphic to the cone of An;n1 . The search for a kind of natural format which explains the duality in question in a more systematic way leads in case An;1 to the following (the reader may check him/herself the correctness of transformations): ! e2 ::: en e1 : e0 e0 e1 C e 2 : : : e0 e1 C e n

10 The general case In this last section we sketch by an example a way how one may understand Hamm’s duality result quite simply by introducing the right formats (which, on the other hand, destroy the motives how they were originally introduced). We look at the case .n; q/ with the a-sequence .4; 2; 3/ such that by the author’s dot diagram

the corresponding b-sequence is .2; 2; 4; 2/. According to the a-sequence we start with a quasi-matrix with fixed entries 1 0 e4 e5 e7 e1 C B C; B e2 e3 e6 A @ e0 where the crosses have to be filled in the correct way. In this case, they have to be successively from left to right e 0 e1 C e 2 C e 3 C e 4 ;

e0 e1 C e2 C e3 C e 5 ;

e0 e1 C e 2 C e 3 C e 6 C e 7 :

178


These are, of course, again generators of the case we discuss, and therefore the corresponding cone in the 8-dimensional space generated by e0 ; : : : ; e7 is defined by all elements 7 X ˛j ej ; j D0

where

˛0 D j C k C l C m;

˛1 D a k l m;

˛2 D b C k C l C m;

˛3 D c C k C l C m;

˛4 D d C k;

˛5 D f C l;

˛6 D g C m;

˛7 D h C m;

when the eleven non negative coefficients a; b; c; d; f; g; h; j; k; `; m are associated to the generators of the cone via the matrix 0 1 a d f h @ A: b c g j k ` m From these inequalities, one deduces for the ˛ that ˛j 0 for j ¤ 1 and that ˛0 C ˛1 0 ;

˛1 C ˛2 0 ;

˛1 C ˛3 0

and ˛1 C ˛4 C ˛5 C ˛6 0 ;

˛1 C ˛4 C ˛5 C ˛7 0:

So, we have twelve inequalities between the ˛, and one can easily check that they form a minimal set of generators. Therefore, the dual cone will be generated by the vectors e0 ;

e2 ;

:::;

e7 ;

e0 C e 1 ; e1 C e 2 ; e1 C e4 C e5 C e6 ;

e1 C e 3 ; e1 C e 4 C e 5 C e 7 :

After replacing e1 by e1 e0 and then interchanging e0 and e1 , we are left with the new generators e0 ;

e1 ;

:::;

e0 e1 C e 2 ;

e7 ; e0 e 1 C e 3 ;

e0 e1 C e 4 C e 5 C e 6 ;

e0 e1 C e 4 C e 5 C e 7 ;

which fit exactly into the corresponding format for the b-sequence .2; 2; 4; 2/: 0 1 e1 e2 e3 e6 e7 @ A: e4 e5 e0


179

References [1]

S. Brohme, Monodromieüberlagerung der versellen Deformation zyklischer Quotientensingularitäten, Dissertation, Hamburg 2002. URN: urn:nbn:de:gbv:18-6733 http://ediss.sub.uni-hamburg.de/volltexte/2002/673/ 168

[2]

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[3]

M. Hamm, Die verselle Deformation zyklischer Quotientensingularitäten: Gleichungen und torische Struktur, Dissertation, Hamburg 2008. URN: urn:nbn:de:gbv:18-37828 http://ediss.sub.uni-hamburg.de/volltexte/2008/3782/ 161, 168

[4]

T. Oda, Convex Bodies and Algebraic Geometry. An introduction to the theory of toric varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) Folge 15, Springer, Berlin etc. 1985. 162, 167

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P. Popescu-Pampu, The geometry of continued fractions and the topology of surface singularities, in Singularities in geometry and topology 2004. Proceedings of the 3 rd Franco–Japanese Conference held at Hokkaido University, Sapporo, September 13–18, 2004. ed. by J.-P. Brasselet and T. Suwa, Advanced Studies in Pure Mathematics 46, Mathematical Society of Japan, Tokyo 2007, 119–195. 165

[6]

O. Riemenschneider, Deformationen von Quotientensingularitäten (nach zyklischen Gruppen), Math. Ann. 209 (1974), 211–248. 166, 167, 170

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J. Stevens, On the versal deformation of cyclic quotient singularities, in Singularity theory and its applications. Part I. Geometric aspects of singularities. Papers from the symposium held at the University of Warwick, Coventry, 1988–1989, ed. by D. Mond and J. Montaldi. Lecture Notes in Mathematics 1462, Springer, Berlin 1991, 302–319.

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J. Stevens, The versal deformation of cyclic quotient singularities, preprint 2009. arXiv:0906.1430 161

Nearby cycles and characteristic classes of singular spaces Jörg Schürmann Mathematisches Institut, Universität Münster Einsteinstr. 62, 48149 Münster, Germany email: [email protected]

Abstract. In this paper we give an introduction to our recent work on characteristic classes of complex hypersurfaces based on some talks given at conferences in Strasbourg, Oberwolfach, and Kagoshima. We explain the relation between nearby cycles for constructible functions or sheaves as well as for (relative) Grothendieck groups of algebraic varieties and mixed Hodge modules, and the specialization of characteristic classes of singular spaces like the Chern-, Todd-, Hirzebruch-, and motivic Chern-classes. As an application we get a description of the differences between the corresponding virtual and functorial characteristic classes of complex hypersurfaces in terms of vanishing cycles related to the singularities of the hypersurface.

1 Introduction A natural problem in complex geometry is the relation between invariants of a singular complex hypersurface X (like Euler characteristic and Hodge numbers) and the geometry of the singularities of the hypersurface (like the local Milnor fibrations). For the Euler characteristic this is for example a special case of the difference between the Fulton and MacPherson Chern classes of X, whose differences are the now well studied Milnor classes of X (see [1], [6], [7], [8], [28], [30], [35], [36], and [46]). Their degrees are related to Donaldson–Thomas invariants of the singular locus (see [3]). A very powerful approach to this type of questions is by the theory of the nearby and vanishing cycle functors. For example a classical result of Verdier [45] says that the MacPherson Chern class transformation [26] and [23] commutes with specialization, which for constructible functions means the corresponding nearby cycles. Here we explain the corresponding result for our motivic Chern and Hirzebruch class transformations as introduced in our joint work with J.-P. Brasselet and S. Yokura [5], i.e. they also commute with specialization defined in terms of nearby cycles. Here one can work either in the motivic context with relative Grothendieck group of varieties [4] and [20], or in the Hodge context with Grothendieck groups of M. Saito’s mixed Hodge modules [31] and [32]. The key underlying specialization result [37] is Work

supported by the SFB 878 “groups, geometry and actions”.

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about the filtered de Rham complex of the underlying filtered D-module in terms of the Malgrange–Kashiwara V -filtration. But here we focus on the geometric motivations and applications as given in our joint work with S. E. Cappell, L. Maxim and J. L. Shaneson [13]. In this paper we work (for simplicity) in the complex algebraic context, since this allows us to switch easily between an algebraic geometric language and an underlying topological picture. Many results are also true in the complex analytic or algebraic context over base field of characteristic zero. First we introduce the virtual characteristic classes and numbers of hypersurfaces and local complete intersections in smooth ambient manifolds. Next we recall some of the theories of functorial characteristic classes for singular spaces; see [26], [2], [10], [5], and [38]. Finally we explain the relation to nearby and vanishing cycles following our earlier results [35] and [36] about different Chern classes for singular spaces. Acknowledgements. This paper is an extended version of some talks given at conferences in Strasbourg, Oberwolfach and Kagoshima. Here I would like to thank the organizers for the invitation to these conferences. I also would like to thank Sylvain Cappell, Laurentiu Maxim and Shoji Yokura for the discussions on our joint work related to the subject of this paper.

2 Virtual classes of local complete intersections Recall that we are working in the complex algebraic context. A characteristic class cl of (complex algebraic) vector bundles over X is a map cl W Vect.X / ! H .X / ˝ R from the set Vect.X / of isomorphism classes of complex algebraic vector bundles over X to some cohomology theory H .X / ˝ R with a coefficient ring R, which is compatible with pullbacks. Here we use as a cohomology theory 8 ˆ H 2 .X; Z/; the usual cohomology in even degrees, ˆ ˆ < H .X / D CH .X /; the operational Chow cohomology of [17], ˆ ˆ ˆ :K0 .X /; the Grothendieck group of vector bundles. We also assume that cl is multiplicative, i.e. cl .V / D cl .V 0 / [ cl .V 00 / for any short exact sequence 0 ! V 0 ! V ! V 00 ! 0

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of vector bundles on X, with [ given by the cup- or tensor-product. Such a characteristic class cl corresponds by the “splitting principle” to a unique formal power series f .z/ 2 RŒŒz with cl .L/ D f .c 1 .L// for any line bundle L on X. Here c 1 .L/ 2 H1 .X / is the nilpotent first Chern class of L, which in the case H .X / D K 0 .X / is given by c 1 .L/ D 1 ŒL_ 2 K 0 .X / (with . /_ the dual bundle). Finally cl should be stable in the sense that f .0/ 2 R is a unit so that cl induces a functorial group homomorphism cl W .K 0 .X /; ˚/ ! .H .X / ˝ R; [/: Let us now switch to smooth manifolds, which will be an important intermediate step on the way to characteristic classes of singular spaces. For a complex algebraic manifold M its tangent bundle TM is available and a characteristic class cl .TM / of the tangent bundle TM is called a characteristic cohomology class cl .M / of the manifold M . We also use the notation cl .M / D cl .TM / \ ŒM 2 H .M / ˝ R for the corresponding characteristic homology class of the manifold M, with ŒM 2 H .M / the fundamental class (or the class of the structure sheaf) in 8 BM ˆ H2 .M /; the Borel–Moore homology in even degrees, ˆ ˆ < H .M / D CH .M /; the Chow group, ˆ ˆ ˆ :G .M /; the Grothendieck group of coherent sheaves. 0 If M is moreover compact, i.e. the constant map k W M ! fptg is proper, one gets the corresponding characteristic number ].M / D k .cl .M // D deg.cl .M // 2 R : Example 2.1 (Hirzebruch 1954). The famous Hirzebruch y -genus is the characteristic number, whose associated characteristic class can be given in two versions (see [21]). (i) The cohomological version, with R D QŒy, is given by the Hirzebruch class cl D Ty corresponding to the normalized power series f .z/ D Qy .z/ D

z.1 C y/ zy 2 QŒyŒŒz: 1 e z.1Cy/

(ii) The K-theoretical version, with R D ZŒy, is given by the dual total Lambdaclass cl D ƒy_ , with X ƒy_ . / D ƒy .. /_ / D Œƒi .. /_ / y i i0

corresponding to the unnormalized power series f .z/ D 1 C y yz 2 ZŒyŒŒz:

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So the y -genus of the compact complex algebraic manifold M is given by X .M; ƒp T M / y p y .M / D p0

D

XX

p0

.1/i dimC H i .M; ƒp T M / y p ;

i0

with T M the algebraic cotangent bundle of M . The equality y .M / D deg.Ty .TM / \ ŒM / 2 QŒy

(gHRR)

is called the generalized Hirzebruch–Riemann–Roch theorem [21]. The corresponding power series Qy .z/ (as above) specializes to 8 ˆ 1 C z; for y D 1, ˆ ˆ ˆ ˆ < z Qy .z/ D 1 e z ; for y D 0, ˆ ˆ ˆ ˆ z ˆ : ; for y D 1. tanh z Therefore the Hirzebruch class Ty .TM / unifies the following important (total) characteristic cohomology classes of TM : 8 ˆ c .TM /; the Chern class for y D 1, ˆ ˆ < (2.1) Ty .TM / D td .TM /; the Todd class for y D 0, ˆ ˆ ˆ :L .TM /; the Thom–Hirzebruch L-class for y D 1. The gRHH-theorem specializes to the calculation of the following important invariants: 1 .M / D e.M / D deg.c .TM / \ ŒM /; the Euler characteristic, 0 .M / D .M / D deg.td .TM / \ ŒM /; the arithmetic genus,

(2.2)

1 .M / D sign.M / D deg.L .TM / \ ŒM /; the signature, which are, respectively, the Poincaré–Hopf or Gauss–Bonnet theorem, the Hirzebruch– Riemann–Roch theorem and the Hirzebruch signature theorem. If X is a singular complex algebraic variety, then the algebraic tangent bundle of X doesn’t exist so that a characteristic (co)homology class of X can’t be defined as before. But if X can be realized as a local complete intersection inside a complex algebraic manifold M, then a substitute for TX is available. Indeed this just means that the closed inclusion i W X ! M is a regular embedding into the smooth algebraic manifold M, so that the normal cone NX M ! X is an algebraic vector bundle over X (compare [17]). Then the virtual tangent bundle of X Tvir X D Œi TM NX M 2 K 0 .X /;


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is independent of the embedding in M (e.g., see Example 4.2.6 in [17]), so it is a well-defined element in the Grothendieck group of vector bundles on X . Of course Tvir X D ŒTX 2 K 0 .X / in case X is a smooth algebraic submanifold. If cl W K 0 .X / ! H .X /˝R denotes a characteristic cohomology class as before, then one can associate to X an intrinsic homology class (i.e. independent of the embedding X ,! M ) defined as: clvir .X / D cl .Tvir X / \ ŒX 2 H .X / ˝ R :

Here ŒX 2 H .X / is again the fundamental class (or the class of the structure sheaf) of X in 8 BM ˆ ˆ ˆH2 .X /; the Borel–Moore homology in even degrees, < H .X / D CH .X /; the Chow group, ˆ ˆ ˆ :G .X /; the Grothendieck group of coherent sheaves. 0 Here \ in the K-theoretical context comes from the tensor product with the coherent locally free sheaf of sections of the vector bundle. Moreover, for the class cl D ƒy_ one has to take R D ZŒy; .1 C y/1 to make it a stable characteristic class defined on K 0 .X /. Let i W X ! M be a regular embedding of (locally constant) codimension r between possible singular complex algebraic varieties. Using the famous deformation to the normal cone, one gets functorial Gysin homomorphisms (compare [17], [44], and [45]) i Š W H .M / ! Hr .X / and i Š W G0 .M / ! G0 .X /: Note that i is of finite tor-dimension, so that the last i Š can also be described as b b i Š D Li W G0 .M / ' K0 .Dcoh .M // ! K0 .Dcoh .X // ' G0 .X /

coming from the derived pullback Li between the bounded derived categories with coherent cohomology sheaves. If M is also smooth, then one gets easily the following important relation between the virtual characteristic classes clvir .X / of X and the Gysin homomorphisms: i Š .cl .M // D i Š .cl .TM / \ ŒM / D cl .NX M / \ clvir .X /: From now on we assume that X D ff D 0g D ffi D 0 j i D 1; : : : ; ng

(2.3)

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is a global complete intersection in the complex algebraic manifold M coming from a cartesian diagram ff D 0g

/M

i

X f0

f D.f1 ;:::;fn /

f0g

/ Cn :

i0

Then NX M ' f .Nf0g Cn / D X Cn is a trivial vector bundle of rank n on X so that 8 dimXsing ;

(2) if X has only isolated singularities (i.e. dimXsing D 0), then X z .Fx I Q// mCyvir .X / mCy .X / D y .H x2Xsing vir D Ty .X / Ty .X / ;

where Fx is the Milnor fiber of the isolated hypersurface singularity .X; x/; (3) if f W M ! C is proper, then H /// D y .H .X t I Q// y .H .X I Q// deg.MHCy .f0H .ŒQM H /// D deg.MHTy .f0H .ŒQM

is the difference between the y -characteristics of a global nearby smooth fiber X t D ff D tg (for 0 ¤ jtj small enough) and of the special fiber X D ff D 0g. Remark 4.13 (Hodge polynomials vs. Hodge spectrum). Let us explain the precise relationship between the Hodge spectrum and the less-studied y -polynomial of the Milnor fiber of a hypersurface singularity. Here we follow notations and sign conventions similar to those in [20]. Denote by mHsmon the abelian category of mixed Hodge structures endowed with an automorphism of finite order, and by K0mon .mHs/ the corresponding Grothendieck ring. There is a natural linear map called the Hodge spectrum, [ ZŒt 1=n ; t 1=n ; hsp W K0mon .mHs/ ! ZŒQ ' n1

such that hsp.ŒH / D

X ˛2Q\Œ0;1/

t˛

X

dim.GrpF HC;˛ /t p :

(4.5)

p2Z

for any mixed Hodge structure H with an automorphism T of finite order, where HC is the underlying complex vector space of H, HC;˛ is the eigenspace of T with eigenvalue exp.2 i˛/, and F is the Hodge filtration on HC . It is now easy to see that the y -polynomial of H is obtained from hsp.ŒH / by equating to 1 the parameter t corresponding to fractional powers ˛ 2 Q \ Œ0; 1/, and by setting the t of integer powers be equal to y. As already explained before, Corollary 4.12 reduces for the value y D 1 of the parameter to the (rationalized version of) Corollary 4.5. Since the ambient space in Theorem 4.9 and Corollary 4.10 need not be smooth, one can generalize in the same way the Corollary 4.7 for a global complete intersection X D ff D 0g D ff1 D 0; : : : ; fn D 0g (of codimension n) in some ambient smooth algebraic manifold M, given by the zero-fiber of an ordered n-tuple of complex algebraic function

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.f / D .f1 ; : : : ; fn / W M ! Cn . Here we leave the details to the reader. It is also very interesting to look at the other specializations of Corollary 4.12 for y D 0 and y D 1. Let us first consider the case when y D 0. Note that in general T0 .X / ¤ td .X / for a singular complex algebraic variety (see [5]). But if X has only Du Bois singularities (e.g. rational singularities, cf. [33]), then by [5] we have T0 .X / D td .X /. So if a global hypersurface X D ff D 0g has only Du Bois singularities, then by Corollaries 3.3 and 4.12 we get H MHT0 .f0H .ŒQM // D 0 2 H .X / ˝ Q:

This vanishing (which is in fact a class version of Steenbrink’s cohomological insignificance of X [41]) imposes interesting geometric identities on the corresponding Todd-type invariants of the singular locus. For example, we obtain the following result. Corollary 4.14. If the global hypersurface X has only isolated Du Bois singularities, then (4.6) dimC Gr0F H n .Fx I C/ D 0 for all x 2 Xsing , with n D dimX. It should be pointed out that in this setting a result of Ishii [22] implies that (4.6) is in fact equivalent to x 2 Xsing being an isolated Du Bois hypersurface singularity. Also note that in the arbitrary singularity case, the Milnor–Todd class H // 2 H .Xsing / ˝ Q T0 .fm .ŒidM // D MHT0 .f0H .ŒQM

carries interesting non-trivial information about the singularities of the hypersurface X . Finally, if y D 1, the formula (4.4) should be compared to the Cappell–Shaneson topological result of (3.2). While it can be shown (compare with [27]) that the normal contribution .lk.V // in (3.2) for a singular stratum V 2 V0 is in fact the signature .Fv / (v 2 V ) of the Milnor fiber (as a manifold with boundary) of the singularity in a transversal slice to V in v, the precise relation between .Fv / and 1 .Fv / is in general very difficult to understand. For X a rational homology manifold, one would like to have a “local Hodge index formula” ‹

.Fv / D 1 .Fv /; which is presently not available. But if the hypersurface X is a rational homology manifold with only isolated singularities, then this expected equality follows from [42], Theorem 11. One therefore gets in this case (by a comparison of the different specialization results for L and T1 ) the following conjectural interpretation of L-classes from [5] (see [13] for more details). Theorem 4.15. Let X be a compact complex algebraic variety with only isolated singularities, which is moreover a rational homology manifold and can be realized as


203

a global hypersurface (of codimension one) in a complex algebraic manifold. Then L .X / D T1 .X / 2 H2 .X I Q/:

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Residues of singular holomorphic distributions Tatsuo Suwa Department of Mathematics, Hokkaido University Sapporo 060-0810, Japan email: [email protected]

Abstract. We present two types of residue theories for singular holomorphic distributions. The first one is for certain Chern polynomials of the normal sheaf of a distribution and the residues arise from the vanishing, by rank reason, of the relevant characteristic classes on the non-singular part. The second one is for certain Atiyah polynomials of vector bundles admitting an action of a distribution and the residues arise from the Bott type vanishing theorem on the non-singular part.

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 2. Holomorphic distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 3. Local Chern classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 4. Chern residues of singular distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 ˇ 5. Atiyah classes and Cech–Dolbeault cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 6. Atiyah residues of singular distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

1 Introduction In this article we review some local invariants (residues) associated to singular holomorphic distributions. For singular foliations, i.e. involutive distributions, various residues are known. They usually arise from localization of some characteristic classes by the Bott vanishing theorem, which depends on involutivity (see [22] for a systematic treatment). Note that they turned out be also closely related to local invariants of holomorphic self-maps (cf. [2] and [3]). Partially

supported by a grant of JSPS.

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Some of those residues coincide with the ones that arise in other contexts where involutivity is not required.Also, if we consider other types of characteristic polynomials, we have a Bott type vanishing theorem without involutivity. Thus the corresponding residues can be defined also for singular distributions. Here we take up two kinds of such localizations. The first one is for the normal sheaf of the distribution. It is rather primitive and comes from the vanishing by rank reason. In case we have involutivity, for a relevant characteristic polynomial, the resulting residue coincides with the Baum–Bott residue in [6]. The second one comes from the fact that Atiyah forms are easier to vanish than the corresponding Chern forms so that we have a Bott type vanishing theorem for certain Atiyah polynomials without involutivity, which leads to a localization theory of Atiyah classes of vector bundles admitting an action of a distribution. In Section 2, we present some basics on singular holomorphic distributions and in Section 3, we review local Chern classes via the Chern–Weil theory adapted to ˇ the Cech–de Rham cohomology. In this context, we also recall the Riemann–Roch theorem for embeddings as given in [23]. Section 4 is an almost thorough revision of [21]. This is based on the observation that the localization considered there arises in fact from a rather primitive fact, i.e. the Chern forms of degree greater than the rank of the vector bundle vanish, and involutivity has nothing to do with it. Thus we define the localization, by rank reason, of some characteristic classes and associated residues of the normal sheaf of the distribution. We show that for singular foliations they coincide with the corresponding Baum–Bott residues, which partially answers the Rationality Conjecture in [6]. We also express the residues in terms of the local Chern class of some sheaf supported on the singular set of the distribution. This allows us to apply the Riemann–Roch theorem for embeddings to compute the residues. ˇ In Section 5, we review the Atiyah classes defined in the Cech–Dolbeault cohomology following [1] and [26]. These classes are originally defined in [4] using complex analytic connections for holomorphic vector bundles. Here we use the construction in [1], which is more appropriate for localization purposes. In Section 6, we recall a Bott type vanishing theorem in [1] for some Atiyah forms, which leads to a localization theory for singular distributions. As an example we discuss Camacho–Sad type residues for the normal bundle of an invariant subvariety of a distribution. For foliations, these are first introduced in [11] to prove the existence of separatrices for holomorphic vector fields on the complex plane and then generalized by several authors, e.g. [17] and [18]. We also discuss the localization problem on singular varieties and give an example. I hoped to include more material on singular distribution as well as on singular contact structures, but was not able to do so. Let me simply list [15] and [20] as literature directly related to characteristic classes of singular holomorphic distributions and thank J. Adachi for precious information on contact structures.

Residues of singular holomorphic distributions

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2 Holomorphic distributions 2.1 Non-singular distributions Let M be a complex manifold of dimension m and TM its holomorphic tangent bundle. A distribution on M is an assignment of a subspace of Tx M to each point x in M; more precisely, it is defined in the following way. Definition 2.1. A non-singular holomorphic tangential distribution on M of rank r is a holomorphic subbundle F of rank r of TM. We call F also the tangent bundle of the distribution and the quotient NF D TM=F the normal bundle of the distribution. A distribution can be dually defined in terms of cotangent bundle T M. Definition 2.2. A non-singular holomorphic cotangential distribution on M of corank s is a holomorphic subbundle G of rank s of T M. In the following we sometimes use the word distribution to refer to the above notions. Definitions 2.1 and 2.2 are equivalent by taking the annihilator of each other. Namely, if F is a distribution of rank r, [ f! 2 Tx M j hv; !i D 0 for all v 2 Tx M g G D Fa D x2M

is a distribution of corank s D m r, where h ; i denotes the paring of vector fields and differential forms. The above G coincides with the dual NF of NF and is called the conormal bundle of the distribution. Likewise, if G is a distribution of corank s, F D G a is a distribution of rank r D m s. A foliation is a distribution F which is involutive, i.e. closed under the bracket operation in TM. Let F be a distribution of rank r on M and V a complex submanifold of dimension n of M. Denoting by W V ,! M the inclusion, we identify T V with the image of its differential W T V ! TM jV . Definition 2.3. We say that F is tangent to V, or leaves V invariant, if F jV T V. In this case F jV is a distribution of rank r on V. We state the above property in terms of conormal bundle. Thus let W T M jV ! T V be the dual of , which is a surjection. For a distribution G T M on M, we set G 0 D .GjV /. Note that it is the restriction of G to V as differential forms. The following proposition is not difficult to see.

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Proposition 2.4. Let G be a distribution of corank s on M and set F D G a . Then F leaves V invariant if and only if G 0 is a subbundle of T V of rank s k, i.e. a distribution of corank s k on V, k D m n.

2.2 Singular distributions Most of the material in this and subsequent subsections are similar to the ones for singular foliations, for which we refer to [22], Chapter VI. Let M be a complex manifold of dimension m. We denote by OM, ‚M and M, respectively, its structure sheaf, tangent sheaf and cotangent sheaf. For simplicity, we assume that M is connected. In general, for a coherent OM -module , we set Sing./ D f x 2 M j x is not OM;x -free g and call it the singular set of . Locally Sing./ is given as follows. By definition, each point of M has a neighborhood U such that there exists an exact sequence of the form r1 ' r2 ! OU ! jU ! 0: OU If we represent the map ' by a matrix .'ij / of holomorphic functions on U, Sing./ \ U D fx 2 U j rank.'ij .x// is not maximalg: Thus Sing./ is an analytic set in M. Away from Sing./, is locally free. Its rank is called the rank of . If the maximal rank of .'ij / is r, the rank of is r2 r. Definition 2.5. A singular holomorphic tangential distribution of rank r on M is a coherent sub-OM -module F of rank r of ‚M. Note that, since ‚M is locally free, the coherence of F here simply means that it is locally finitely generated. We call F the tangent sheaf of the distribution and the quotient NF D ‚M =F the normal sheaf of the distribution so that we have the exact sequence (2.1) 0 ! F ! ‚M ! NF ! 0: The singular set S.F / of a distribution F is defined to be the singular set of the coherent sheaf NF : S.F / D Sing.NF /: Note that Sing.F / S.F /. Away from S.F /, F defines a non-singular distribution of rank r, i.e. there is a rank r subbundle F0 of TM0 , M0 D M n S.F /, such that F jM0 D OM0 .F0 /. We say that F is reduced, if for any open set U in M, .U; ‚M / \ .U n S.F /; F / D .U; F /:


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As in the case of singular foliations, it can be shown that, if F is reduced, then codim S.F / 2 and that, if F is locally free and if codim S.F / 2, then F is reduced. In particular, if F is locally free of rank r, in a neighborhood of each point in M it is generated by r holomorphic vector fields v1 ; : : : ; vr , without relations, on U. The set S.F / \ U is the set of points where the vector fields fail to be linearly independent. In this case, (2.1) gives a locally free resolution of NF and the general theory of determinantal varieties tells us that dim S.F / r 1. Again, singular distributions can be defined in terms of holomorphic 1-forms. Definition 2.6. A singular holomorphic cotangential distribution of corank s on M is a coherent sub-OM -module G of rank s of M. Definitions 2.5 and 2.6 are related as follows. If a rank r distribution F is given, denoting by F a the annihilator of F , F a D f v 2 OM j hv; !i D 0 for all v 2 F g; which may be also written as HomOM .NF ; OM /, we set G D F a . Then G is a corank s D m r reduced distribution and we have S.G / S.F /. Conversely, if a corank s distribution G is given, we let F be the annihilator G a of G . Then F is a rank r D m s reduced distribution and we have S.F / S.G /. Thus if we consider only reduced distributions, by the above correspondence, the two definitions are equivalent. Moreover in this case, the singular sets S.F / and S.G / are the same.

2.3 Singular distributions on singular varieties Let M be a complex manifold of dimension m and V a possibly singular analytic variety in M of dimension n. We denote by V the ideal sheaf in OM of germs of holomorphic functions vanishing identically on V. Thus the quotient sheaf OV D OM = V is the sheaf of germs of holomorphic functions on V. We denote by Sing.V / the singular set of V and V 0 D V n Sing.V / the regular part. We consider the sheaf ‚M .logV / of logarithmic vector fields of V : ‚M .logV / D f v 2 ‚M j v. V / V g: Note that a germ of vector field v in ‚M is in ‚M .logV / if and only if it is tangent to V 0 . We define the tangent sheaf ‚V of V to be the image of the sheaf homomorphism ‚M .logV / ˝ OV ! ‚M ˝ OV: The sheaf ‚V may also be defined as the dual of the sheaf of holomorphic 1-forms V on V. Namely, recall that there is an exact sequence V = V2 ! M ˝ OV ! V ! 0;

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where .Œf / D df ˝ 1. Then ‚V coincides with HomOV .V ; OV / so that we have the exact sequence 0 ! ‚V ! ‚M ˝ OV ! NV ; where NV D HomOV . V = V2 ; OV /. The restriction of NV to V 0 is the sheaf of germs of holomorphic sections of the normal bundle of V 0 in M. Let F be a singular distribution of rank r on M and assume that (1) F leaves V invariant, i.e. F ‚M .logV /, and that (2) V 6 S.F /. The singular distribution FV on V induced from F is defined to be the image of the sheaf homomorphism F ˝ OV ! ‚M ˝ OV. Note that from the condition (1) above, FV is a subsheaf of ‚V.

3 Local Chern classes 3.1 Chern–Weil theory for virtual bundles For the Chern–Weil theory of characteristic classes of complex vector bundles, we refer to [6], [8], [19], and [22]. Let M be a C 1 manifold and E a C 1 complex vector bundle of rank ` over M. For an open set U in M, we denote by Ar .U / the complex vector space of complex r .U; E/ the vector space of “E-valued valued C 1 r-forms on U. Also, we denote by A V r 1 r-forms” on U,i.e. C sections of the bundle .TRc M / ˝ E on U, where .TRc M / denotes the dual of the complexification of the real tangent bundle TR M of M. Thus A0 .U / is the ring of C 1 functions and A0 .U; E/ is the A0 .U /-module of C 1 sections of E on U. Recall that a connection for E is a C-linear map r W A0 .M; E/ ! A1 .M; E/ satisfying the Leibniz rule: r.f s/ D df ˝ s C f r.s/

for f 2 A0 .M / and s 2 A0 .M; E/:

Note that E always admits a connection. If r is a connection for E, it induces a C-linear map r 0 W A1 .M; E/ ! A2 .M; E/ satisfying r 0 .! ˝ s/ D d! ˝ s ! ^ r.s/

for ! 2 A1 .M / and s 2 A0 .M; E/:

The composition K D r 0 ı r W A0 .M; E/ ! A2 .M; E/


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is called the curvature of r. It is not difficult to see that K is A0 .M /-linear so that we may think of it as a C 1 2-form with coefficients in the bundle Hom.E; E/. Definition 3.1. We set

c .r/ D det.I C A/;

p 1 K; AD 2

and call it the total Chern form of r. It is shown that c .r/ is a closed form on M. The p-th Chern form c p .r/ of r is the component of c .r/ of degree 2p. Thus we may write p 1 p p c .r/ D p .K/; 2 where p is the p-th elementary symmetric polynomial. We call p 1 p p p c D 2 the p-th (elementary) Chern polynomial. We refer to [8] and [22] for the construction of the following “difference forms”. Here we use the sign convention of [22]. Proposition 3.2. Suppose we have r C 1 connections r0 ; : : : rr for E. Then there exists a .2p r/-form c p .r0 ; : : : ; rr /, alternating in the r C 1 entries and satisfying r X

y ; : : : ; rr / C .1/r dc p .r0 ; : : : ; rr / D 0: .1/ c p .r0 ; : : : ; r

D0

In particular, if we have two connections r0 and r1 , there is a .2p 1/-form c p .r0 ; r1 / satisfying dc p .r0 ; r1 / D c p .r1 / c p .r0 /: Thus, if r is a connection for E, the class of c p .r/ in the de Rham cohomology 2p .M / does not depend on the choice of r. HdR 2p Definition 3.3. The p-th Chern class c p .E/ of E is the class of c p .r/ in HdR .M /, where r is a connection for E. The total Chern class is

c .E/ D 1 C c 1 .E/ C C c ` .E/: Remark 3.4. The class c p .E/ is in the image of the canonical homomorphism 2p H 2p .M; Z/ ! H 2p .M; C/ ' HdR .M /;

see [25] for detailed discussions on this matter.

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More generally let ' be a symmetric series. We may write ' as a series in elementary Chern polynomials ; ' D P .c 1 ; c 2 ; : : : /. For a connection r for E, we set '.r/ D P .c 1 .r/; c 2 .r/; : : : /; which is a closed form. We also have the difference forms as in Proposition 3.2 .M /, which coincides with and we define '.E/ to be the class of '.r/ in HdR 1 2 P .c .E/; c .E/; : : : /. As an example, the Segre polynomials s 1 ; s 2 ; : : : are defined by .1 C s 1 C s 2 C /.1 C c 1 C c 2 C / D 1: Then for a connection r for E, we have the (total) Segre form s .r/ D c .r/1 and the Segre class s .E/ D c .E/1 of E. If we have a complex vector P bundle Ei on M, for each i D 0; : : : ; q, we may consider the “virtual bundle” D qiD0 .1/i Ei . Letting r .i/ be a connection for Ei , i D 0; : : : ; q, we denote by r the family of connections .r .q/ ; : : : ; r .0/ / and define its total Chern form c .r / by c .r / D

q Y

c .r .i/ /.i/ ;

iD0

where .i/ D .1/i . The p-th Chern form c p .r / is the component of c .r / of degree 2p. More generally, for a symmetric series ', we write ' D P .c 1 ; c 2 ; : : : / as before and set '.r / D P .c 1 .r /; c 2 .r /; : : : /. If we have a finite number of families of connections r D .r.q/ ; : : : ; r.0/ /, D 0; : : : ; r, we have the difference form '.r0 ; : : : ; rr / as in Proposition 3.2 (cf. [22], Chapter II, 8). In particular, for two families of connections, d'.r0 ; r1 / D '.r1 / '.r0 /:

(3.1)

Thus the class of '.r / in H .M / is well-defined. We denote it by './ and call it the characteristic class of with respect to '. In particular, the total Chern class c ./ is the class of c .r / and is also given by c ./ D

q Y

c .Ei /.i/ :

iD0

The p-th Chern class c p ./ is the component of c ./ in H 2p .M / and is the class of c p .r /. Now let hq

h2

h1

0 ! Eq ! ! E1 ! E0 ! 0

(3.2)

be a sequence of vector bundles on M and, for each i, let r .i/ be a connection for Ei . We say that the family .r .q/ ; : : : ; r .0/ / is compatible with the sequence if, for each i,


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the following diagram is commutative: r .i /

A0 .M; Ei /

/ A1 .M; Ei /

hi

1˝hi

A0 .M; Ei1 /

r .i 1/

/ A1 .M; Ei1 /:

We have the following “vanishing by exactness” (cf. [6], Lemma (4.22)). Lemma 3.5. If the sequence (3.2) is exact, there exists always a family r D .r .q/ ; : : : ; r .0/ / of connections compatible with the sequence and for such a family we have c p .r / 0 for p > 0: Thus inPthe above situation, we have c .r / D 1, and, in particular, c ./ D 1, with D qiD0 .1/i Ei . In fact, the above holds for the difference form of a finite number of families of connections compatible with (3.2). For a symmetric series ' without constant term, we also have a similar vanishing '.r / D 0. From this we have the following result. Proposition 3.6. Suppose the sequence (3.2) is exact. Let ' be a symmetric polynomial and r D .r .q/ ; : : : ; r .0/ /, a family of connections compatible with (3.2). Then { / D '.r .0/ / '.r and, in particular, L D '.E0 /; './ { denotes the family of connections .r .q/ ; : : : ; r .1/ / for the virtual bundle where r P L D qiD1 .1/i1 Ei . Similar identities hold for the other “partitions” of the virtual bundle and for the difference forms of families of connections.

ˇ 3.2 Characteristic classes in the Cech–de Rham cohomology ˇ The Cech–de Rham cohomology is defined for an arbitrary covering of a manifold M, however for simplicity here we only consider coverings of M consisting of two open sets. For the general case and details, we refer to [9], [16], [22], and [25]. In Section 5.2 ˇ below we recall the Cech–Dolbeault cohomology for coverings with arbitrary number of open sets.

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Let M be a C 1 manifold of (real) dimension m0 and U D fU0 ; U1 g an open covering of M. We set U01 D U0 \ U1 and define the vector space Ar .U/ as Ar .U/ D Ar .U0 / ˚ Ar .U1 / ˚ Ar1 .U01 /: Thus an element in Ar .U/ is given by a triple D .0 ; 1 ; 01 /. We define the operator D W Ar .U/ ! ArC1 .U/ by D D .d0 ; d1 ; 1 0 d01 /: ˇ Then we have D ı D D 0. The Cech–de Rham cohomology HDr .U/ of U is the cohomology of the complex .A .U/; D/. Note that there is a natural isomorphism r .M /: HDr .U/ ' HdR

(3.3)

ˇ The Cech–de Rham cohomology is also equipped with the cup product, which is defined on the cochain level by assigning to in Ar .U/ and in As .U/ the cochain Y in ArCs .U/ given by Y D .0 ^ 0 ; 1 ^ 1 ; .1/r 0 ^ 01 C 01 ^ 1 /:

(3.4)

r .M / via the The cup product is compatible with the usual one in H r .M; C/ ' HdR isomorphism (3.3). ˇ If M is oriented and compact, we may define the integration on the Cech–de Rham m0 cohomology HD .U/ and the cup product followed by the integration describes the Poincaré duality:

0

H r .M; C/ ' HDr .U/ ! H m r .U/ ' Hm0 r .M; C/: ˇ Next we define the relative Cech–de Rham cohomology and describe the Alexander duality. Let S be a closed subset of M. Letting U0 D M n S and U1 an open neighborhood of S, we consider the covering U D fU0 ; U1 g of M. We set Ar .U; U0 / D f D .0 ; 1 ; 01 / 2 Ar .U/ j 0 D 0 g: Then we see that if is in Ar .U; U0 /, D is in ArC1 .U; U0 /. This gives rise ˇ Rham to another complex .A .U; U0 /; D/ and we define the r-th relative Cech–de r cohomology HD .U; U0 / of the pair .U; U0 / to be the cohomology of this complex. Note that there is a natural isomorphism HDr .U; U0 / ' H r .M; M n SI C/: Note that the cup product of a cochain in A .U/ and a cochain in A .U; U0 / is in A .U; U0 / and this induces a natural HD .U/-module structure on HD .U; U0 /, which is compatible with the usual H .M /-module structure on H .M; M n S/. SupposeRM is oriented and S is compact (M may not be). Then we may define the integration M W HDm .U; U0 / ! C. From (3.4) we see that the cup product induces a pairing Ar .U; U0 / Amr .U1 / ! Am .U; U0 /, which, followed by the integration,


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gives a bilinear pairing Ar .U; U0 / Amr .U1 / ! C: If we further assume that U1 is a regular neighborhood of S, this induces the Alexander duality

A W H r .M; M n SI C/ ' HDr .U; U0 / ! H mr .U1 ; C/ ' Hmr .S; C/: (3.5) The following proposition, which is rather obvious from the above construction, is fundamental in the localization theory. Proposition 3.7. If M is compact, we have the commutative diagram H r .M; M n SI C/

j

o A

Hmr .S; C/

/ H r .M; C/ o P

i

/ Hmr .M; C/;

where i and j denote the inclusions S ,! M and .M; ;/ ,! .M; M nS/, respectively. P Let M be a C 1 manifold, D qiD0 .1/i Ei a virtual bundle over M and ' a symmetric series, as before. Also let U D fU0 ; U1 g be an open covering of M. Choosing a family of connections r D .r.q/ ; : : : ; r.0/ / for on U , D 0; 1, we ˇ have a Cech–de Rham cochain '.r / D .'.r0 /; '.r1 /; '.r0 ; r1 //:

(3.6)

ˇ By (3.1), this is a cocycle and defines a class in the Cech–de Rham cohomology HD .U/, which corresponds to the class './ via the isomorphism (3.3). Moreover, if we may choose r0 so that '.r0 / 0, the cocycle '.r / defines a class in the relative cohomology HD .U; U0 /. This idea is used in the localization theory of characteristic classes of virtual bundles. In the next subsection, we give such an example.

3.3 Local Chern classes and characters We discuss the localization theory of characteristic classes by exactness of vector bundle sequences. For details of this and the subsequent subsections, we refer to [23]. Let M be a C 1 manifold and S a closed set in M. Letting U0 D M n S and U1 a neighborhood of S in M, we consider the covering U D fU0 ; U1 g of M, as before. Suppose that (3.2) is a sequence complex of vector bundles over M which is exact on U0 . Then we will see below that, for each p > 0, there is a natural localization

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cSp ./Pin H 2p .M; M n S/ of the Chern class c p ./ in H 2p .M / of the virtual bundle D qiD0 .1/i Ei . Let r0 be a family of connectionsPcompatible with (3.2) on U0 and r1 an arbitrary family of connections for D qiD0 .1/i Ei on U1 . Then the class c p ./ is represented by the cocycle c p .r / D .c p .r0 /; c p .r1 /; c p .r0 ; r1 // in A2p .U/. By Lemma 3.5, we have c p .r0 / D 0 and thus the cocycle is in A2p .U; U0 / and it defines a class cSp ./ in HD2p .U; U0 /. It is sent to c p ./ by the canonical homomorphism j . It is not difficult to see that the class cSp ./ does not depend on the choice of the family of connections r0 compatible with (3.2) or on the choice of the family of connections r1 . If ' is a symmetric series without constant term, we may also define the localized class 'S ./ of './. The localized Chern character chS ./ of the virtual bundle as above can be defined in this context. Thus in general, let r be a connection for a complex vector bundle E of rank `. The Chern character form of r is defined by p 1 A ch .r/ D tr.e /; A D K; 2 where K is the curvature of r. If we set t p .r/ D tr.Ap /, it is a closed 2p-form on M and we may write X t p .r/ ch .r/ D ` C : pŠ p1 The forms c p D c p .r/ and t p D t p .r/ are related by Newton’s formula: t p c 1 t p1 C c 2 t p2 C .1/p p c p D 0;

p 1:

HdR .M /

is the Chern character ch .E/ of E. More generThe classP of ch .r/ in ally, let D qiD0 .1/i Ei be a virtual bundle over M and r D .r .q/ ; : : : ; r .0/ / a family of connections for . We set ch .r / D

q X .1/i ch .r .i/ / iD0

and define the Chern character ch ./ of to be the class of ch .r /. Let S, U, , r0 and r1 be as in the beginning of this subsection. Then the class ch ./ in HD .U/ is represented by the cocycle ch .r / D .ch .r0 /; ch .r1 /; ch .r0 ; r1 // in A .U/. Noting that the alternating sum of the ranks of Ei is zero if M n S ¤ ;, by Lemma 3.5 we have ch .r0 / 0 and we see that the cocycle is in A .U; U0 /. Its class in the relative cohomology HD .U; U0 / is the localized Chern character chS ./ of . It is sent to ch ./ by the homomorphism j W HD .U; U0 / ! HD .U/.


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Remark 3.8. The local Chern characters defined as above have all the necessary properties and should coincide with the ones in [14]. Hence they are in the cohomology H .M; M n SI Q/ with Q coefficients. Also, the local Chern classes above are in the image of H .M; M n SI Z/ ! H .M; M n SI C/. See also [7] for local Chern characters. Now let M be a complex manifold and denote by AM the sheaf of germs of real analytic functions on M. Let be a coherent OM -module and suppose that the support S of is compact. Taking a relatively compact open neighborhood U of S in M, there is a complex of real analytic vector bundles on U as (3.2) such that, on the sheaf level, the sequence 0 ! AU .Eq / ! ! AU .E0 / ! AU ˝OU ! 0 is exact [5]. We call such a sequence a resolution of by vector bundles. We define the Chern character ch ./ of by ch ./ D ch ./;

D

q X .1/i Ei : iD0

Then ch ./ does not depend on the choice of the resolution. We set U0 D U n S, U1 D U and U D fU0 ; U1 g. Since the sequence (3.2) is exact on U0 , we have the local Chern character chS ./ in HD .U; U0 / ' H .U; U n S/ ' H .M; M n S/. We finish this subsection by recalling the Todd class in our context, which will be used in the subsequent sections. Let r be a connection for a complex vector bundle E of rank ` on a C 1 manifold M. The Todd form of r is defined by p A 1 K; td.r/ D det ; AD I e A 2 where K is the curvature of r. The constant term in td.r/ is 1 so that the form is invertible. It is closed and its class in HdR .M / is the Todd class td.E/ of E. We have the following fundamental relation (cf. [12], III, Corollary 5.4), which is one of the essential ingredients in the proof of the Riemann–Roch theorem for embeddings presented and used below:

` X .1/i ch .ƒi r / D td.r/1 c ` .r/; iD0

where r denotes the connection for E dual to r and ƒi r the connection for ƒi E induced by r . We also set ƒ0 E D M C and ƒ0 r D d (the exterior derivative). See [13], Theorem 10.1.1, for the above formula in cohomology.

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3.4 Riemann–Roch theorem for embeddings Let M be a complex manifold of dimension m and V a compact analytic subvariety of pure dimension n in M. Let k D m n. Recall that ([10], see also [25]) we have the Poincaré homomorphism P W H r .V / ! H2nr .V / and the Thom homomorphism T W H r .V / ! H rC2k .M; M n V / so that the following diagram is commutative: H r .V /

/ H rC2k .M; M n V /

T

o A

P

H2nr .V /

id

/ H2nr .V /

Recall also that a subvariety V of codimension k in M is a local complete intersection (abbreviated as LCI) in M if the ideal sheaf V in OM of functions vanishing on V is locally generated by k functions. In this case, the normal sheaf NV D HomOV . V = V2 ; OV / is a locally free OV -module of rank k. We denote by NV the associated vector bundle, which gives a natural extension of the normal bundle NV 0 in M of the regular part V 0 of V to the whole V. We say that a subvariety V of codimension k in M is an LCI defined by a section, if there exist a holomorphic vector bundle N of rank k over M and a holomorphic section s of N such that the local components of s locally generate V. In this case V is an LCI and we have NV D N jV. Let W V ,! M denote the embedding and let be a coherent OV -module. The direct image Š is a coherent OM -module, which is simply extended by zero on M n V, and we have the localized Chern character chV .Š / in H .M; M n V I Q/. The following localized version of the Riemann–Roch theorem for embeddings is ˇ proved on the level of Cech–de Rham cocycles in [23]. Theorem 3.9. Let V be a compact subvariety in M and a coherent OV -module. Suppose (i) V is non-singular, or (ii) V is an LCI defined by a section and is locally free. Then we have chV .Š / D T .ch ./ Y td1 .NV // in H .M; M n V I Q/: Here we emphasize that M may not be compact. See [23], Remark 3.6, for other works related to the above.


221

4 Chern residues of singular distributions 4.1 Localization by rank reason Let M be a complex manifold of dimension m and F a singular distribution of rank r on M. Let NF be the normal sheaf so that we have the exact sequence (2.1). Let S D S.F / be the singular set of the distribution. There is a rank r subbundle F0 of TM0 , M0 D M n S, such that F jM0 D O.F0 /. If we set NF0 D TM0 =F0 , it is a vector bundle of rank m r D s and NF jM0 D O.NF0 /. Observe that, for an arbitrary connection r for NF0 , c p .r/ 0 for p > s: Thus for a Chern monomial ' D c p1 c pk , if pi > s for some i, '.r/ 0. This leads to the following result. Theorem 4.1. In the above situation, suppose S is compact. Then there is a natural localization 'S .NF ; F / in H 2d .M; M nS/ of '.NF / in H 2d .M /, d D p1 C Cpk . Proof. Let U be a relatively compact regular neighborhood of S in M and set U0 D U n S, U1 D U , and U D fU0 ; U1 g. Take a locally free resolution of NF on U (we omit to write down the sheaf A of germs of real analytic functions): 0 ! Eq ! ! E0 ! NF ! 0:

(4.1)

Let r be a connection for NF0 on U0 and r0.i/ a connection for each Ei on U0 such that the family .r0.q/ ; : : : ; r0.0/ ; r/ is compatible with (4.1) on U0 . Let r1.i/ be a connection for each Ei on U1 and set r D .r.q/ ; : : : ; r.0/ /, D 0; 1. If ' is a polynomial as above, then by Proposition 3.6 and the above observation we have '.r0 / D '.r/ D 0 and the cocycle '.r / D .0; '.r1 /; '.r0 ; r1 // is in A2d .U; U0 /. Now we claim that, if we start with another connection D for NF0 , we get a cocycle cohomologous to the above. In fact, let D0.i/ be a connection for each Ei on U0 such that the family .D0.q/ ; : : : ; D0.0/ ; D/ is compatible with (4.1) on U0 . Let D1.i/ be a connection for each Ei on U1 and set D D .D.q/ ; : : : ; D.0/ /, D 0; 1. Again we have '.D0 / D '.D/ D 0 and we have a cocycle '.D / D .0; '.D1 /; '.D0 ; D1 // in A2p .U; U0 /. We have '.D / '.r / D .0; '.D1 / '.r1 /; '.D0 ; D1 / '.r0 ; r1 //: On the other hand, considering the difference form '.r0 ; D0 ; D1 / for the three families of connections r0 , D0 and D1 , we have '.D0 ; D1 / '.r0 ; D1 / C '.r0 ; D0 / C d'.r0 ; D0 ; D1 / D 0:

222

Tatsuo Suwa

Again by Proposition 3.6 and by rank reason '.r0 ; D0 / D '.r; D/ D 0. Considering also the difference form '.r0 ; r1 ; D1 /, '.r1 ; D1 / '.r0 ; D1 / C '.r0 ; r1 / C d'.r0 ; r1 ; D1 / D 0 so that '.D / '.r / D D ;

D .0; '.r1 ; D1 /; '.r0 ; D0 ; D1 / '.r0 ; r1 ; D1 //:

Thus the class of '.r / in HD2d .U; U0 / ' H 2d .U; U n S/ ' H 2d .M; M n S/ does not depend on the choice of connections involved. We can also show that it does not depend on the choice of resolution in a similar manner. Thus the class of '.r /, denoted by 'S .NF ; F /, is well-defined. Suppose S has a finite number of connected components .S /, then via the Alexander isomorphism M

H2m2d .S /; A W H 2d .M; M n S/ ! H2m2d .S/ D the above localization 'S .NF ; F / defines a local invariant in H2m2d .S ; Q/ (in fact, in the image of H2m2d .S ; Z/ ! H2m2d .S ; Q/) for each . We call it the residue of F on NF at S with respect to ' and denote by Res' .F ; NF I S /. If M is compact, from Proposition 3.7, we have the residue formula: X . / Res' .F ; NF I S / D P .'.NF //; (4.2)

where W S ,! M denotes the inclusion. If F is a singular foliation, i.e. if F0 in involutive, there is an “action” of F0 on NF0 and, if r is an “F0 -connection” for NF0 , we have the Bott vanishing '.r/ D 0 for a homogeneous symmetric polynomial ' of degree d > s (cf. Section 6.1, in particular Remark 6.4 below). The Baum–Bott residue with respect to ' is defined exactly the same way as above, using an F0 -connection as r. Here we emphasize that the Baum– Bott residue is defined for an arbitrary ' whose degree is greater than s and in general is in the homology with complex coefficients, while the above residue is defined for ' containing c p with p > s in each of its terms, but is in the homology with rational coefficients, if the coefficients of ' are rational. From the construction we have the following, which shows the rationality of the relevant Baum–Bott residues (cf. [6], Rationality conjecture): Proposition 4.2. If F is a foliation, the above residue coincides with the Baum–Bott residue.


223

4.2 Residues and the local Chern classes In this subsection, we show that the residues defined in the previous subsection are related to the local Chern class of some sheaf supported on the singular set of the distribution. This way, in some cases we can compute the residues using the Riemann– Roch theorem for embeddings (cf. Theorem 3.9). The contents of this subsection are essentially in [21]. Note that involutivity is not really needed there. Let G be a reduced singular distribution of corank s on a complex manifold M of dimension m and set F D G a and G D M =G . Taking the duals of 0 ! G ! M ! G ! 0; we have the exact sequence

0 ! F ! ‚M ! G ! Ext1O .G ; O/ ! 0:

(4.3)

Note that, by definition, the support of Ext1O .G ; O/ is in S D S.G / D S.F /, which is assumed to be compact with a finite number of connected components .S /. Comparing with (2.1), we have the exact sequence 0 ! NF ! G ! Ext1O .G ; O/ ! 0:

(4.4)

Recall that there is a rank r D ms subbundle F0 of TM0 such that F jM0 D O.F0 /. We set NF0 D TM0 =F0 : Hereafter we assume that G is locally free of rank s. Thus there is a vector bundle G of rank s on M with G D OM .G/. Note that we may think of G as a subbundle of T M only away from S. We express the Chern classes of NF in terms of those of G D OM .G / and Ext1O .G ; O/ using (4.4). In the sequel we denote Ext1O .G ; O/ simply by E. We have the local Chern classes cSp .E/ in H 2p .M; M n S/ for p > 0. If we denote by j W H 2p .M; M n S/ ! H 2p .M / the canonical homomorphism, j cSp .E/ D c p .E/ is the p-th Chern class of the coherent sheaf E. Accordingly, we have the local Segre class sSp .E/ in H 2p .M; M n S/ for each p > 0, so that j sSp .E/ D s p .E/ is the p-th Segre class of E in H 2p .M /. For simplicity we assume that S is connected. Basically, we have c .NF / D c .G E/ D c .G / s .E/ with some localized components. Definition 4.3. For each integer p, 1 p m, we set 8 p c .G / C c p1 .G / s 1 .E/ C C c 1 .G / s p1 .E/ C s p .E/; ˆ ˆ ˆ ˆ < 1 p s; c p .G E/ D ˆ ˆ c s .G / sSps .E/ C C c 1 .G / sSp1 .E/ C sSp .E/; ˆ ˆ : s < p m:

224

Tatsuo Suwa

Note that, for p s, c p .G E/ is in H 2p .M; Q/, while for p > s, c p .G E/ is in H 2p .M; M n SI Q/. Let A W H 2d .M; M n S/ ! H2m2d .S/ denote the Alexander isomorphism. Theorem 4.4. For a Chern monomial ' D c p1 c pk with pi > s for some i, the product c p1 .G E/ c pk .G E/ is in H 2d .M; M n S/, d D p1 C C pk , and we have Res' .F ; NF I S/ D A.c p1 .G E/ : : : c pr .G E//; which is in H2m2d .S; Q/, in fact in the image of H2m2d .S; Z/ ! H2m2d .S; Q/. Proof. Let U and U D fU0 ; U1 g be as in the proof of Theorem 4.1. Combining (4.4) and (4.1), we have a locally free resolution defining the local Chern class of E: 0 ! Eq ! ! E0 ! G ! E ! 0: We may identify NF0 and G on U0 . Let r be a connection for G on U. Let r0.i/ be a connection for each Ei on U0 so that the family .r0.q/ ; : : : ; r0.0/ ; r/ is compatible with (4.1) on U0 . Let r1.i/ be a connection for each Ei on U1 and set y D .r.q/ ; : : : ; r.0/ ; r/, D 0; 1. The point here is that we take, for G , the same r connection r on U0 and U1 . Then we have .c .r0 /; c .r1 /; c .r0 ; r1 // y 0 /; s .r y 1 /; s .r y 0 ; r y 1 //; D .c .r/; c .r/; 0/ Y .s .r which proves the theorem. As noted above, in some cases the residues can be computed using the Riemann– Roch theorem for embeddings. The following result is proved in [21] this way (involutivity is not necessary as noted above). Proposition 4.5. In the above situation, suppose that s D 1 and that S D fpg is an isolated point. Then we have Resc m .F ; NF I fpg/ D .1/m .m 1/Š dim Ext 1O .G ; O/ in H0 .fpg; Q/ D Q; where we denoted OM;p and G ;p simply by O and G . P Note that if .z1 ; : : : ; zm / is a coordinate system on M around p and ! D m iD fi dzi is a generator of G near p, then the set of common zeros of the fi ’s is fpg and we have dim Ext 1O .G ; O/ D dim O=.f1 ; : : : ; fm /: The following result is proved in [15] using a Koszul complex associated to the distribution.


225

Proposition 4.6. If G is locally free of rank one, then c m .NF / D .1/m .m 1/Š c m .M ˝ G /: Thus the residue in Proposition 4.5 also arises from the localization of the top Chern class of M ˝ G by a section, i.e. the section locally corresponding to W G ' O ! M given by .1/ D ! (cf. [24]). See [21] for more examples. In the next subsection we give an example for which the singular set of the distribution is non-isolated and singular.

4.3 An example We consider the 1-form ! D z dx C z dy y dz on C3 D f.x; y; z/g. It defines a corank one singular distribution on C3 with singular set fy D z D 0g. As generators of its annihilator, we may take the vector fields v1 D y

@ @ Cz @y @z

and

v2 D

@ @ : @x @y

We extend the distribution to the projective space. Let P 3 denote the complex projective space of dimension three with homogeneous coordinates D .0 W 1 W 2 W 3 /. It is covered by four open sets W .i/,0 i 3, given by i ¤ 0. We take the original affine space C3 as W .0/ with x D 1 =0 , y D 2 =0 and z D 3 =0 . Let G be the corank one distribution on P 3 naturally obtained as an extension of the above. (0) On W .0/ , G is defined by !0 D z dx C z dy y dz as given before. (1) On W .1/ , we set x1 D 0 =1 , y1 D 3 =1 and z1 D 2 =1 . Then G is defined by !1 D y1 dx1 x1 z1 dy1 C x1 y1 dz1 : (2) On W .2/ , we set x2 D 3 =2 , y2 D 0 =2 and z2 D 1 =2 . Then G is defined by !2 D y2 dx2 x2 z2 dy2 C x2 y2 dz2 : (3) On W .3/ , we set x3 D 2 =3 , y3 D 1 =3 and z3 D 0 =3 . Then G is defined by !3 D z3 dx3 C z3 dy3 y3 dz3 : We see that G is a reduced distribution of corank one. It is locally free and from !i D .j =i /3 !j in W .i/ \ W .j / , we see that it is .H3 /˝3 as a line bundle, where H3 denotes the hyperplane bundle on P 3 . The singular set S D S.G / is defined by 0 2 D 0 3 D 1 3 D 0 and has three irreducible components S1 D f2 D 3 D 0g, S2 D f0 D 3 D 0g and S3 D f0 D 1 D 0g, each of which is a projective line P 1 . We also let P1 D .0 W 1 W 0 W 0/,

226

Tatsuo Suwa

which is the intersection point of S1 and S2 and P2 D .0 W 0 W 1 W 0/, which is the intersection point of S2 and S3 . Let F D G a denote the annihilator of G , which is a singular distribution of rank two on P 3 with the singular set S.F / D S.G / D S. We try to find the residues Res' .F ; NF I S/ in H .S/ for ' D c 2 , c 1 c 2 and c 3 . For this, we first find the local Chern class cS .E/, E D Ext 1 .G ; O/, using Theorem 3.9, which in our case reads chS .Š E/ D T .ch .E/ Y td1 .NS //

in H .P 3 ; P 3 n SI Q/:

(4.5)

Note that we see below that the hypothesis (2) of the theorem is satisfied in our case. Let T0 , T1 , T2 and T3 be tubular neighborhoods of S1 n fP1 g, S2 n fP2 g, S2 n fP1 g and S3 n fP2 g, respectively, sufficiently small so that Ti W .i/ and that none of the three distinct Ti ’s intersects. We set T D T0 [ [ T3 , which is a neighborhood of S in P 3 . Let S denote the ideal sheaf of S. Then it is generated by .f0 ; g0 / D .z; y/ on W .0/ , .f1 ; g1 / D .y1 ; x1 z1 / on W .1/ , .f2 ; g2 / D .x2 z2 ; y2 / on W .2/ and .f3 ; g3 / D .y3 ; z3 / on W .3/ . Thus we see that S is a local complete intersection, although it is not a complete intersection. Let Aij denote the 2 2 matrix defined by .fi ; gi / D .fj ; gj /Aj i . Then the locally free sheaf S = S2 is defined by the system of transition matrices fAij g. We compute 1 1 0 0 01 1 1 1 0 0 C B z2 B 2 0C B C 1 C ; A23 D B x2 C: A01 D @ x (4.6) A ; A12 D B @ @ 1A 1A 1 0 0 0 z12 x2 x2 Let NS be the normal bundle of S in P 3 , which is the vector bundle corresponding to the locally free sheaf Hom. S = S2 ; OS /. Since it is defined by the system ft A1 ij g D 2 2 2 fAj i g, by (4.6), we see that NS jS1 D H1 ˚ H1 , NS jS2 D H1 ˚ H1 and NS jS3 D H12 ˚ H1 , H1 being the hyperplane bundle on P 1 . It is not difficult to see that we may construct a vector bundle N of rank two on T extending NS and a regular section of N which defines S. We try to find td1 .NS /. We have H .S/ D H 0 .S/ ˚ H 2 .S/ D Q ˚ H 2 .S1 / ˚ H 2 .S2 / ˚ H 2 .S3 / and we may write c .NS / D 1 C .c 1 .NS jS1 /; c 1 .NS jS2 /; c 1 .NS jS3 // D 1 C .3 1 ; 4 1 ; 3 1 /; where we set 1 D c 1 .H1 /. Thus we have 1 1 td1 .NS / D 1 c 1 .NS / D 1 .3 1 ; 4 1 ; 3 1 /: 2 2

(4.7)

We now try to find E D Ext1 .G ; O/. For this, we use the exact sequence (4.3). We have G D OP 3 .H3˝3 / and G jW .i / ' OW .i / , the correspondence being given


227

by ' $ fi D '.!i /. We compute @

@x

@

@x1

@

@x2

@

@x3

D z;

@

@y

@

D y1 ;

@y1

@

D y2 ; D z3 ;

@y2

@

@y3

@

D z;

@z

@

D x1 z1 ;

D x2 z2 ;

D z3 ;

@z1

@

@z2

@

@z3

D y;

on W .0/ ;

D x1 y1 ; on W .1/ ; D x2 y2 ; on W .2/ ; D y3

on W .3/ :

Thus we see that E D OS .H3˝3 jS /, which is locally free of rank one, and ch .E/ D 1 C c 1 .E/ D 1 C .3 1 ; 3 1 ; 3 1 /:

(4.8)

From (4.7) and (4.8), we have 1 ch .E/ Y td1 .NS / D 1 .3 1 ; 2 1 ; 3 1 /: 2 and by (4.5),

1 chS .Š E/ D T 1 .3 1 ; 2 1 ; 3 1 / : 2

From this we have ch1S .Š E/ D 0; ch2S .Š E/ D .1 ; 2 ; 3 /; 1 ch3S .Š E/ D .3 C 2 C 3/ D 4; 2 where i denotes a generator of each component of H 4 .P 3 ; P 3 n S/ ' Q ˚ Q ˚ Q. Note that H 6 .P 3 ; P 3 n S/ ' Q. By the Newton formula, cS1 .E/ D ch1S .iŠ E/ D 0; cS2 .E/ D ch2S .iŠ E/ D .1 ; 2 ; 3 /; cS3 .E/ D 2 ch3S .iŠ E/ D 8: Thus we have dS1 .E/ D cS1 .E/ D 0; dS2 .E/ D cS2 .E/ D .1 ; 2 ; 3 /; dS3 .E/ D cS3 .E/ D 8

228

and

Tatsuo Suwa

c 1 .G E/ D c 1 .G / C dS1 .E/ D c 1 .G / D 3 3 ; c 2 .G E/ D c 1 .G / dS1 .E/ C dS2 .E/ D dS2 .E/; c 3 .G E/ D c 1 .G / dS2 .E/ C dS3 .E/;

where 3 D c 1 .H3 /. Note that c 1 .G E/ is in H 2 .P 3 /, while c 2 .G E/ and c 3 .G E/ are in the relative cohomology H .P 3 ; P 3 n S/. From (4.3) we see that Resc 2 .F ; NF I S/ D A.c 2 .G E/// D ŒS;

where A W H 4 .P 3 ; P 3 n S/ ! H2 .S/ is the Alexander isomorphism. Resc 1 c 2 .F ; NF I S/ D A.c 1 .G E/c 2 .G E/// D c 1 .G E/ a A.c 2 .G E// D 3 3 a ŒS D 9; Resc 3 .F ; NF I S/ D A.c 3 .G E// D A.c 1 .G / dS2 .E/ C dS3 .E// D 3 3 a ŒS C 8 D 17;

where A W H 6 .P 3 ; P 3 n S/ ! H0 .S/ D Q.

ˇ 5 Atiyah classes and Cech–Dolbeault cohomology 5.1 Atiyah classes For details of this subsection, we refer to [1]. Let M be a complex manifold of dimension m and E a holomorphic vector bundle of rank ` over M.Also let r be a connection for E (cf. Section 3.1). Note that r is a local operator. Thus, if s .`/ D .s1 ; : : : ; s` / is a frame (` C 1 sections linearly independent everywhere) of E on an open set U, we have the connection matrix D .ij / with entries ij 1-forms on U defined by r.si / D

` X

j i ˝ sj :

j D1

Definition 5.1. A connection r for E is of type .1; 0/ if the entries of the connection matrix with respect to a holomorphic frame are forms of type .1; 0/. In this case, we also say that r is a .1; 0/-connection.


229

Note that the above property of r does not depend on the choice of a holomorphic frame and that a holomorphic vector bundle always admits a .1; 0/-connection. Let r be a .1; 0/-connection for E and K its curvature. The curvature matrix P D .ij /, which is defined by K.si / D j`D1 j i ˝ sj , is related to the connection matrix by D d C ^ . Thus we have the decomposition D 2;0 C 1;1 with 2;0 D @ C ^

and

N 1;1 D @:

(5.1)

Accordingly we have a decomposition K D K 2;0 C K 1;1 ; where K 2;0 and K 1;1 are, respectively, a .2; 0/-form and a .1; 1/-form with coefficients in Hom.E; E/. Thus we may define, for each elementary symmetric polynomial p , N p D 1; 2; : : : ; `, a C 1 .p; p/-form p .K 1;1 / on M, which is @-closed, being locally N@-exact by (5.1). Definition 5.2. We set

p 1 p p .K 1;1 / a .r/ D 2 N and call it the p-th Atiyah form of r, which is a @-closed .p; p/-form on M. p

The following is proved using the construction of Chern difference forms (cf. Proposition 3.2), in fact the form ap .r0 ; : : : ; rr / is the .p; p r/-component of c p .r0 ; : : : ; rr /, see [26] for details. Proposition 5.3. Suppose we have r C 1 .1; 0/-connections r0 ; : : : ; rr for E. Then there exists a .p; p r/-form ap .r0 ; : : : ; rr /, alternating in the r C 1 entries and satisfying r X

N p .r0 ; : : : ; rr / D 0: y ; : : : ; rr / C .1/r @a .1/ ap .r0 ; : : : ; r

D0

In particular, if r D 1, we have N p .r0 ; r1 /: ap .r1 / ap .r0 / D @a Thus, if r is a .1; 0/-connection for E, the class of ap .r/ in H Np;p .M / does not @ depend on the choice of r. Definition 5.4. The p-th Atiyah class ap .E/ of E is the class of ap .r/ in the Dolbeault cohomology H Np;p .M /, where r is a .1; 0/-connection for E. @

Remark 5.5. The Atiyah form ap .r/ is the .p; p/-component of the corresponding Chern form c p .r/. In particular, am .r/ D c m .r/.

230

Tatsuo Suwa

2p The p-th Chern class c p .E/ of E is the class of c p .r/ in HdR .M /. Thus, although we may not be able to compare the Atiyah and Chern classes directly on the cohomology level (unless M is compact Kähler), we may do so on the form level. If r is a metric connection for E, i.e. a .1; 0/-connection compatible with some Hermitian metric on E, then its curvature is of type .1; 1/ and thus we have ap .r/ D c p .r/ for all p. If M is compact Kähler, the Hodge decomposition gives a canonical injection 2p .M / and we have, by the above, h.ap .E// D c p .E/. h W H Np;p .M / ! HdR @

Let ' be a homogeneous symmetric polynomial of degree d . We may write ' as a polynomial in elementary symmetric polynomials ; ' D P .1 ; 2 ; : : : /. For a .1; 0/-connection r for E, we set '.r/ D P .c 1 .r/; c 2 .r/; : : : /; N which is a closed 2d -form and ' A .r/ D P .a1 .r/; a2 .r/; : : : /, which is a @-closed .d; d /-form and is the .d; d /-component of '.r/. For two .1; 0/-connections r and r 0 , we have the difference forms '.r; r 0 / and ' A .r; r 0 / as above and we have the 2d .M / and ' A .E/ in H Nd;d .M /. classes '.E/ in HdR @

ˇ 5.2 Cech–Dolbeault cohomology ˇ In this and subsequent sections, we recall the theory of Cech–Dolbeault cohomology. For details we refer to [26]. The treatment of relative cohomologies in Section 5.3 below is slightly more general than [26]. Let M be a complex manifold of dimension m. For an open set U of M, we denote by Ap;q .U / the vector space of C 1 .p; q/-forms on U. Let U D fU˛ g˛2I be an open covering of M, indexed by an ordered set I . We set I .r/ D f.˛0 ; : : : ; ˛r / j ˛0 < < ˛r ; ˛ 2 I g and denote by C r .U; Ap;q / the direct product Y C r .U; Ap;q / D

Ap;q .U˛0 :::˛r /;

.˛0 ;:::;˛r /2I .r/

where we set U˛0 :::˛r D U˛0 \ \ U˛r . Thus an element in C r .U; Ap;q / assigns to each .˛0 ; : : : ; ˛r / in I .r/ a form ˛0 :::˛r in Ap;q .U˛0 :::˛r /. The coboundary operˇ ator ı W C r .U; Ap;q / ! C rC1 .U; Ap;q / is defined as in the usual Cech theory. This together with the operator @N W C r .U; Ap;q / ! C r .U; Ap;qC1 /


231

makes C .U; Ap; / a double complex for each p D 0; : : : ; n. The simple complex x Thus associated to this is denoted by .Ap; .U/; D/. M 0 C r .U; Ap;q / Ap;q .U/ D q 0 CrDq

and the differential x DD x q W Ap;q .U/ ! Ap;qC1 .U/ D is given by x ˛0 :::˛r D .D/

r X

N ˛ :::˛ ; .1/ ˛0 :::˛O :::˛r C .1/r @ r 0

(5.2)

D0

where y means the letter under it is to be omitted. We denote the q-th cohomology of x by H p;q .U/ and we call it the Cech–Dolbeault ˇ cohomology of U of .Ap; .U/; D/ x D type .p; q/. We refer to [26] for the proof of the following. Theorem 5.6. The restriction map Ap;q .M / ! C 0 .U; Ap;q / Ap;q .U/ induces an isomorphism

p;q H Np;q .M / ! HD x .U/: @

We define the “cup product” 0

0

0

0

Ap;q .U/ Ap ;q .U/ ! ApCp ;qCq .U/ 0

0

0

0

by assigning to in Ap;q .U/ and in Ap ;q .U/ the element Y in ApCp ;qCq .U/ given by r X . Y /˛0 :::˛r D .1/.pCq/.r/ ˛0 :::˛ ^ ˛ :::˛r : D0

Then this induces the cup product 0

0

0

0

p;q p ;q pCp ;qCq HD .U/ ! HD .U/ x .U/ HD x x

compatible, via the isomorphism of Theorem 5.6, with the product in the Dolbeault cohomology induced by the exterior product of forms. ˇ Now we recall the integration on the Cech–Dolbeault cohomology. Let M and U D fU˛ g˛2I be as above and fR˛ g˛2I a system of honey-comb cells adapted to U (see [16] and also [22]). Suppose M is compact, then each R˛ is compact and we may define the integration Z W Am;m .U/ ! C M

by the sum

Z D M

m X rD0

X .˛0 ;:::;˛r /2I .r/

Z R˛0 :::˛r

˛0 :::˛r

232

Tatsuo Suwa

for in Am;m .U/. Then it induces the integration on the cohomology Z m;m W HD .U/ ! C; x M

which is compatible, via the isomorphism of Theorem 5.6, with the usual integration on the Dolbeault cohomology H Nm;m .M /. Also the bilinear pairing @

p;q

A

.U/ A

mp;mq

.U/ ! Am;m .U/ ! C

defined as the composition of the cup product and the integration induces the Kodaira– Serre duality

p;q mp;mq .U/ ' H Nmp;mq .M / : KS W H Np;q .M / ' HD x .U/ ! HD x @

@

(5.3)

ˇ 5.3 Relative Cech–Dolbeault cohomology Let S be a closed set in M. Let U 0 be an open neighborhood of S in M and let U0 D fU˛ g˛2I 0 be an open covering of U 0 . Let U0 D M nS and consider the covering U D fU˛ g˛2I of M, where I D f0g t I 0 with the order 0 < ˛ for all ˛ in I 0 . We denote by Ap;q .U; U0 / the subspace of Ap;q .U/ consisting of elements with 0 D 0 so that we have the exact sequence 0 ! Ap;q .U; U0 / ! Ap;q .U/ ! Ap;q .U0 / ! 0: x maps Ap;q .U; U0 / into Ap;qC1 .U; U0 /. Denoting by H p;q .U; U0 / We see that D x D x we have the long exact sethe q-th cohomology of the complex .Ap; .U; U0 /; D/, quence p;q p;q p;q ! H Np;q1 .U0 / ! HD x .U; U0 / ! HD x .U/ ! H N .U0 / ! : @

@

p;q HD x .U/

In view of the fact that

'

H Np;q .M /, @

we set

p;q H Np;q .M; M n S/ D HD x .U; U0 /: @

Suppose S is compact (M may not be) and let fR˛ g be a system of honey-comb cells adapted to U. Then we may assume that each R˛ is compact for ˛ in I 0 and we have the integration on Am;m .U; U0 / given by Z D M

XZ ˛2I 0

˛ C

R˛

m X rD1

X .˛0 ;:::;˛r /2I .r/

Z R˛0 :::˛r

This again induces the integration on the cohomology Z m;m W HD .U; U0 / ! C: x M

˛0 :::˛r :


233

It is not difficult to see that the cup product Ap;q .U/Amp;mq .U/ ! Am;m .U/ induces a pairing Ap;q .U; U0 / Amp;mq .U0 / ! Am;m .U; U0 /, which, followed by integration, gives a bilinear pairing Ap;q .U; U0 / Amp;mq .U0 / ! C: This induces a homomorphism p;q mp;mq .U0 / D H Nmp;mq .U 0 / ; AxW H Np;q .M; M n S/ D HD x .U; U0 / ! HD x @ @ (5.4) N which we call the @-Alexander homomorphism. From the above construction, we have the following result.

Proposition 5.7. If M is compact, the following diagram is commutative: H Np;q .M; M n S/

j

/ H p;q .M / N

@

@

Ax

o KS

H Nmp;mq .U 0 / @

i

/ H mp;mq .M / ; N @

where i W U 0 ,! M denotes the inclusion. Remark 5.8. Suppose that S is a compact complex submanifold of M and that there exists a holomorphic retraction r W U 0 ! S, i.e. a holomorphic map with r ı D 1S , where W S ,! U 0 is the embedding. Then the following diagram is commutative: H Np;q .M; M n S/

j

/ H p;q .M / N

@

@

r ıAx

o KS

H Nmp;mq .S/ @

i ı

/ H mp;mq .M / ; @N

where r and denote the transposed of the pull-backs r W H Nmp;mq .S/ ! H Nmp;mq .U 0 / @

@

and W H Nmp;mq .U 0 / ! H Nmp;mq .S/; @

@

respectively.

ˇ 5.4 Atiyah classes in Cech–Dolbeault cohomology Let U D fU˛ g be an open covering of M as in Section 5.2.Also, let E be a holomorphic vector bundle over M. For each ˛, we choose a .1; 0/-connection r˛ for E on U˛ ,

234

Tatsuo Suwa

and for the collection r D .r˛ /˛ , we define the element ap .r / in Ap;p .U/ D ˚prD0 C r .U; Ap;pr / by ap .r /˛0 :::˛r D ap .r˛0 ; : : : ; r˛r /: x p .r / D 0 by the identity in Proposition 5.3. See [26] for the proof Then we have Da of the following result. p;p Proposition 5.9. The class Œap .r / in HD x .U/ does not depend on the choice of the collection of .1; 0/-connections r and corresponds to the Atiyah class ap .E/ by the isomorphism of Theorem 5.6.

More generally, for a homogeneous symmetric polynomial ' of degree d, we may d;d define ' A .r / in Ad;d .U/ and the class Œ' A .r / in HD x .U/, which corresponds A to ' .E/ by the isomorphism of Theorem 5.6. Let S, U0 and U be as in Section 5.3. If we may choose r0 so that ' A .r0 / 0, d;d the cocycle ' A .r / defines a class in the relative cohomology HD x .U; U0 /. This idea is used in the localization theory of Atiyah classes of holomorphic bundles.

6 Atiyah residues of singular distributions We review the results in [1] from a slightly different viewpoint and also discuss the localization problem on singular varieties.

6.1 Actions of distributions Let M be a complex manifold of dimension m and F a non-singular distribution of rank r, i.e. a subbundle of TM of rank r. Definition 6.1. A (holomorphic) action of F on a holomorphic vector bundle E over M is a C-bilinear map ˛ W A0 .M; F / A0 .M; E/ ! A0 .M; E/ satisfying the following conditions, for f in A0 .M /, u in A0 .M; F / and s in A0 .M; E/: (i) ˛.f u; s/ D f ˛.u; s/, (ii) ˛.u; f s/ D u.f /s C f ˛.u; s/, and (iii) ˛.u; s/ is holomorphic whenever u and s are. A vector bundle E with an action of F is called an F -bundle.


235

Definition 6.2. Let ˛ be an action of F on E. An F -connection (or ˛-connection, if it is necessary to specify the action) for E is a connection for E satisfying (i) rs.u/ D ˛.u; s/, for s 2 A0 .M; E/ and u 2 A0 .M; F /, and (ii) r is of type .1; 0/. From the fact that an action is a local operation, we see that an F -bundle always admits an F -connection. We note that the above material can be equivalently treated in the language of partial holomorphic connections instead of actions (cf. [3] and [1]), the condition (iv) in Remark 6.4 below corresponding to the fact that the partial connection is flat. We have the following Bott type vanishing theorem for F -connections. A proof of the part (1) of the following is given in [3] and the part (2) is proved in [1] in the context of partial connections. Theorem 6.3. Let M be a complex manifold of dimension m and F a non-singular distribution of rank r on M. Also let E be an F -bundle and r0 ; : : : ; rq F -connections for E. For a homogeneous symmetric polynomial ' of degree d , we have: (1) if d > m r C Œ 2r , then '.r0 ; : : : ; rq / 0I (2) if d > m r, then ' A .r0 ; : : : ; rq / 0: Remark 6.4. If F is involutive and if the action satisfies (iv) ˛.Œu; v; s/ D ˛.u; ˛.v; s// ˛.v; ˛.u; s//, we have '.r/ D 0 for ' with d > m r. This is usually referred to as the Bott vanishing theorem. Let M be a complex manifold of dimension m and V a complex submanifold of dimension n of M. Let NV be a normal bundle of V in M so that we have the exact sequence 0 ! T V ! TM jV ! NV ! 0: Let F be a distribution of rank r. Recall that F leaves V invariant in F jV T V (Definition 2.3). The following is proved in [17] (see also [18]) for the case of foliations. In fact the involutivity of F is not necessary and a proof is given in [1] in terms of partial connections. Here we reproduce the proof in our context. Theorem 6.5. Let V be a complex submanifold of M. Let F TM be a distribution of rank r leaving V invariant. Then there exists a holomorphic action of F jV on the normal bundle NV.

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Tatsuo Suwa

Proof. Let u and be C 1 sections of F jV and NV, respectively. Take sections uQ of F Q V / D , where jV means the restriction as and vQ of TM such that uj Q V D u and .vj a section. Define ˛ W A0 .V; F jV / A0 .V; NV / ! A0 .V; NV / by ˛.u; / D .Œu; Q vj Q V /: We now show that this does not depend on the choice of uQ or v. Q First, let uQ be a section of F with uj Q D 0. Take a frame . u Q ; : : : ; u Q / of F on an open set U and write V 1 r Pr 1 as uQ D iD1 aQ i uQ i with aQ i (C ) functions on U. We set ui D uQ i jV and ai D aQ i jV, i D 1; : : : ; r. Then .u1 ; : : : ; ur / is a frame of F jV on U \ V and, from the condition uj Q V D 0, we have ai D 0, i D 1; : : : ; r. We compute Œu; Q vj Q V D

r X

.ai ŒuQ i ; vj Q V v. Q aQ i /jV ui / D

iD1

r X

v. Q aQ i /jV ui ;

iD1

Q vj Q V / D 0. which is a section of F jV T V. Hence .Œu; Q V is a section of T V Second, let vQ be a section of TM with .vj Q V / D 0. Then vj such that Œu; Q vj Q V D Œuj Q V ; vj Q V is a section of T V, since F jV T V. Hence we have .Œu; Q vj Q V / D 0. It is then straightforward to check that ˛ is a holomorphic action. Corollary 6.6. In the above situation, let r be an FV -connection for NV. Then, for a symmetric homogeneous polynomial ' of degree d > n r, we have ' A .r/ 0.

6.2 Localization and residues Let M be a complex manifold of dimension m and F a singular distribution of rank r on M with singular set S D S.F /. There is a rank r subbundle F0 of TM jM0 , M0 D M n S, such that F jM0 D OM0 .F0 /. Let U0 and U be as in Section 5.3. Let E be a holomorphic vector bundle on M with an action of F0 on M0 . Let r0 be an F0 -connection for E on U0 D M0 and r˛ an arbitrary .1; 0/-connection for E for each ˛ > 0. For a symmetric polynomial ' homogeneous of degree d , we have ˇ the Cech–Dolbeault cocycle ' A .r / in Ad;d .U/. If d > m r, by Theorem 6.3, d;d A d;d ' .r / is in A .U; U0 / and defines a class in HD x .U; U0 /, which we denote by A A 'U 0 .E; F / and call the localization of ' .E/ by F at U 0 . N Suppose S is compact. Then its image by the A-Alexander homomorphism d;d md;md HD .U0 / ' H Nmd;md .U 0 / x .U; U0 / ! HD x @

is denoted by Res' A .F ; EI U / and called the residue of F for E at U 0 with respect to ' A . 0


237

If S has a finite number of connected components .S /, we break up U0 accordingly ; U0 D [ U0 so that we have a decomposition M md;md md;md 0 .U / D HDx .U0 / HD x

md;md .U0 / ' H Nmd;md .U 0 / and we have the residue Res' A .F ; EI U 0 / in HD x @ for each . From Proposition 5.7, we have the following residue theorem.

Theorem 6.7. In the above situation, if M is compact X .i / Res' A .F ; EI U 0 / D KS.' A .E// in H Nmd;md .M / ; @

where i W U 0 ,! M denotes the inclusion.

6.3 Atiyah classes on singular varieties In this subsection, we discuss Atiyah classes on singular varieties, similarly as for Chern classes (cf. [22], Chapter VI, 4). Let M be a complex manifold of dimension m and V a subvariety of pure dimension n of M. Let Sing.V / denote the singular set of V and V 0 D V n Sing.V / the regular part. First we assume that V is compact and let Uz be a neighborhood of V in M. Also, z˛ g˛2I a system of honey-comb let U D fUz˛ g˛2I be an open covering of Uz and fR 0 z˛0 ˛p . We cells adapted to U such that the regular part V of V is transverse to each R set z˛0 ˛p \ V: R˛0 ˛p D R Then we may define the integration Z n;n W HD x .U/ ! C V

as in [22], Chapter IV, 2. Also the bilinear pairing Ap;q .U/ Anp;nq .U/ ! An;n .U/ ! C defined as the composition of the cup product and the integration over V induces the Kodaira–Serre homomorphism on V : p;q np;nq KSV W H Np;q .Uz / ' HD .U/ ' H Nnp;nq .Uz / ; x .U/ ! HD x @

@

238

Tatsuo Suwa

which is not an isomorphism in general. The above homomorphism KSV sends the p;q np;nq .U/ class Œ in HD x .U/ to the functional assigning to each class Œ in HD x the value Z Y : V

Now suppose V may not be compact. Let S be a compact set in V. Let Uz0 be a neighborhood of V0 D V n S in M, Uz 0 a neighborhood of S in M and U0 D fUz˛ g˛2I 0 an open covering of Uz 0 . Set I D f0g [ I 0 , with the order 0 < ˛ for all ˛ 2 I 0 , and consider the open covering U D fUz˛ g˛2I of Uz D Uz0 [ Uz 0 , which is a neighborhood of V in M. p;q z We define Ap;q .U; Uz0 / and HD x .U; U0 / as in Section 5.3. Then we have the integration Z n;n z W HD x .U; U0 / ! C: V

The cup product induces the pairing Ap;q .U; Uz0 / Anp;nq .U0 / ! An;n .U; Uz0 /; which, followed by integration, gives a bilinear pairing Ap;q .U; Uz0 / Anp;nq .U0 / ! C: N This induces the “@-Alexander homomorphism over V ”: p;q np;nq z .U0 / ' H Nnp;nq .Uz 0 / ; AxV W HD x .U; U0 / ! HD x @

which is not an isomorphism in general. p;q z The homomorphism AxV sends the class Œ in HD x .U; U0 / to the functional which mp;mq 0 .U / the value assigns to each class Œ in HDx Z Y : V

From the above construction, we have the following: Proposition 6.8. In the above situation, if V is compact, the following diagram is commutative: p;q z HD x .U; U0 /

j

/ H p;q .U/ ' H p;q .Uz / o x N D

H Np;q .M /

@

AxV

@

KSV

H Nnp;nq .Uz 0 / @

i

/ H np;nq .Uz / N @

where i W Uz 0 ,! Uz denotes the inclusion.

KSM

/ H np;nq .M / ; N @


239

Moreover, if M is compact Kähler, we have the following commutative diagram / H pCq .M / o

H Np;q .M / @

@

H pCq .M /

PV

KSV

H Nnp;nq .M /

id

/ H2.npq/ .M / o

o PM

V

H2.mpq/ .M /

where V denotes the intersection product with V (cf. [25], 7.2). Remark 6.9. Suppose S contains Sing.V / and V0 D V n S admits a neighborhood Uz0 in M with a holomorphic retraction r W Uz0 ! V0 (cf. Remark 5.8). In this situation, to define 0 in a cochain in Ap;q .U/, we only need to define it on V0 , as we may take the pull-back by r. Again, let V be a variety of dimension m in a complex manifold M. First suppose V is compact and let Uz and U be as above. For a holomorphic vector bundle E over Uz and a homogeneous symmetric polynomial ' of degree d, we have the characterd;d d;d z .U /. We also have the class KSV .' A .E// in istic class ' A .E/ in HD x .U/ ' H@N md;md z H .U / . @N

Now let F be a singular distribution of rank r on M satisfying conditions (1) and (2) in Section 2.3 and let FV be the singular distribution on V induced from F . Set S D .S.F / \ V / [ Sing.V/ and V0 D V n S, which is a submanifold of M. There is a subbundle F0 of TM on M n S.F / defining F away from S.F /. We denote by FV0 the subbundle F0 jV0 of T V0 . Then it defines FV away from V. Let Uz0 , Uz 0 , U, and Uz D Uz0 [ Uz 0 be as above. We assume that Uz0 is a Stein neighborhood of V0 admitting a holomorphic retraction r W Uz0 ! V0 (cf. Remark 6.9). For a holomorphic vector bundle E over Uz and a homogeneous symmetric polynomial ' of degree d, the characteristic class ' A .E/ is represented by the cocycle z / in Ad;d .U/, where r z is a collection .rQ ˛ / of connections, each r z ˛ being a ' A .r z connection for E on U˛ . Note that it is sufficient if r0 is defined only on V0 D V n S, since by our assumption, the bundle EjUz0 is isomorphism to r EjV0 and we may take z 0 , the pull-back r r0 . as r Suppose there is an action of FV0 on EjV0 and let r0 be an FV0 -connection for z 0 / D r '.r0 / D 0, by Theorem 6.3, and EjV0 . Then, if d > n r, we have '.r z / is in Ad;d .U; Uz0 / and it defines a localization ' A .E; FV / the above cocycle ' A .r z0 U in H d;d .U; Uz0 / of ' A .E/ in H d;d .U/. We denote the class AV .' A .E; FV // in x D

x D

z0 U

md;md .U0 / ' H Nmd;md .Uz 0 / by Res' A .FV ; EI Uz 0 / and call it the residue HD x @ of ' A .E/ at Uz 0 with respect to FV. The residue is a functional described as above. If S has a finite number of connected components .S /, we break up U0 accordingly,

240

Tatsuo Suwa

U0 D [ U0 , so that we have a decomposition M md;md md;md 0 .U / D HDx .U0 / HD x

md;md .U0 / ' H Nmd;md .Uz 0 / and we have the residue Res' A .FV ; EI Uz 0 / in HD x @ for each . From Proposition 6.8, we have the following theorem.

Theorem 6.10. In the above situation, if V is compact, then X .i / Res' A .FV ; EI Uz 0 / D KSV .' A .E// in H Nmd;md .Uz / ; @

where i W Uz 0 ,! Uz denotes the inclusion.

6.4 An example We take up again the singular distribution F on P 3 considered in Example 4.3. Recall that it is an extension to P 3 of the distribution defined by the 1-form ! D z dx C z dy y dz in the affine space C3 D f.x; y; z/g. It leaves the plane fz D 0g invariant. From ! ^ d! D z dx ^ dy ^ dz, we see that ! defines a contact structure on C3 with singular set fz D 0g (Martinet hypersurface). We will see that the (first) Atiyah class of the normal bundle of the (projectivized) Martinet hypersurface is localized at the singular set of the corresponding distribution. The extended distribution F leaves the singular hypersurface V D f0 3 D 0g in P 3 invariant and we work on V. In fact F also leaves the hyperplane f3 D 0g invariant and this case is treated in [1]. Recall that the singular set S D S.F / of F has three irreducible components Si , i D 1; 2; 3. Note that S2 is the singular set of V, in fact V is a union of two projective planes crossing normally along S2 . We also recall that P1 D .0 W 1 W 0 W 0/ is the intersection point of S1 and S2 and P2 D .0 W 0 W 1 W 0/ is the intersection point of S2 and S3 . There is a subbundle F0 of rank two of T P 3 on P 3 n S such that F D O.F0 / away from S. The normal bundle NV 0 of the regular part V 0 of V canonically extends to a line bundle NV on V and then onto the whole P 3 , which is the line bundle LV associated to the divisor V. Since P 3 is compact Kähler, we know that the first Atiyah class a1 .LV / in H N1;1 .P 3 / D H 2 .P 3 / ' C coincides with first Chern class c 1 .LV /, which @ is 2c 1 .H3 /. Let V0 D V n S. Then it is in V 0 and has two connected components V0;1 and V0;2 , given by 3 D 0 and 0 D 0, respectively. Let Uz0;1 and Uz0;2 be tubular neighborhoods of V0;1 and V0;2 , respectively, sufficiently small so that they are disjoint. The restriction of the projection from the point .0 W 0 W 0 W 1/ to the plane f3 D 0g gives a holomorphic retraction r1 W Uz0;1 ! V0;1 and the restriction of the projection from the point .1 W 0 W 0 W 0/ to the plane f0 D 0g gives a holomorphic retraction


241

r2 W Uz0;2 ! V0;2 . Thus if we set Uz0 D Uz0;1 [ Uz0;2 , it is a tubular neighborhood of V0 with a holomorphic retraction r W Uz0 ! V0 . Let Uz1 , Uz2 and Uz3 be tubular neighborhoods of S1 n fP1 g, S2 n fP1 ; P2 g and S3 n fP2 g, respectively, sufficiently small so that they are pairwise disjoint. Also let Uz4 and Uz5 be balls around P1 and P2 , respectively, sufficiently small so that they are disjoint. Then U D fUz0 ; : : : ; Uz5 g is a covering of V and U0 D fUz1 ; : : : ; Uz5 g is that of S. We set Uz D Uz0 [ [ Uz5 and Uz 0 D Uz1 [ [ Uz5 . Let FV0 D F0 jV0 . We have NV0 D LV jV0 and it admits an FV0 -connection (Theorem 6.5), which can be lifted to Uz0 . Thus we have the localization a1 .LV jUz ; F / in 1;1 1;1 z 1;1 1 0 z HD z / in H@N .U / and its residue in HD x .U; U0 / of a .LV jU x .U / , which we will compute. Note that the Chern class c 1 .LV jUz / is not localized in this context. Let r be an FV0 -connection for NV0 on V0 and let r0 D r r. Let r1 ; : : : ; r5 z0 ; : : : ; R z5 g be a system of honey-comb be connections for LV on Uz1 ; : : : ; Uz5 . Let fR cells adapted to U whose boundaries are transverse to V, which will be given more zi \ V. specifically below. We set Ri D R We set D .i ; ij / with i D a1 .ri / and ij D a1 .ri ; rj / (note that the 1;1 0 form a1 .ri ; rj ; rk / is a .1; 1/-form and is zero). Let Œ be a class in HD x .U / represented by D . i ; ij /. Then the residue is a functional assigning to Œ the integral Z ^ D V

5 Z X iD1

C

Z i ^ i C

Ri

X

1i<j 5

Z

0i ^ i

R0i

Z

i ^ ij C ij ^ j

Rij

(6.1)

0i ^ ij : R0ij

Note that the possible combinations for .i; j / in the second term are .1; 4/, .2; 4/, .2; 5/, and .3; 5/. Since Uzi , 1 i 5, are Stein, as in [1], Lemma 9.9, we see that 1;1 0 each class Œ in HD x .U / is represented by a cocycle of the form D .0; ij /, where N i: i D @

ij D ij C i j ; With this the right hand side of (6.1) becomes Z X Z a1 .ri / ^ ij 1i<j 5

Rij

(6.2)

a1 .r0 ; ri / ^ ij ;

(6.3)

R0ij

the possible combinations of .i; j / being as above. Note that by symmetry, we need to consider only .1; 4/ and .2; 4/, the case for .3; 5/ being same as .1; 4/ and the case for .2; 5/ as .2; 4/. Thus to express the residue, we only need to find a1 .ri / and a1 .r0 ; ri / for i D 1; 2.

242

Tatsuo Suwa

Proposition 6.11. By a suitable choice of connections, a1 .ri / D 0;

0 i 2;

p 1 dx1 dz1 ; a .r0 ; r1 / D 2 x1 z1 z1 8p ˆ 1 dx1 dz1 1 ˆ ˆ 1 C on V0;1 ; ˆ < 2 z 1 x1 z1 1 a .r0 ; r2 / D p ˆ 1 ˆ dy1 ˆ ˆ .1 z1 / C dz1 on V0;2 : : 2 y1 1

Proof. Since the integrals are to be performed on Rij and R0ij that are in V0 , it suffices to determine connections for NV on V0 . Moreover, for .i; j / D .1; 4/ and .2; 4/, they are in W .1/ . Thus we fix r1 and r2 on the affine coordinate system .x1 ; y1 ; z1 /. We have the exact sequence, for i D 1; 2,

0 ! T V0;i ! T P 3 jV0;i ! NV0;i ! 0: We may take 1 D . @y@1 / as a frame of NV0;1 and 2 D . @x@1 / as a frame of NV0;2 . Let r1 be the connection trivial with respect to 1 . Then we have a1 .r1 / D 0 in W .1/ . The normal bundle NV of the hypersurface V has a frame , which is the canonical extension of the frame @x1@y1 on the regular part on W .1/ . Let r2 be the connection trivial with respect to . Then we have a1 .r2 / D 0 in W .1/ . Note that, on V0;1 , we have D x1 1 1 and on V0;2 , D y1 2 . 1

To compute the difference forms a1 .ri ; rj /, we make the following observation (cf. Section 5.1). Let i be the connection matrix (form, in this case) of ri with respect to some holomorphic frame of NV0 . Then, since the i ’s are of type .1; 0/, p 1 1 1 (6.4) a .ri ; rj / D c .ri ; rj / D .j i /: 2 Let r0 be an FV0 -connection for NV0 and compute its connection forms with respect to 1 and . On W .1/ , we may take the vector fields v1 D x1

@ @ C @x1 @z1

and

v2 D y1

@ @ C z1 @y1 @z1

as generators of F . We set u1;1 D v1 jV0;1 D x1 u1;2 D v1 jV0;2 D

@ @ @ C ; u2;1 D v2 jV0;1 D z1 ; @x1 @z1 @z1

@ ; @z1

u2;2 D v2 jV0;2 D y1

@ @ C z1 : @y1 @z1


243

In the above the restrictions are as sections. We find r0 on V0;1 . The connection form 0 of r0 with respect to 1 is of the form 0 D f dx1 C g dz1 , as it is of type .1; 0/. Then on the one hand we have r0 .1 /.u1;1 / D .x1 f C g/ 1 and r0 .1 /.u2;1 / D z1 g 1 . On the other hand by definition, h

r0 .1 /.u1;1 / D x1 and

h

r0 .1 /.u2;1 / D y1

@ @ i @ C ; jV D 0; @x1 @z1 @y1 0

@ i @ @ C z1 ; jV D 1 : @y1 @z1 @y1 0

Hence we get 0 D

dx1 dz1 ; x1 z1 z1

which gives the expression for a1 .r0 ; r1 / by (6.4). By similar computations we can find the connection form 0 of r0 with respect to , noting that, on V0;1 , it is of the form D f dx1 C g dz1 and, on V0;2 , it is of the form D f dy1 C g dz1 . Now we give the domains of integration more specifically. Let ı be a sufficiently small positive number and set z4 D f 2 P 3 j j0 j2 C j2 j2 C j3 j2 ı 2 j1 j2 g; R z5 D f 2 P 3 j j0 j2 C j1 j2 C j3 j2 ı 2 j2 j2 g; R z4 ; z1 D f 2 P 3 j j2 j2 C j3 j2 ı 2 j0 j2 g n Int R R z2 D f 2 P 3 j j0 j2 C j3 j2 ı 2 j1 j2 g \ f 2 P 3 j j0 j2 C j3 j2 ı 2 j2 j2 g R z5 /; z4 [ Int R n .Int R z5 ; z3 D f 2 P 3 j j0 j2 C j1 j2 ı 2 j3 j2 g n Int R R z0 D Uz0 n R

5 [ iD1

zi [ Int R

[

zij : Int R

ij

Now we express the domains R14 , R014 , R24 and R024 of integration explicitly. Note that the domains in question are all contained in W .1/ D f.x1 ; y1 ; z1 /g. In the ı2 sequel, we let ı 0 be a positive number with ı 0 2 D 1Cı 2. First, R14 D fy1 D 0; jx1 j2 C jz1 j2 D ı 2 ; jz1 j ıjx1 jg D fy1 D 0; jx1 j2 C jz1 j2 D ı 2 ; jz1 j ıı 0 g;

244

Tatsuo Suwa

which is a part of the boundary @R4 (in 3 D 0) of R4 and has the orientation opposite to that of @R4 . We also have R014 D f.x1 ; y1 ; z1 / j y1 D 0; jx1 j D ı 0 ; jz1 j D ıı 0 g; which coincides with @R14 and is oriented so that d argx1 ^ d argz1 is positive. Next, R24 is given by R24 D fx1 y1 D 0; jx1 j2 C jy1 j2 C jz1 j2 D ı 2 ; jx1 j2 C jy1 j2 ı 2 jz1 j2 g: The set R024 has two connected components and can be expressed as a disjoint union R024 D fjx1 j D ıı 0 ; y1 D 0; jz1 j D ı 0 g t fx1 D 0; jy1 j D ıı 0 ; jz1 j D ı 0 g: The first part is oriented so that d argx1 ^ d argz1 is negative and the second part so that d argy1 ^ d argz1 is negative. We have similar expressions for R035 and R025 . Now we consider the commutative diagram (cf. Proposition 6.8): 1;1 z HD x .U; U0 /

j

/ H 1;1 .Uz / o N @

AV

H N1;1 .Uz 0 / @

H N1;1 .P 3 / ' H 2 .P 3 / o

KSV

PV

/ H 1;1 .Uz / N

/ H 1;1 .P 3 / ' H2 .P 3 / o N

@

Id

@

@

H 2 .P 3 / o PP 3

V

H4 .P 3 /:

We denote by i the composition of the first two maps of the second row and try to 1;1 find i Resa1 .F ; NV I S/. Recall that the map H N1;1 .P 3 / ! HD x .U/ is induced by @

7! . i ; ij / D . ; 0/. Note that H N1;1 .P 3 / ' C, which is generated by the class of @ p 1 N @@ log kk2 : 0 D 2 On each affine coordinate system, we have 0 D

xN dx C yN dy C zN dz i i i i i i : @N 2 2 2 2 1 1 C jxi j C jyi j C jzi j

1 p

Thus we may take, as 1 , 2 and 4 in (6.2), the following forms. • Since Uz1 is in W .0/ , we may set 1 D

xN dx C yN dy C zN dz : 2 1 1 C jxj2 C jyj2 C jzj2 1 p

• Since Uz2 is both in W .1/ and W .2/ , for the sake of symmetry, we set xN 1 dx1 C yN1 dy1 C zN1 dz1 1 1 xN 2 dx2 C yN2 dy2 C zN2 dz2 p : C 2 D 2 2 1 1 C jx1 j2 C jy1 j2 C jz1 j2 1 C jx2 j2 C jy2 j2 C jz2 j2


245

• Since Uz4 is in W .1/ , we may set 4 D

xN 1 dx1 C yN1 dy1 C zN1 dz1 1 p : 2 1 1 C jx1 j2 C jy1 j2 C jz1 j2

On V1.0/ , z D y1 D x2 D 0 and we compute p 1 dx1 ; 14 D 1 4 D 2 x1 p 1 1 dz1 : 24 D 2 4 D 2 2 z1 On V2.0/ , x1 D y2 D 0 and we compute 24

p 1 1 dz1 D 2 4 D : 2 2 z1

Using Proposition 6.11, we compute (we omit the constant a1 .r0 ; r1 / ^ 14 D

1

a .r0 ; r2 / ^ 24 D

dz1 dx1 ^ ; z1 x1 8 1 dx1 dz1 1 ˆ ˆ ˆ < 2 .1 z 1 / x1 ^ z1 ; ˆ 1 dy1 dz1 ˆ ˆ : .1 z1 / ^ ; 2 y1 z1

p1 2 2

)

on V0;1 ; on V0;2 :

Therefore i Resa1 .F ; NV I S/ is a functional which assigns to the canonical generator Œ 0 two times the value Z

Z a1 .r0 ; r1 / ^ 14

R014

a1 .r0 ; r2 / ^ 24 D 1 C R024

1 1 C D2 2 2

so that i Resa1 .F ; NV I S/.Œ 0 / D 4, as expected. The above computation appears to suggest that the residue Resa1 .F ; NV I S/ is in fact “ŒS1 C 2ŒS2 C ŒS3 ”.

References [1]

M. Abate, F. Bracci, T. Suwa, and F. Tovena, Localization of Atiyah classes, to appear in Rev. Math. Iberoam. 208, 228, 234, 235, 240, 241

[2]

M. Abate, F. Bracci, and F. Tovena, Index theorems for holomorphic self-maps, Ann. of Math. (2) 159 (2004), 819–864. 207

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[3]

M. Abate, F. Bracci, and F. Tovena, Index theorems for holomorphic maps and foliations, Indiana Univ. Math. J. 57 (2008), 2999–3048. 207, 235

[4]

M. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 84 (1957), 181–207. 208

[5]

M. Atiyah and F. Hirzebruch, Analytic cycles on complex manifolds, Topology 1 (1961), 25–45. 219

[6]

P. Baum and R. Bott, Singularities of holomorphic foliations, J. Differential Geometry 7 (1972), 279–342. 208, 212, 215, 222

[7]

P. Baum, W. Fulton, and R. MacPherson, Riemann–Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 101–145. 219

[8]

R. Bott, Lectures on characteristic classes and foliations, Notes by L. Conlon, with two appendices by J. Stasheff, in Lectures on algebraic and differential topology. Delivered at the Second Latin American School in Mathematics, Mexico City, July 1971. Dedicated to the memory of Heinz Hopf, ed. by S. Gitler, Lecture Notes in Mathematics 279, Springer, Berlin and New York 1972, 1–94. 212, 213

[9]

R. Bott and L. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer, New York and Berlin 1982. 215

[10] J.-P. Brasselet, Définition combinatoire des homomorphismes de Poincaré, Alexander et Thom pour une pseudo-variété, in Caractéristique d’Euler–Poincaré. E.N.S. Seminar, 1978–1979. ed. by J.-L. Verdier, Astérisque 82-83, Société Mathématique de France, Paris 1981, 71–91. 220 [11] C. Camacho and P. Sad, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math. (2) 115 (1982), 579–595. 208 [12] R. F. Harvey and H. B. Jr. Lawson, A theory of characteristic currents associated with a singular connection, Astérisque 213, Société Mathématique de France 1993. 219 [13] F. Hirzebruch, Topological methods in algebraic geometry, Die Grundlehren der Mathematischen Wissenschaften 131, Springer-Verlag New York, New York 1966. 219 [14] B. Iversen, Local Chern classes, Ann. Scient. Éc. Norm. Sup. 9 (1976), 155–169. 219 [15] T. Izawa, Residues of codimension one singular holomorphic distributions, Bull. Braz. Math. Soc. (N.S.) 39 (2008), 401–416. 208, 224 ˇ [16] D. Lehmann, Systèmes d’alvéoles et intégration sur le complexe de Cech–de Rham, Publications de l’IRMA, 23, No VI, Université de Lille I, Lille 1991. 215, 231 [17] D. Lehmann, Résidus des sous-variétés invariantes d’un feuilletage singulier, Ann. Inst. Fourier (Grenoble) 41 (1991), 211–258. 208, 235 [18] D. Lehmann and T. Suwa, Residues of holomorphic vector fields relative to singular invariant subvarieties, J. Differential Geom. 42 (1995), 165–192. 208, 235 [19] J. Milnor and J. Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton University Press and University of Tokyo Press, Princeton and Tokyo 1974. 212 [20] R. S. Mol, Classes polaires associées aux distributions holomorphes de sous-espaces tangents, Bull. Braz. Math. Soc. (N.S.) 37 (2006), 29–48. 208 [21] T. Suwa, Residues of complex analytic foliation singularities, J. Math. Soc. Japan 36 (1984), 37–45. 208, 223, 224, 225


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[22] T. Suwa, Indices of vector fields and residues of singular holomorphic foliations,Actualités Mathématiques, Hermann, Paris 1998. 207, 210, 212, 213, 214, 215, 231, 237 [23] T. Suwa, Characteristic classes of coherent sheaves on singular varieties, in Singularities– Sapporo 1998. Proceedings of the International Symposium on Singularities in Geometry and Topology held at Hokkaido University, Sapporo, July 6–10, 1998, ed. by J.-P. Brasselet and T. Suwa, Advanced Studies in Pure Mathematics 29, Kinokuniya, Tokyo 2000, 279–297. 208, 217, 220 [24] T. Suwa, Residues of Chern classes on singular varieties, Singularités Franco–Japonaises. Papers from the 2 nd Franco–Japanese Singularity Conference held in Marseill–Luminy, September 9–13, 2002, ed. by J.-P. Brasselet and T. Suwa, Séminaires et Congrés 10, Société Mathématique de France, Paris 2005, 265–285. 225 [25] T. Suwa, Residue theoretical approach to intersection theory, in Real and complex singularities. Papers from the 9 th International Workshop held at the University of São Paulo, São Carlos, July 23–28, 2006., ed. by M. J. Saia and J. Seade, Contemporary Mathematics 459, American Mathematical Society, Providence, RI, 2008, 207–261. 213, 215, 220, 239 N ˇ [26] T. Suwa, Cech–Dolbeault cohomology and the @-Thom class, in Singularities–NiigataToyama 2007. Proceedings of the 4 th Franco-Japanese Symposium and the workshop held in Niigata, August 27–31, 2007, ed. by J.-P. Brasselet, S. Ishii, T. Suwa, and M. Vaquie, Advanced Studies in Pure Mathematics 56, Mathematical Society of Japan, Tokyo 2009, 321–340. 208, 229, 230, 231, 234

Two birational invariants in statistical learning theory Sumio Watanabe Precision and Intelligence Laboratory, Tokyo Institute of Technology 4259 Nagatsuta, Midori-ku, Yokohama, 226-8503 Japan email: [email protected]

Abstract. This paper introduces a recent advance in the research between algebraic geometry and statistical learning theory. A lot of statistical models used in information science contain singularities in their parameter spaces, to which the conventional theory can not be applied. The statistical foundation of singular models was been left unknown, because no mathematical base could be found. However, recently new theory was constructed based on algebraic geometry and algebraic analysis. In this paper, we show that statistical estimation process is determined by two birational invariants, the real log canonical threshold and the singular fluctuation. As a result, a new formula is derived, which enables us to estimate the generalization error without any knowledge of the information source. In the discussion, a relation between mathematics and the real world is introduced to pure mathematicians.

1 Introduction The purpose of this paper is to introduce a singularity problem in statistics to researchers of mathematics. A lot of statistical models used in artificial intelligence, information science, neuroscience, natural language processing, and bioinformatics contain singularities in their parameter spaces. A singularity is most important in such models because it corresponds to the knowledge to be discovered. In other words, singularities determine the statistical estimation and statistical hypothesis test, hence we need singular theory on which new statistical theory will be established. Let .; B; P / be a probability space and X W ! RN be a random variable whose probability distribution is represented by q.x/dx. Let X1 ; X2 ; :::; Xn be independent random variables which are subject to the same probability distribution as X . The expectation operators EX Œ and EŒ are defined by Z EX ŒF .X / D F .x/q.x/dx; Z EŒF .X1 ; X2 ; : : : ; Xn / D F .x1 ; x2 ; :::; xn /

n Y iD1

q.xi /dxi ;

250

Sumio Watanabe

for functions F .x/ and F .x1 ; x2 ; :::; xn /. Let W be a subset of Rd. A statistical model is defined by a pair .p.x j w/; '.w//, where p.x j w/ is a probability density function of x 2 RN for a given parameter w 2 W Rd and '.w/dw is a probability density function on W. The expectation operator Ew Œ is defined by Z n Y F .w/ p.Xi jw/ˇ '.w/dw Ew ŒF .w/ D

iD1

Z Y n

; ˇ

p.Xi jw/ '.w/dw

iD1

where 0 < ˇ < 1 is a constant. Note that Ew Œ depends on X1 ; X2 ; : : : ; Xn , hence Ew ŒF .w/ is not a constant but a random variable. The distribution-valued random variable p .x/ Ew Œp.x j w/ is called a predictive distribution. Main Problem. Let G and T be real valued random variables defined by G D EX Œlog p .X /; 1X log p .Xi /: n n

T D

iD1

The two random variables G and T are called generalization and training error, respectively. We ask what mathematical properties of the triple .q.x/; p.x j w/; '.w// determine the asymptotic behavior of G and T as n ! 1. In this paper, we show that the above problem has a close relation to singularity theory. On the other hand, it has direct applications in statistics. In Section 8, the reason why such a problem is important in statistics is explained. Example 1 (Regular case). Assume that N D d , W D Rd and that p.x j w/ D

1 1 2 exp ; kx wk 2 .2/N=2

q.x/ D p.x j 0/; '.w/ D

1 1 2 kwk exp : 2 .2/d=2

In this case, the set of parameters which make p.xjw/ be equal to q.x/ is trivial, fw 2 W I p.x j w/ D q.x/g D f0g:

Two birational invariants in statistical learning theory

251

Such a case can be analyzed by the conventional statistical method. The entropy S and the empirical entropy Sn of q.x/ respectively defined by S D EX Œlog q.X /; 1X log q.Xi /: n n

Sn D

iD1

Then EŒSn D S, and 1 1

1 X

2

p Xi C Op 2 ; 2n n n n

GDSC

iD1

1 1

1 X

2

p Xi C Op 3=2 ; T D Sn 2n n n n

iD1

where Yn D Op .1=n / if and only if n Yn converges in law to some random variable when n ! 1. Moreover, it is easy to show that ˛

˛

lim EŒn.G S/ D

n!1

d ; 2

d lim EŒn.T Sn / D : 2

n!1

By this example, we have three conjectures. (1) The main problem of this paper has a relation to the central limit theorem. (2) The asymptotic behaviors of G and T are determined by some invariants such as the dimension of the triple .q.x/; p.x j w/; '.w//. (3) There is some symmetry between EŒG and EŒT . Example 2 (Singularities in statistics). Assume that N D 2, W D R4 , x D .x1 ; x2 /, and w D .a; b; c; d / and that p.xjw/ D

1 1 2 q .x / exp a sin.bx / c sin.dx /k ; kx 0 1 2 1 1 2 .2/1=2

q.x/ D p.x j 0/; '.a; b; c; d / D

a2 C b 2 C c 2 C d 2 1 exp ; .2/2 2

where q0 .x1 / is an arbitrary probability density function. In this case, the set of parameters which make p.x j w/ be equal to q.x/ is a real algebraic set, fw 2 R4 I p.x j w/ D q.x/g D fw 2 R4 I ab C cd D ab 3 C cd 3 D 0g;

252

Sumio Watanabe

which contains a singularity at the origin. Although the same cases often occur in statistics, no method has been found by which G and T can be analyzed. In this paper, we show that the asymptotic behaviors of G and T are determined by the two birational invariants of singularities. Acknowledgment. The author would like to thank a lot of mathematicians for their kind teaching mathematical concepts. Without singularity theory, our research results could not be obtained. Modern mathematics is necessary even in information science.

2 Preparation In this section, we prepare mathematical definitions and assumptions, and explain the singularity problem in statistics more precisely. The function L.w/ is defined by L.w/ D EX Œlog p.X jw/: By using the definition S D EX Œlog q.X /, it follows that L.w/ S for arbitrary w 2 W. It is assumed that there exists w0 2 W o that minimizes L.w/, where W o is the largest open set that is contained in W. A nonnegative function K.w/ is defined by K.w/ D L.w/ L.w0 /: Then the set W0 D fw 2 W I K.w/ D 0g is not the empty set. It is assumed that, for arbitrary w 2 W0 , p.x j w/ is the same probability distribution, which is referred to as p0 .x/. Also we use notations, L0 D L.w0 / D EX Œlog p0 .X /: To study the problem, we need some assumptions. Assumptions. Let W be a compact set and assume that the largest open set contained in W is not the empty set. It is defined by several analytic functions 1 .w/; 2 .w/; : : : ; l .w/, W D fw 2 Rd I 1 .w/ 0; 2 .w/ 0; : : : ; l .w/ 0g: Let W be an open set such that W W Rd . Let s 3 be a constant and ° Z 1=s ± jf .x/js q.x/dx Ls .q/ D f I kf k

0 such that Z . sup jf .x; w/j2 /. sup p.xjw//dx < 1: w2W

K.w/ 0. Therefore, we obtain the asymptotic expansion [27] and [28], for n ! 1 Z log exp.nK.w//'.w/dw D log n .m 1/ log log n C O.1/; where O.1/ is a bounded function of n.

4 Singular fluctuation Definition 3. If there exists a constant A > 0 such that, for an arbitrary w 2 W EX Œf .X; w/ AEX Œf .X; w/2 ; then f .X; w/ is said to have a relatively finite variance. Since W is compact and K.w/ D EX Œf .X; w/, this definition is ensured if it holds in the region W W D fw 2 W I K.w/ < g; where > 0 is sufficiently small constant. The following lemmas can be proved immediately. Lemma 4.1. If q.x/ is realizable by p.x j w/, then f .X; w/ has a relatively finite variance. Lemma 4.2. If q.x/ is regular for p.x j w/, then f .X; w/ has a relatively finite variance.

256

Sumio Watanabe

If q.x/ is unrealizable by and singular for p.x j w/, then f .X; w/ may and may not have a relatively finite variance. The case when f .X; w/ does not have a relatively finite variance is studied in Section 6. In this section, we study the case when f .X; w/ has a relatively finite variance. By using the resolution of singularities, K.g.u// is a normal crossing function, K.g.u// D u2k . By using K.g.u// D EX Œf .X; g.u// AEX Œf .X; g.u//2 ; there exists an Ls .q/-valued analytic function a.x; u/ such that f .x; g.u// D a.x; u/uk : Therefore EX Œa.X; u/ D uk . Remark that such a function a.x; u/ is well-defined on M, whereas a.x; g 1 .w// is not on W in general. This is one of the reasons why resolution of singularities is necessary in statistics. Let us define a random process n .u/ on M by n 1 X fa.Xi ; u/ uk g: n .u/ D p n iD1 Let B.M/ be a Banach space defined by B.M/ D ff .u/ is continuous I kuk sup jf .u/j < 1g: u2M

Since M is a compact set, B.M/ is a Polish space; in other words, it is a complete and separable metric space. By using a.x; u/ is an Ls .q/-valued analytic function of u, it follows that the set of random processes fn gnD1;2;::: is uniformly tight in B.M/. Then by applying Prohorov’s theorem, n .u/ converges in law to a tight Gaussian process .u/ on B.M/, by using the uniqueness of a Gaussian process that satisfies E Œ.u/ D 0; E Œ.u/.u0 / D EX Œa.X; u/a.X; u0 / EX Œa.X; u/EX Œa.X; u0 /; where E Œ shows the expectation value over the random process .u/. Such a process n .u/ is called an empirical process or it is said that n .u/ satisfies the central limit theorem on the Banach space. An expectation operation h i on M is defined by Z 1 Z p dt D.u/du F .u; t/t 1 exp.ˇt C ˇ t.u// : hF .u; t/i D 0 Z 1 Z p dt D.u/du t 1 exp.ˇt C ˇ t.u// 0

The value hF .u; t/i is a functional of .u/, hence it is also a random variable, but it does not depend on n.


257

Definition 4. The singular fluctuation > 0 is defined by D

p ˇ E EX Œhta.X; u/2 i h t a.X; u/i2 : 2

In general, depends on ˇ. Lemma 4.3. The singular fluctuation is a birational invariant. Proof. Let us define a random variable V by V D

n X

fEw Œ.log p.Xi jw//2 Ew Œlog p.Xi jw/2 g:

iD1

Then we can prove that V converges in law to a random variable by the same way as the proof of the following theorem and it satisfies ˇ EŒV ; n!1 2

D lim

which shows that is a birational invariant.

Then we have the following theorem. Theorem 4.4 (Main theorem). Assume that f .X; w/ has a relatively finite variance. Then both random variables n.G L0 / and n.T Ln / converge in law. Also the following convergences hold: lim EŒn.G L0 / D

C ; ˇ

lim EŒn.T Ln / D

: ˇ

n!1

n!1

Outline of proof. For the complete version of the proof, see [28] and [29], and [32]. Firstly, we prove the convergences in law. We define a distribution Yn .w/dw by Yn .w/dw exp.nˇKn .w// '.w/ dw: Then by using the fact that 1 Kn .g.u// D u2k p n .u/ uk ; n

258

Sumio Watanabe

we have Yn .w/dw D Yn .g.u//jg 0 .u/jdu 2k

p

k

D e nˇ u C nˇ u n .u/ uh b.u/du Z 1 p dt ı.t n u2k /uh e ˇ tC tˇ n .u/ b.u/du D 0

.log n/m1 D.u/du Š n

Z

1

dt t 1 e ˇ tC

p

tˇ .u/

;

0

where Š shows the asymptotic expansion of Schwartz distribution as n ! 1. Therefore the convergence in law holds, p p Ew Œ. nf .x; w//s ! h. t a.x; u//s i for s 0. Here the relations of the parameters are w D g.u/; t D nK.w/ D nu2k ; f .x; w/ D a.x; u/uk : The generalization error is h i 1 1 G D EX log Ew Œ1 f .X; w/ C f .X; w/2 C op 2 n 1 1 1 D EX Ew Œf .X; w/ EX Ew Œf .X; w/2 C EX ŒEw Œf .X; w/2 C op ; 2 2 n where op .1=n/ is a random variable which satisfies the convergence in probability, n op .1=n/ ! 0. Hence G converges in law. The convergence of T in law can be proved by the same way. Secondly, let us show the convergences of expectation values. The Gaussian process .u/ can be represented by 1 X cj .u/gj .u/ D j D1

where fgj g are independent random variables, each of which gj is subject to the standard normal distribution. Then 1 X E Œ.u/.u0 / D cj .u/cj .u0 /: j D1

Let us introduce a generating function Z h i Fn .˛/ D EEX log exp.˛f .X; w/ ˇnKn .w//'.w/dw :


259

Then EŒG D L0 C Fn .1/ Fn .0/; EŒT D L0 C Fn1 .1 C ˇ/ Fn1 .ˇ/; 00 .ˇ/: EŒV D n Fn1

We need the asymptotic behavior of Z Zn .s/ D f .x; w/s exp.ˇnHn .w//'.w/dw; where s 0 is a real value. For example, h Z .1/ i n ; Fn0 .0/ D E Zn .0/ h Z .1/ i2 h Z .2/ i n n Fn00 .0/ D E CE : Zn .0/ Zn .0/

(4.1) (4.2)

By the same method as above,

where

R

Z p .log n/m1 s=2 Zn .s/ Š D.u; t/t exp.ˇ t.u// ; n Cs=2

D.u; t/ is defined by the integration over the manifold, Z Z 1 Z D.u; t/ D dt duD.u/ t 1 exp.ˇt/: 0

Let us define y r; s/ D Z.q;

Z

p D.u; t/ .u/q t r=2 a.x; u/s exp.ˇ t.u//:

Then Zn .s/ Š

.log n/m1 y Z.0; s; s/: n Cs=2

(4.3)

Firstly, since EX Œa.X; u/ D uk , y 1; 1/ D Z.0; y 2; 0/: EX ŒZ.0; Secondly, by using the partial integration of t Z Z 1 p 1 1 ˇ tCˇ pt.u/ dt t e ˇ tCˇ t.u/ D dt t e ˇ 0 0 Z p 1 1 1=2 C dt t .u/e ˇ tCˇ t .u/ ; 2 0 it follows that

y 0; 0/ C 1 Z.1; y 1; 0/: y 2; 0/ D Z.0; Z.0; ˇ 2

260

Sumio Watanabe

And lastly, by using the partial integration over the Gaussian process .u/, "Z # p 1 h Z.1; X y 1; 0/ i t 1=2 e ˇ t.u/ D E D.u; t/ cj .u/gj Z E p 0 y 0; 0/ 0 Z.0; j D1 D.u0 ; t 0 /e ˇ t .u / "Z

1 X

D.u; t/

D E

j D1

D ˇEX E

@ Z cj .u/ @gj

h Z.0; y 2; 2/ i

y 0; 0/ Z.0;

ˇEX E

t 1=2 e ˇ

p

t.u/

D.u0 ; t 0 /e ˇ

p

#

t 0 .u0 /

h Z.0; y 1; 1/ i2

y 0; 0/ Z.0;

D 2; where we used E Œ.u/.u0 / D EX Œa.X; u/a.X; u0 / on the set fuI K.g.u// D 0g. Then by using eq. (4.3), lim

sup

n!1 0˛1Cˇ

jFn.3/ .˛/jn D 0;

0 .0/jn D 0; lim jFn0 .0/ Fn1

n!1

00 .0/jn D 0: lim jFn00 .0/ Fn1

n!1

At last, by (4.1) and (4.2), 1 1 EŒG D L0 C Fn0 .0/ C Fn00 .0/ C o ; 2 n

EŒT D L0 C Fn0 .0/ C

1 2ˇ C 1 00 ; Fn .0/ C o 2 n

EŒV D nFn00 .0/ C o.1/; which completes the theorem. In the special case ˇ D 1, the predictive distribution p .x/ is called Bayes estimation in statistics. The average expectation value of the generalization error is given by 1 EŒG D L0 C C o ; n n where o.1=n/ is a smaller order term than 1=n. This function shows the accuracy of Bayes estimation, which is called the learning curve in statistical learning theory. The learning curve is in inverse proportion to the number of random samples, and its coefficient is equal to the real log canonical threshold. Therefore, the learning curve is determined by the singularities. From Theorem 4.4, by eliminating and , we obtain the following formula.


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Corollary 4.5 (Equation of state in statistical estimation). Assume that f .X; w/ has a relatively finite variance. The following formula holds. EŒG D EŒT C

1 ˇ EŒV C o : n n

(4.4)

This equation holds for arbitrary triple .q.x/; p.x j w/; '.w//. The method how to apply this formula to the practical problems is explained in Section 8.

5 Two birational invariants Theorem 4.4 shows that the learning process is determined by the two birational invariants. In this section, we discuss the mathematical properties of them.

5.1 Real log canonical threshold The original concept of the real log canonical threshold was found in the problem of the asymptotic expansion [8] of the singular Schwartz distribution ı.t f .x// as t ! 0 for a polynomial f .x/ that satisfies rf .x/ D 0. Remark that, if rf .x/ D 0 on f .x/ D 0, then ı.f .x// is not well-defined. It was proved that resolution of singularities gives a general solution to this problem [4]. The Bernstein–Sato polynomial was found to study this problem from the algebraic analysis point of view; see [5], [12], [19], [17], and [20]. The log canonical threshold plays an important role in higher dimensional algebraic geometry [13]. Its algebraic property was studied in [14], [15], [21], and [10]. Application to the oscillating integral was proposed in [26]. The relation between the real log canonical threshold and Bayes integral was found [27], which was applied to several statistical models; see [6], [28], [3], [33], and [34]. It is well known that the log canonical threshold shows the relative quantity of two algebraic varieties W and W0 , or it is defined for a pair .W; W0 /. In statistical learning theory, they correspond to the set of parameters W and the set of the optimal parameters W0 . In this paper, we have shown that the learning curve is determined by the mathematical relation between W and W0 .

5.2 Singular fluctuation From the theoretical point of view, it is still unknown what singular fluctuation is. For a given function f .x; w/, its average and covariance are defined by K.w/ D EX Œf .X; w/; .w; w 0 / D EX Œf .X; w/f .X; w 0 /:

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If two sets of triples .qi .x/; pi .x j w/; 'i .w// .i D 1; 2/ have the same K.w/ and .w; w 0 /, then they have the same real log canonical threshold and singular fluctuation. It is well known that the central limit theorem is determined by the average and the covariance of the original random variable. Singular fluctuation may express the variance information on the functional space.

6 Case study In this section, we show some examples of the real log canonical thresholds and singular fluctuations.

6.1 Regular and realizable case If q.x/ is regular for and realizable by p.xjw/, and if '.w/ > 0 at W0 , then D D d=2, where d is the dimension of the parameter space. In this case, does not depend on ˇ. Example 1 is a special case.

6.2 Regular and unrealizable case If q.x/ is regular for but unrealizable by p.xjw/, and if '.w/ > 0 at W0 , then D

d ; 2

D

1 tr.IJ 1 /; 2

can be proven [30], where I , J are d d matrices respectively defined by Z I D rw log p.x j w0 /rw log p.x j w0 /q.x/dx; Z J D

2 rw log p.x j w/q.x/dx:

In this case, does not depend on ˇ. Note that there are both cases > and < .

6.3 Singular and realizable case In order to construct the model selection algorithm or the hypothesis testing in statistics, this case is most important.


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In general, and depend on q.x/ and p.x j w/. Moreover, depends on ˇ. For example, w D fak ; bk I k D 1; 2; : : : ; H g H 1 2 X 1 p.x; y j w/ D q .x/ exp y a sin.b x/ 0 k k 2 .2/1=2 kD1

and q.x/ D p.x; yj0/, then the real log canonical threshold is equal to that of K0 .w/ D

H X H X hD1

2

ak .bk /2h1 :

kD1

In this case, a resolution of singularities was found [3] and p p Œ H 2 C Œ H C 1 p ; D 4Œ H C 2 where Œx shows the maximum integer that is not larger than x. Example 2 is a special case of this. Hence the real log canonical threshold of Example 2 is equal to 2=3. In practical applications, statistical models have the same type function as K0 .w/. In other words, its Newton diagram is degenerate and its complexity grows when the dimension of the parameter becomes large. In the above case, it follows that p H ; Š 4 when H is sufficiently large. It is the future study to clarify the behavior of the real log canonical threshold when the complexity and the dimension of the polynomial tends to infinity. This problem might have relation to random matrix theory.

6.4 Singular and unrealizable case If f .X; w/ does not have a relatively finite variance, then there exists an example, in which Theorem 4.4 does not hold. For example, q.x; y/ D p.x; yja/ D

1 1 exp .x 2 C y 2 / ; 2 2 1 p 1 exp f.x a/2 C .y a4 a2 C 1/2 g ; 2 2

where a 2 R1 is the parameter. It is easy to check that f .X; w/ does not have a relatively finite variance. By the direct calculation, we have EŒG Š L0 C EŒT Š L0

1

2 1

2

1 Q ; ˇ n2=3

C

1 Q ; ˇ n2=3

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Sumio Watanabe 7=6

2 where Q D p . 76 /, therefore Theorem 4.4 does not hold [32]. However, the 2 equation of state in statistical estimation holds,

1 ˇ EŒV C o 2=3 : n n The equation of state in statistical estimation may hold more universally.

EŒG D EŒT C

7 Problems in probability theory In this section, we discuss the mathematical problem in probability theory. From the viewpoint of probability theory, this paper is based on two strong assumptions. The former is that w 7! f .x; w/ is an Ls .q/-valued analytic function, and the latter is that the set of parameters W is compact. The assumption that f .x; w/ is analytic is necessary because the resolution theorem is employed. If it is not an analytic function, it is unknown whether the theorem can be generalized or not. It is the future study to generalize the results of this paper to non-analytic functions. The assumption that W is a compact set is necessary to show that B.M/ is a Polish space and that n .u/ is a uniformly tight process. The convergence in law n .u/ ! .u/ was proved using Prohorov’s theorem. If W is not compact, then M is not compact in general, hence the convergence in law of n .u/ can not be proved in general. It is also the future study to prove the convergences in law of G and T without using the convergence in law of n .u/.

8 Application to statistics In this section, the application of singularity theory to statistics is introduced to researchers of mathematics. Readers who are not interested in applications can skip this section.

8.1 Background of the problem Firstly, we discuss the background of the problem. The probability distribution q.x/dx from which random variables X1 ; X2 ; :::; Xn are taken is called a true distribution or a true information source. In real world problems, the true distribution is unknown in general, and only a set of sample values of X1 ; X2 ; : : : ; Xn can be observed. To estimate the unknown true distribution, a pair .p.x j w/; '.w// is employed, which is called a probabilistic model, a statistical model, or a learning machine. One of the


265

main purposes of statistical learning theory is to establish the method how to evaluate the probabilistic model compared to the unknown true distribution. The predictive distribution p .x/ Ew Œp.xjw/ is the estimated probability density function for a given set of samples and a given model .p.xjw/; '.w//. In practical applications, although people never know the true distribution q.x/, they want to evaluate how accurately p .x/ approximates q.x/. It seems impossible to make such an evaluation. The random variable G is a quantitative measure of the accuracy of p .x/. If p .x/ D q.x/, the generalization error G takes the minimum value which is equal to the entropy of q.x/. If otherwise, G is larger than the entropy. Hence the smaller G means that p .x/ is more accurate for q.x/. However, in order to obtain G, we have to calculate EX Œ but this integration can not be performed without the true distribution q.x/. Instead of G, we can calculate the training error T, using only samples and the model. We ask whether we can estimate G from T. The equation of state in statistical estimation, Corollary 4.5, EŒG D EŒT C.ˇ=n/EŒV claims that EŒG can be obtained from EŒT and EŒV , where T and V can be calculated only samples and the model without any knowledge of the true distribution. Since this formula holds for an arbitrary set .q.x/; p.x j w/; '.w//, one can evaluate the statistical model .p.x j w/; '.w//.

8.2 New results Secondly, let us discuss what the original points of this paper are. If a true distribution is regular for and realizable by a statistical model, then the equation of state in statistical estimation is equivalent to the AIC (Akaike information criterion); see [1]. If a true distribution is regular for and unrealizable by a statistical model, then it is equivalent to TIC (Takeuchi information criterion). Therefore, if a true distribution is regular for a statistical model, then the obtained result of this paper contains the conventional results. If a true distribution is singular for a statistical model, there has been no formula by which we can estimate the generalization error. Hence the equation of state is the first result by which we can estimate the generalization error in singular cases. Table 1 shows the mathematical difference between the regular and singular statistical theory. The regular theory studies the probability distribution on the parameter space, whereas the singular theory does that on the functional space.

9 Conclusion In this paper, we introduced two birational invariants by which we can estimate the generalization error without any knowledge about the true distribution. Singularity theory is essential to statistical learning theory.

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Sumio Watanabe Table 1. Regular and singular algebra geometry analysis probability theory Fisher inform. matrix Cramer–Rao inequality Maximum likelihood Bayes a posteriori log canonical threshold singular fluctuation Bayes marginal information criterion examples

Regular linear algebra differential real-valued central limit theorem positive definite holds asymptotic normal asymptotic normal d=2 d=2 .d=2/ log n AIC, TIC exponential polynomial regression linear prediction

Singular ring and ideal algebraic function-valued empirical process semi-positive def. no meaning singular singular log n equation of state mixtures neural networks hidden Markov

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Frobenius morphisms of noncommutative blowups Takehiko Yasuda Department of Mathematics Graduate School of Science, Osaka University Toyonaka, Osaka 560-0043, Japan email: [email protected]

Abstract. We define the Frobenius morphism of certain class of noncommutative blowups in positive characteristic. Thanks to a nice property of the class, the defined morphism is flat. Therefore we say that the noncommutative blowups in this class are Kunz regular. One of such blowups is the one associated to a regular Galois alteration. As a consequence of de Jong’s theorem, we see that for every variety over an algebraically closed field of positive characteristic, there exists a noncommutative blowup which is Kunz regular. We also see that a variety with F-pure and FFRT (finite F-representation type) singularities has a Kunz regular noncommutative blowup which is associated to an iteration of the Frobenius morphism of the variety.

1 Introduction The Frobenius morphism is arguably the most important notion in the algebraic geometry of positive characteristic and used almost everywhere. Concerning the singularity theory, Kunz’s theorem is classical [11]: A scheme is regular if and only if its Frobenius morphism is flat. The main aim of this article is to define the Frobenius morphism of certain class of noncommutative blowups in positive characteristic and to see that the defined morphism is flat. Here the noncommutative blowup that we mean is basically the same as the noncommutative crepant resolution in [16] and the noncommutative desingularization in [2] except that we remove some assumptions, especially the finiteness of global dimension. Let k be a field of characteristic p > 0. Recently it was found in [15] that if X D Spec R belongs to some classes of singularities over k, then for sufficiently e large e, the endomorphism ring EndR .R1=p /, whose elements are differential ope erators on R1=p , has finite global dimension and is regarded as a noncommutative resolution of X. This article derives from the author’s attempt to know where the rege ularity of EndR .R1=p / comes from and to show its regularity for a broader class of singularities. However the regularity which we will consider in this article is the flatness of Frobenius rather than the finiteness of global dimension. It is because the This

work was supported by Grant-in-Aid for Young Scientists (20840036) from JSPS.

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former seems to the author simpler and compatible with EndR .R1=p /. We however e consider not only noncommutative blowups of the form EndR .R1=p /. Let X; Y be integral normal Noetherian schemes over k with finite Frobenius morphisms and f W Y ! X a finite dominant morphism. We associate to f a noncommutative blowup NCB.Y =X /, which is the pair of the endomorphism ring EndOX .OY / and the left EndOX .OY /-module OY . (More generally we will consider the noncommutative blowup associated to a coherent sheaf. However the examples in which we are interested are associated to finite morphisms of schemes.) Also we regard this as the category of left EndOX .OY /-modules with the distinguished object OY . Let Ye ! Y be the e-times iteration of the Frobenius morphism. We say that f is F-steady if for every e 0, the structure sheaves of Ye and Y locally have, as OX -modules, the same summands (for details, see Section 4). For instance, if Y is regular, then f is F-steady. Given an F-steady morphism Y ! X, we define the Frobenius morphism of NCB.Y =X /, which is flat by construction. Hence we say that NCB.Y =X / is Kunz regular. If k is algebraically closed, from de Jong’s theorem [4], every k-variety X admits a Galois alteration Y ! X with Y regular. It uniquely factors as Y ! Yx ! X such that Yx is a normal variety, Y ! Yx is finite and Yx ! X is a modification. Then the associated noncommutative blowup NCB.Y =X / D NCB.Y =Yx / is Kunz regular. Thus every variety admits a noncommutative blowup which is Kunz regular (Corollary 4.5). Another interesting example of noncommutative blowups is the one associated to an iterated Frobenius morphism Xe ! X of a normal scheme X. In the affine case, e this corresponds to the above-mentioned ring EndR .R1=p /. If X has only F-pure and FFRT (finite F-representation type) singularities, then for sufficiently large e, Xe ! X is F-steady and the associated noncommutative blowup NCB.Xe =X / is Kunz regular (see Section 5). The FFRT singularity was introduced in [13] and proved to have Dmodule theoretic nice properties; see [13], and [14]. Our result is yet another such property.

1.1 Convention Throughout the paper, we work over a fixed base field k unless otherwise noted. We mean by a scheme a separated Noetherian scheme over k. In Sections 4, 5 and 6, we additionally assume that k has characteristic p > 0 and that every scheme is F-finite, that is, the Frobenius morphism is finite. If f W Y ! X is an affine morphism of schemes and M is a quasi-coherent sheaf on Y , then by abuse of notation, we denote the push-forward f M again by M.

Frobenius morphisms of noncommutative blowups

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2 Noncommutative schemes 2.1 Pseudo-schemes Following [1], p. 235, we first define the category PS. Definition 2.1. A pseudo-scheme is the pair .A; M / of a k-linear abelian category A and an object M 2 A. A morphism f W .A; M / ! .B; N / of pseudo-schemes is the equivalence class of pairs .f ; / of a k-linear functor f W B ! A which admits a right adjoint f W A ! B and an isomorphism W f M Š N . Here two such pairs .f ; / and ..f /0 ; 0 / are equivalent if there is an isomorphism f Š .f /0 which is compatible with and 0 . The composition of morphisms is defined in the obvious way. We denote the category of pseudo-schemes by PS. A morphism f is said to be flat if its pull-back functor f is exact. For a scheme X, we denote by Qcoh.X / the category of quasi-coherent sheaves on X. We have a natural functor .scheme/ ! PS; X 7! X ps D .Qcoh.X /; OX /: From a theorem of Gabriel [5], we can reconstruct X from X ps (which was generalized to the non-Noetherian case by Rosenberg [12]). Theorem 2.2 (Reconstruction of schemes). If X ps Š Y ps , then X Š Y . We can also reconstruct morphisms. Proposition 2.3 (Reconstruction of morphisms). The functor X 7! X ps is faithful. Proof. Suppose that f W Y ! X D Spec A be a morphism of schemes with X affine. Then f ps determines a k-algebra map A D End.A/ ! .OY / D End.OY / and so determines f. Next suppose that f W Y ! X be an arbitrary morphism of schemes. Then applying f to the structure sheaves of integral closed subschemes of Y, we see that f ps uniquely determines f as the map of sets. For each affine open subset W U ,! X, applying f to the sheaves M, M 2 Qcoh.U /, we see that f ps uniquely determines the scheme morphism f jf 1 .U / W f 1 .U / ! U. As a consequence, f ps uniquely determines f. Hence the functor is faithful. The above results allow us to identify a scheme X (resp. a scheme morphism f ) with X ps (resp. f ps ).

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2.2 Noncommutative schemes Definition 2.4. Let Z be a scheme. A finite NC (noncommutative) scheme over Z is the pair X D .A; M/ of a coherent sheaf A of OZ -algebras and a coherent sheaf M of left A-module. We denote by Qcoh.X / D Qcoh.A/ the category of quasi-coherent left A-modules and set X ps D .Qcoh.X /; M/. Like a scheme, we often identify X and X ps . A morphism X D .A; M/ ! X 0 D .A0 ; M0 / of finite NC schemes over Z is a morphism X ps ! .X 0 /ps defined by the functor N ˝A0 W Qcoh.X 0 / ! Qcoh.X / for some coherent sheaf N of .A; A0 /-bimodules and an isomorphism N ˝A0 M0 Š M: Note that the functor has the right adjoint HomA .N ; / and indeed defines a morphism in PS. We do not construct the correct category of finite NC schemes over different schemes in this article. Instead we will work in the ambient category PS.

3 Alterations and noncommutative blowups Definition 3.1. A morphism Y ! X of integral schemes is called an alteration (resp. modification) if it is generically finite, dominant and proper (resp. birational and proper). An alteration Y ! X is said to be normal (resp. regular) if Y is so. A finitebirational factorization of a normal alteration Y ! X is a factorization of Y ! X into a finite and dominant morphism Y ! Yx and a modification Yx ! X with Yx normal. (This is clearly unique up to isomorphism if exist.) A normal alteration is said to be factorizable if it admits a finite-birational factorization. An alteration f W Y ! X is said to be Galois if there exists a finite group G of automorphisms of Y such that f is G-equivariant under the trivial G-action on X and the field extension K.Y /G =K.X / is purely inseparable. Lemma 3.2. Let f W Y ! X be a normal Galois alteration. Suppose that the quotient algebraic space Y =G is a scheme. In the case where k has positive characteristic, we suppose that Y =G is F-finite. Then f is factorizable. Proof. In characteristic 0, the natural morphism Y =G ! X is birational, hence f is factorizable. Let us suppose that k has characteristic p > 0. We take e 2 Z0 such e that .K.Y /G /p K.X /. Let Y =G be the quotient variety and Y =G ! .Y =G/e the e morphism corresponding to the inclusion OYp=G ,! OY =G of sheaves, though this is not a morphism of k-schemes unless k is perfect. Let Yx be the normalization of .Y =G/e


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x in K.X /. Then we S through Y . To see this, we take affine open S claim that f factorizes coverings Y D Spec Si and X D Spec Ri such that for eachSi, f .Spec Si / e Spec Ri and Spec Si is stable under the G-action. Then .Y =G/e D Spec..SiG /p /. S e If we denote the normalization of .SiG /p in K.X / by Sxi , then Yx D Spec Sxi . For each i, we have Sxi D Si \ K.X /. Since Si Sxi Ri , the claim holds. Definition 3.3. For a torsion-free coherent sheaf M on an integral scheme X, we write EM D EM=X D EndOX .M/: We define the NC blowup of X associated to M, NCB.M=X /, to be the finite NC scheme .EM ; M/ over X. We define the projection NCB.M=X / ! X by the functor M ˝OX ; where we think of M as a .EM ; OX /-bimodule. For a dominant finite morphism f W Y ! X of integral schemes, we put EY =X D EOY =X and NCB.Y =X / D NCB.OY =X /: Since OY is a subring of EY =X , we have the induced functor Qcoh.EY =X / ! Qcoh.OY /; which is identical to OY EY =X ˝EY =X . We call the corresponding morphism Y ! NCB.Y =X / the coforgetful morphism. This is obviously flat. The composition of the coforgetful morphism and the projection, Y ! NCB.Y =X / ! X; is exactly the original morphism Y ! X. Definition 3.4. For a factorizable normal alteration Y ! X, if Y ! Yx ! X is the finite-birational factorization, then we define the associated NC blowup, NCB.Y =X /, to be NCB.Y =Yx /. Remark. The normality assumption in the above definition is not really necessary, but just for simplicity. Every factorizable normal alteration Y ! X factors also as Y ! NCB.Y =X / ! X:

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4 Frobenius morphisms In this section, we shall define the Frobenius morphism for some class of noncommutative blowups. From now on, we suppose that the base field k has characteristic p > 0. We also suppose that every scheme is F-finite.

4.1 Equivalent modules Definition 4.1. Let R be a commutative complete local Noetherian ring. Then every finitely generated R-module is uniquely the direct sum of finitely many indecomposable R-modules. We say that R-modules M and N are equivalent if M ˚b M ˚a Li i and N Š Li i M Š i

i

for some indecomposable R-modules Li and positive integers ai and bi . We say that coherent sheaves M and N on a scheme X are equivalent if for every yX;x -modules. y x and Nyx are equivalent O point x 2 X, the complete stalks M Given coherent sheaves M and N on a scheme X. We think of M as an .EM ; OX /bimodule and similarly for N . Then Hom.M; N / D HomOX .M; N / is an .EN ; EM /bimodule. Lemma 4.2. Let L, M and N be coherent sheaves on X which are mutually equivalent. (i) We have a natural isomorphism of .EN ; EL /-bimodules Hom.M; N / ˝EM Hom.L; M/ Š Hom.L; N /: In particular Hom.M; N / ˝EM Hom.N ; M/ Š EN : Hence the functors Hom.M; N / ˝EM W Qcoh.EM / ! Qcoh.EN / Hom.N ; M/ ˝EN W Qcoh.EN / ! Qcoh.EM / are equivalences which are inverses to each other. (ii) We have a natural isomorphism of .EN ; OX /-bimodules Hom.M; N / ˝EM M Š N : Proof. These are well-known to the specialists.


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(i) There exists a natural morphism Hom.M; N / ˝EM Hom.L; M/ ! Hom.L; N /: It is easy to see that the morphism is an isomorphism after the completion at each point of X. Hence the morphism is an isomorphism. (ii) The proof is similar to the above.

4.2 F-steady modules and Frobenius morphisms For a scheme X, we write the e-iterated k-linear Frobenius as F e D FXe W Xe ! X: Sometimes we simply call this the e-th Frobenius of X. A key observation is that the morphism F e factors as Xe ! NCB.Xe =X / ! X (see Definition 3.3). Definition 4.3. Let X be an integral normal scheme and M a reflexive coherent sheaf (that is, M__ Š M). Denote by Me the sheaf on Xe corresponding to M via the obvious identification Xe D X. Then Me is identical as an OX -module to the pushforward of M by the e-iterated absolute Frobenius. We say that M is F-steady if for every e, M and Me are equivalent OX -modules. For a finite dominant morphism f W Y ! X of integral normal schemes, we say that f is F-steady if OY is an F-steady OX -module. Example. If Y is regular, then f is F-steady. Indeed being flat over OY , OYe is locally isomorphic to OY˚r , r > 0, as an OY -module and hence also as an OX -module. From Lemma 4.2, for an F -steady sheaf M, we have an isomorphism NCB.Me =X / Š NCB.Me0 =X /; e; e 0 0: We also define a morphism NCB.Me =Xe / ! NCB.Me =X /; which we call the coforgetful morphism, as follows. We think of OX as a subring of e OXe D OX1=p in the obvious way. Then EMe =Xe is a subring of EMe =X . Hence we have a natural morphism NCB.Me =Xe / ! NCB.Me =X / defined by EMe =Xe EMe =X ˝EMe =X . Definition 4.4. Let M be an F-steady sheaf on X. We define the e-th Frobenius of NCB.M=X / to be the composite cofor.

e F e D FNCB.M=X/ W NCB.Me =Xe / ! NCB.Me =X / ! NCB.M=X /:

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By construction, this is flat, which we call the Kunz regularity of NCB.M=X /. (Recall that from Kunz [11], a scheme is regular if and only if its Frobenius morphisms are flat.) The morphism is also directly defined by the functor EMe =Xe HomOX .M; Me /

˝EM=X :

Corollary 4.5. Suppose that k is algebraically closed and that X is an arbitrary kvariety. Take a regular Galois alteration Y ! X with Y quasi-projective. (Such an alteration exists from de Jong’s theorem [4], 7.3). Then the associated noncommutative blowup NCB.Y =X / is Kunz regular. Proof. We first note that Y ! X is factorizable. Let Y ! Yx ! X be the finitebirational factorization. Since Y is regular, the morphism Y ! Yx is F-steady. Hence NCB.Y =Yx / D NCB.Y =X / is Kunz regular. Remark. Bondal and Orlov [2] conjectured the following: Let Y ! X be a finite morphism of varieties such that X has canonical singularities and Y is regular. Then the derived category of EY =X -modules is a minimal categorical desingularization. Their conjecture and the above corollary seem somehow related.

4.3 Compatibility of Frobenius morphisms In this subsection, to justify our definition of the Frobenius morphism, we show some compatibility of it (see also Section 6). We suppose that M is an F-steady reflexive coherent sheaf on an integral normal scheme X. Proposition 4.6. The diagram NCB.Me =Xe / proj.

Fe

Xe

/ NCB.M=X /

Fe

proj.

/X

is commutative. Proof. From Lemma 4.2, we have isomorphisms of .EMe =Xe ; OX /-bimodules Me ˝OXe OXe Š Me Š HomOX .M; Me / ˝EM=X M: The left hand side defines the composite morphism F e ı proj., while the right hand side defines proj. ı F e. Hence the proposition follows.


277

Lemma 4.7. For e 0 e 0, the diagram NCB.Me0 =Xe / o

cofor.

NCB.Me =Xe /

/ NCB.Me 0 =X /

cofor.

o

/ NCB.Me =X /

is commutative. Proof. There exists a natural morphism ˛ W HomOXe .Me ; Me0 / ˝EMe =Xe EMe =X ! HomOX .Me ; Me0 /; ˝

7! ı :

We claim that this is an isomorphism, which proves the lemma. Let U X be an open subset such that X n U has codimension 2 and MjU is locally free. Then locally on U, we have an isomorphism of OXe -modules Me0 Š Me˚r for some r. Hence locally on U, the source and target of ˛ are both isomorphic to ˚r . It is now easy to see that ˛ is an isomorphism over U. EM e =X Moreover both hand sides are flat right EMe =X -modules and hence locally isomor˚l ˚l for some l. From the normality assumption, EM phic to direct summands of EM e =X e =X is a reflexive OX -module (see [7], Proposition 1.6) and so are its direct summands. So ˛ is an isomorphism all over X. We have proved the claim and the lemma. Corollary 4.8. For e; e 0 0, the diagram NCB.MeCe0 =Xe / o

Fe

NCB.Me =Xe /

/ NCB.Me 0 =X /

Fe

o

/ NCB.M=X /

is commutative. Proof. If e 0 e, then from Lemmas 4.2 and 4.7, the diagram NCB.MeCe0 =X /

5 kkk kkk k k kkk kkk

NCB.M e 0 =X / k5 k k kk kkk kkk k k k NCB.Me0 =Xe / NCB.M e =X / k5 k k kk kkk kkk k k k % z NCB.Me =Xe / NCB.M=X /

NCB.MeCe0 =Xe /

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Takehiko Yasuda

is commutative. Now the corollary follows from our definition of the Frobenius morphism. If e 0 < e, then similarly the diagram NCB.MeCe0 =X /


NCB.M e =X / 5 kkk k k kkk kkk kkk NCB.Me =Xe / NCB.M e 0 =X / k5 k k kk kkk kkk k k k % z NCB.Me0 =Xe / NCB.M=X /

NCB.MeCe0 =Xe /

is commutative and the corollary follows. Corollary 4.9. For e; e 0 0, the diagram e0

F / NCB.Me =Xe / NCB.MeCe0 =XQeCe0 / QQQ QQQ QQQ Q Fe 0 QQQ eCe QQQ F Q( NCB.M=X / 0

0

is commutative. Namely we have F eCe D F e ı F e . In particular, the e-th Frobenius of NCB.M=X / is the e-iterate of the first Frobenius. Proof. This follows from the commutativity of the diagram NCB.MeCe0 =X /


0 =Xe / NCB.M NCB.M eCe e =X / 5 jjj5 kkk j k j k j j kkk jjjj kkk kkk jjjj NCB.MeCe0 =XeCe0 / NCB.Me =Xe / NCB.M=X /:

Proposition 4.10. Let f W Y ! X be a finite dominant morphism with Y regular. Then the diagram


Fe

Ye

279

/Y

cofor.

cofor.

NCB.Ye =Xe /

Fe

/ NCB.Y =X /

is commutative. Proof. Because OYe is a locally free OY -module, the canonical map OYe ˝OY EY =X ! HomOX .OY ; OYe /: is an isomorphism, which proves the proposition.

5 D-blowups Among NC blowups, especially interesting are the ones associated to Frobenius morphisms of schemes. Definition 5.1. For an integral scheme X, we define the e-th D-blowup of X as DBe .X / D NCB.Xe =X /. Remark. The D-blowup can be regarded as the noncommutative counterpart of the F-blowup (see [15]). Definition 5.2 (Hochster–Roberts [8]). Let X D Spec R be an integral scheme. We say that R and X are F-pure if R ,! Re splits as an R-module map. Definition 5.3 (Smith–Van den Bergh [13]). Suppose that R is a complete local Noetherian domain so that the Krull–Schmidt decomposition holds for finitely generated R-modules. Then R and Spec R are said to be FFRT (finite F-representation type) if there are finitely many indecomposable R-modules Mi , i D 1; : : : ; n, such L ˚r that for any e, Re is isomorphic to niD1 Mi i , ri 0, as an R-module. Proposition 5.4. Let R be a complete local Noetherian normal domain. Suppose that X D Spec R is F-pure and FFRT. Then for sufficiently large e, the Frobenius morphism FXe W Xe ! X is F-steady. Proof. Let Mi , i D 1; : : : ; n, be the irredundant set of indecomposable modules as in the above definition.L Then there exists e0 such that for every e e0 , Re is isomorphic, ˚r as an R-module, to niD1 Mi i , ri > 0. Hence Xe ! X is F-steady. As a corollary, we obtain the following.

280

Takehiko Yasuda

Corollary 5.5. Let X be an integral normal scheme with F-pure and FFRT singularities. Namely the completion of every local ring of X is F-pure and FFRT. Then for sufficiently large e, DBe .X / is Kunz regular. Example. Normal toric singularities and tame quotient singularities are F-pure and FFRT. See [14] for other examples.

6 Comparing Frobenius morphisms of commutative and noncommutative blowups Let X D Spec R be an integral normal affine scheme and M a finitely generated z is F-steady. Let g W Z ! X be reflexive R-module such that the associated sheaf M z , that is, M D g M z =tors is locally free. a modification which is a flattening of M Lemma 6.1. We have .M/ D M . Proof. There exists an open subset U X such that X n U has codimension 2 and z is locally free on U. Since X is normal, from Zariski’s main theorem, .g M/jU D M z jU / D M z jU. It follows that the natural morphism M z ! g M is an injection g g .M z is reflexive, this is into a torsion-free sheaf which is an isomorphism over U. Since M an isomorphism. Therefore we have z / D M: .M/ D .g M/ D .M

Set E D EndR .M / and E D EM=Z . Then from the preceding lemma, E D .E/. Since M is locally free, the projection h W NCB.M=Z/ ! Z; which is defined by M ˝OZ , is an isomorphism. For F 2 Qcoh.E/ D Qcoh.NCB.M=Z//, .F / is a left E-module. Thus we have a left exact functor ˆ W Qcoh.Z/ ! E-mod F 7! .h F /: Put Ee D EndRe .Me /. Similarly we have a functor ê W Qcoh.Ze / ! Ee -mod:


281

Proposition 6.2. The diagram Qcoh.Z/

/ E-mod

ˆ

.F e /

Qcoh.Ze /

.F e /

ê

/ Ee -mod

is commutative up to isomorphism of functors. Proof. We claim that for F 2 Qcoh.E/, there exists a natural isomorphism HomR .M; Me / ˝E .F / Š .HomOZ .M; Me / ˝E F /: Obviously there exists a natural morphism from the left-hand side to the right-hand side. Since the claim is local on X, to show this, we may suppose that R is a complete local ring. Then the claim easily follows from the definition of equivalent modules. We have natural isomorphisms ..F e / ı ˆ/.F / D HomR .M; Me / ˝E .M ˝OZ F / Š .HomOZ .M; Me / ˝E M ˝OZ F / Š .Me ˝OZ F /

(Lemma 4.2)

Š .Me ˝OZe OZe ˝OZ F / Š .Me ˝OZe .F e / F / Š .ê ı .F e / /.F /: Thus the proposition holds. We have the right derived functor of ˆ Rˆ W D C .Qcoh.Z// ! D C .E-mod/: Similarly for ê . Corollary 6.3. The diagram D C .Qcoh.Z//

Rˆ

.F e /

D C .Qcoh.Ze //

/ D C .E-mod/ .F e /

Rê

is commutative up to isomorphism of functors.

/ D C .Ee -mod/

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Takehiko Yasuda

Proof. We have .F e / ı Rˆ Š R..F e / ı ˆ/ Š R.ˆ ı .F e / / Š .Rˆ/ ı .F e / : The functor Rˆ maps D b .Coh.Z// into D b .E-modfg /. Here Coh.Z/ denotes the category of coherent sheaves and E-modfg that of finitely generated left E-modules. As shown in [6], [10], [3], and [16], in some situations, the functor Rˆ W D b .Coh.Z// ! D b .E-modfg / is an equivalence, a kind of Fourier–Mukai transform. Then through this equivalence, the Frobenius morphisms on both hand sides correspond to each other at the level of derived category. Example. Let G SLd .k/ be a small finite subgroup of order prime to p with d D 2; 3. Set R D kŒx1 ; : : : ; xd G and X D Spec R. Let Y be either RAdk or Xe for e 0, and let Z be the universal flattening of Y ! X, which is isomorphic to the G-Hilbert scheme of Ito–Nakamura [9] (for the case Y D Xe , see [15], and [17]). If we put M to be the coordinate ring of Y , then the above functor is an equivalence (for instance, see [15], and [16]).

References [1]

M. Artin and J. J. Zhang, Noncommutative projective schemes. Adv. Math. 109 (1994), 228–287. 271

[2]

A. Bondal and D. Orlov. Derived categories of coherent sheaves, in Proceedings of the International Congress of Mathematicians. Invited lectures. Held in Beijing, August 20–28, 2002, Vol. II, ed. by T. Li, Higher Education Press, Beijing 2002, 47–56. 269, 276

[3]

T. Bridgeland, A. King, and M. Reid, The McKay correspondence as an equivalence of derived categories. J. Amer. Math. Soc. 14 (2001), 535–554 (electronic). 282

[4]

A. J. de Jong, Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. (83) (1996), 51–93. 270, 276

[5]

P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448. 271

[6]

G. Gonzalez-Sprinberg and J.-L. Verdier, Construction géométrique de la correspondance de McKay, Ann. Sci. École Norm. Sup. (4) 16 (1983). 282

[7]

R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), 121–176. 277

[8]

M. Hochster and J. L. Roberts, The purity of the Frobenius and local cohomology, Advances in Math. 21 (1976), 117–172. 279

[9]

Y. Ito and I. Nakamura, McKay correspondence and Hilbert schemes, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 135–138. 282

[10] M. Kapranov and E. Vasserot, Kleinian singularities, derived categories and Hall algebras, Math. Ann. 316 (2000), 565–576. 282


283

[11] E. Kunz, Characterizations of regular local rings for characteristic p, Amer. J. Math. 91 (1969), 772–784. 269, 276 [12] A. L. Rosenberg, Noncommutative schemes, Compositio Math. 112 (1998), 93–125. 271 [13] K. E. Smith and M. Van den Bergh, Simplicity of rings of differential operators in prime characteristic, Proc. London Math. Soc. (3) 75 (1997), 32–62. 270, 279 [14] S. Takagi and R. Takahashi, D-modules over rings with finite F -representation type, Math. Res. Lett. 15 (2008), 563–581. 270, 280 [15] Y. Toda and T. Yasuda, Noncommutative resolution, F -blowups and D-modules, Adv. Math. 222 (2009), 318–330. 269, 279, 282 [16] M. Van den Bergh, Non-commutative crepant resolutions, in The legacy of Niels Henrik Abel. Papers from the Abel Bicentennial Conference held at the University of Oslo, Oslo, June 3–8, 2002, ed. by O. A. Laudal and R. Piene, Springer , Berlin 2004, 749–770. 269, 282 [17] T. Yasuda, Universal flattening of Frobenius, Amer. J. Math. 134 (2012), 349–378. 282

Bivariant motivic Hirzebruch class and a zeta function of motivic Hirzebruch class Shoji Yokura Department of Mathematics and Computer Science Faculty of Science, Kagoshima University 1-21-35 Korimoto, Kagoshima 890-0065, Japan email: [email protected]

Abstract. The Euler–Poincaré characteristic is a generalization of the cardinality (or counting) and its higher homological extension for singular varieties as a natural transformation (what could be put in as its “categorification”) is MacPherson’s Chern class transformation. This transformation furthermore has two main developments: a bivariant-theoretic analogue and a generating series of it, i.e. a zeta function. The motivic Hirzebruch class is a unified theory of the three well-known characteristic classes of singular varieties, i.e. the above MacPherson’s Chern class transformation, Baum–Fulton–MacPherson’s Riemann–Roch and Cappell– Shaneson’s L-class transformation, which extends Goresky–MacPherson’s L-class. In this paper we discuss a bivariant-theoretic analogue and a zeta function of the motivic Hirzebruch class.

Contents 1. 2. 3. 4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Counting: from cardinality to Hodge–Deligne polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 288 A “categorification” of an additive homology class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Motivic characteristic classes: the most sophisticated categorification of additive-multiplicative homology classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .304 5. A bivariant-theoretic analogue of the motivic Hirzebruch class Ty . . . . . . . . . . . . . . . . . . 306 6. A zeta function of the motivic Hirzebruch class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

1 Introduction It is safe to say that theories of characteristic classes are super generalizations of “counting points of a finite set” (see e.g. [62]), in other words the notion of cardinality Partially supported by Grant-in-Aid for Scientific Research (No. 21540088), the Ministry of Education, Culture, Sports, Science and Technology (MEXT), and JSPS Core-to-Core Program 18005, Japan.

286

Shoji Yokura

has been generalized to theories of characteristic classes of singular spaces. On the other hand it has another development into the so-called “generating function” or “zeta function” in a much fancier way. Namely we have: cardinality

characteristics

characteristic classes

A

/ zeta functions

B

/ zeta functions

C

/ zeta functions

where A, B, and C are described below. A: Hasse–Weil zeta functions, for which the Weil conjecture was solved affirmatively by Deligne in [16] and [17] (see [54]). In the Hasse–Weil zeta function just the cardinality of a finite set is used. Now the cardinality is generalized or extended as the notion of “characteristic” and we have B:

• the zeta function for the Euler–Poincaré characteristic was studied by Macdonald [35]; • the zeta function for the arithmetic genus and more generally for the Hirzebruch y -characteristic was studied by Moonen [41]; • the zeta function for the signature was studied by Zagier [67]. These three important characteristics, i.e. Euler–Poincaré characteristic, arithmetic genus and signature have class versions, which are all described as a natural transformation from its corresponding covariant functor to the homology functor. They are respectively the Chern–MacPherson class [36], Baum–Fulton–MacPherson’s Riemann–Roch or Todd class [4] and Cappell–Shaneson’s L-class [12] (also see [60]) which extends Goresky–MacPherson’s homology L-class as a natural transformation (see [8] and also [60]). The zeta functions of these three characteristic classes are the following:

C:

• the zeta function for the Chern–MacPherson class was studied by Ohmoto [43] (see also [42]); • the zeta function for Baum–Fulton–MacPherson’s Riemann–Roch or Todd class was studied by Moonen [41]; • the zeta function for Thom–Hirzebruch L-class for smooth manifolds was studied by Zagier [67].

Bivariant motivic Hirzebruch class and a zeta function of motivic Hirzebruch class 287

As the Hirzebruch’s y -characteristic unifies the above three characteristics (1 , the Euler–Poincaré characteristic, 0 , the arithmetic genus, and 1 the signature), the motivic Hirzebruch class Ty , see [8] and [9], in a sense unifies the above three characteristic classes. Thus it is quite natural to consider the zeta function of the motivic Hirzebruch class as a class version of Moonen’s zeta function of the Hirzebruch y characteristic. On the other hand, Fulton and MacPherson introduced the theory of Bivariant Theory [21] (also see [19]) and they showed the existence of a bivariant-theoretic analogue of Baum–Fulton–MacPheron’s Riemann–Roch W G0 .X / ! H .X I Q/:

td W K0 .X ! Y / ! H.X ! Y /: Furthermore they conjectured (or posed as a problem) the existence of a bivarianttheoretic analogue of the Chern–MacPherson class and J.-P. Brasselet [7] proved it affirmatively:

c W F .X ! Y / ! H.X ! Y /: Thus it is also quite natural to think of the existence of a bivariant-theoretic analogue of the above motivic Hirzebruch class, since it unifies the Chern–MacPherson class and Baum–Fulton–MacPheron’s Riemann–Roch. If we get a bivariant-theoretic analogue of the motivic Hirzebruch class, then we could speculate a reasonable bivarianttheoretic analogue of the Cappell–Shaneson’s L-homology class, which is not yet available, as far as the author knows. Thus the above flow of diagrams extends as follows: characteristic classes

motivic characteristic classes

bivariant motivic characteristic classes

/ zeta functions

C

/ motivic zeta functions

/

?

As we will see, when it comes to thinking of zeta functions of the motivic characteristic classes, it seems quite natural to consider generalizing the above story or thoughts to an arbitrary natural transformation of two covariant functors on a reasonably nice category so that one can consider such a generalized zeta function associated to a given covariant functor or a given natural transformation. In such a more general category it is not necessarily guaranteed that the symmetric product X .n/ and hence the projection n W X n ! X .n/ exist. We formulate a general and formal zeta function of natural transformation associated to a covariant functor and a natural transformation in such a way that it specializes to usual zeta functions.

288

Shoji Yokura

In the present paper we first deal with bivariant-theoretic analogues of the motivic Hirzebruch classes and then the zeta functions of them. As indicated in the above diagram, however, we do not know what is the implication of “a bivariant zeta function” corresponding to the bivariant motivic Hirzebruch class.

Acknowledgements The author would like to thank the organizers of The 5th Franco–Japanese Symposium on Singularities (FJ2009) for such a wonderful organization. He would also like to thank P. Aluffi, M. Banagl, J.-P. Brasselet, D. Eisenbud, S. Ishii, S.-I. Kimura, L. Maxim, T. Ohmoto, Y. B. Rudyak, J. Schürmann, and T. Yasuda for useful discussions and comments, and the referee for pointing out typos and so on.

2 Counting: from cardinality to Hodge–Deligne polynomials For a finite field k, counting k-points of varieties gives rise to the following homomorphism from the Grothendieck ring K0 .V .k// of varieties: ] W K0 .V .k// ! Z defined by ].ŒX / D ].X.k//: This homomorphism is extended to the following formal power series via the symmetric product X .n/ D X n =Sn , where the symmetric group Sn acts on the Cartesian product X n of n-copies of X as permutations of factors: H W .X; t/ D

1 X

].X .n/ .k//t n 2 ZŒŒt:

nD0

This function is called the Hasse–Weil zeta function of X and it is a rational function due to Dwork. M. Kapranov [29] modified the Hasse–Weil zeta function just a bit to define the following motivic zeta function: .X; t/ D

1 X

ŒX .n/ .k/t n 2 K0 .V .k//ŒŒt;

nD0 Kap

which shall be denoted by ŒX .t/, called the Kapranov motivic zeta function. More generally, for any ring homomorphism W K0 .V .k// ! R he considered the following zeta function .X; t/ D

1 X nD0

.ŒX .n/ .k//t n 2 RŒŒt:


This zeta function is called the Kapranov zeta function associated to . The above ring homomorphism is a “motivic” generalization of the counting or cardinality. Indeed, certainly the counting ].A/ of points of a finite set A satisfies the following basic properties: • A Š A0 (bijection or equipotent) H) ].A/ D ].A0 /, • ].A/ D ].A n B/ C ].B/ for B A, • ].A B/ D ].A/ ].B/, • ].pt/ D 1.

(Here pt denotes one point.)

If we consider the following “topological counting” ]top on the category Top of topological spaces such that ]top .X / 2 Z and it satisfies the following four properties: • X Š X 0 (homeomorphism = Top-isomorphism) H) ]top .X / D ]top .X 0 /, • ]top .X / D ]top .X n Y / C ]top .Y / for Y X, • ]top .X Y / D ]top .X / ]top .Y /, • ]top .pt/ D 1; then one can show that if such a ]top exists, then we must have that ]top .R1 / D 1;

hence

]top .Rn / D .1/n :

Hence if X is a finite C W -complex with n .X / denoting the number of open n-cells, then X .1/n n .X / D .X / ]top .X / D n

is the Euler–Poincaré characteristic of X. Namely, sloppily speaking, the topological counting ]top is uniquely determined and it is the topological Euler–Poincaré characteristic. Now, let us consider such a counting on the category V of algebraic varieties: • X Š X 0 (V -isomorphism) H) ]alg .X / D ]alg .X 0 /, • ]alg .X / D ]alg .X n Y / C ]alg .Y / for a closed subvariety Y X, • ]alg .X Y / D ]alg .X / ]alg .Y /, • ]alg .pt/ D 1: If such an “algebraic” counting ]alg exists, then it follows from the decomposition of the n-dimensional complex projective space P n D C0 t C1 t t Cn1 t Cn that we must have ]alg .P n / D 1 y C y 2 y 3 C C .y/n

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Shoji Yokura

where y D ]alg .C1 / 2 Z. In fact, it follows from Deligne’s theory of mixed Hodge structures that the following Hodge–Deligne polynomial (cf. [15]) X W .1/i .1/pCq dimC .GrpF GrpCq Hci .X; C//up v q u;v .X / D i;p;q0

satisfies the above four properties, namely any Hodge–Deligne polynomial u;v with uv D y is such a ]alg . The Hirzebruch y -characteristic is nothing but y;1 and the most important and interesting ones are the following: y D 1: 1 D , the topological Euler–Poincaré characteristic, y D 0: 0 D a , the arithmetic genus, y D 1: 1 D , the signature. These three important characteristics are extended as higher class analogues, i.e. “categorified” or natural transformations of additive-multiplicative homology classes, as explained in the following section. In fact they are most sophisticated ones and well-known characteristic classes: y D 1: Chern–MacPherson class c W F .X / ! H .X I Z/: y D 0: Baum–Fulton–MacPheron’s Riemann–Roch W G0 .X / ! H .X I Q/ y D 1: Cappell–Shaneson’s L-homology class ! W .X / ! H .X I Q/: It turns out (see [8] and [62]) that the Hodge–Deligne polynomial u;v W K0 .V / ! ZŒu; v can be extended as a class version only when u D y; v D 1, just like Hirzebruch– Riemann–Roch was extended by A. Grothendieck as a natural transformation from the covariant functor of coherent sheaves to the rational homology theory, which is called Grothendieck–Riemann–Roch. Namely only the Hirzebruch y -characteristic y W K0 .V / ! ZŒy can be extended as a class version Ty W K0 .V =/ ! H .I QŒy/: This is called the motivic Hirzebruch class and it “unifies” the above three characteristic classes (see later sections).


3 A “categorification” of an additive homology class 3.1 An additive homology class Let Top be the category of topological spaces and continuous maps. Let C be another category of topological spaces equipped with possibly extra more geometric structures, e.g. such as differentiable manifolds, almost complex manifolds, complex algebraic varieties, real algebraic varieties, complex analytic varieties, etc. Let f W C ! Top be the forgetful functor, i.e. the functor forgetting those extra more geometric structures. Definition 3.1 (An additive homology class on the category C ). Let AB be the category of abelian groups and let C be a possibly more geometric category as above. Let H W C ! AB be the homology covariant functor. If an element ˛.X / 2 H .X / is defined uniquely (up to isomorphism) for an object X, and it satisfies that Š

(i) for an isomorphism f W X ! X 0 , f ˛.X / D ˛.X 0 / (ii) ˛.X t Y / D ˛.X / C ˛.Y /; then the element ˛.X / is called an additive homology class of X. Furthermore, if the additive homology class ˛ is non-trivial and satisfies the following “cross product” formula (iii) ˛.X Y / D ˛.X / ˛.Y /; then it is called an additive-multiplicative homology class of X. Remark 3.2. (i) It is easy to see that the non-triviality of ˛ means that ˛.pt/ D 1; namely, the non-trivial additive-multiplicative homology class is a normalized one. (ii) In fact, the homology functor H can be replaced by any covariant functor H W C ! AB equipped with the cross product structure such that the coefficient ring H .pt/ is a domain. In this case we call such an additive class an additive H -class. However, we stick to the homology theory for the sake of simplicity. (iii) Almost all topological invariants are additive classes: the Euler–Poincaré characteristic on Top, all the characteristic classes and the characteristic numbers on the category C 1 of differentiable manifolds and the category AC of almost complex manifolds, all the characteristic classes and characteristic numbers of singular varieties, etc. They are all in fact additive-multiplicative classes. Fulton’s canonical class, Fulton–Johnson’s Chern class and Milnor class are additive but not multiplicative. (iv) The Euler–Poinaré characteristic is an integer, but we still call it a class. Note that any additive-multiplicative homology class of a compact 0-dimensional manifold, i.e. a finite set, is nothing but the cardinality of the finite set. In this sense, an additivemultiplicative homology class is a generalization of counting points.

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3.2 A natural transformation associated to an additive homology class Given an additive homology class on a category C with the forgetful functor f W C ! Top; the correspondence ˛ W M 7! ˛.M / 2 H .M / is not “categorical” as it is, although the receiver of the correspondence ˛ is the covariant homology functor, but the source of the correspondence ˛ is not a functor. We can make it a natural transformation from a certain functor to the covariant homology functor ˛ W .X / ! H .X /: This will be called a categorification of the correspondence ˛ . Here is one simple answer for the functor

.

Definition 3.3. Let X be a topological space and let C be a category of topological spaces equipped with possibly extra more geometric structures and let f W C ! Top be the forgetful functor. Let f

M prop .C ! Top=X / p

be the monoid consisting of isomorphism classes ŒV ! X of proper morphisms p W V ! X (more precisely, p W f.V / ! X 2 HomTop .f.V /; X /), where V 2 Obj.C /. Here h W V ! X and h0 W V 0 ! X (more precisely, h W f.V / ! X and h0 W f.V 0 / ! X ) are called isomorphic over X 2 Obj.Top/ • if there is an isomorphism W V ! V 0 2 HomC .V; V 0 / and • if h0 ı f./ D h in HomTop .f.V /; X /. The addition and zero are defined by h

h0

hCh0

X C ŒV 0 ! X D ŒV t V 0 ! X , • ŒV ! • 0 D Œ ! X : Then we define

f

K prop .C ! Top=X / f

to be the Grothendieck group of the monoid M prop .C ! Top=X /, which shall be provisionally called the Grothendieck–Thom relative group over X.

Bivariant motivic Hirzebruch class and a zeta function of motivic Hirzebruch class 293 f

Proposition 3.4. (i) K prop .C ! Top=X / is a covariant functor on the category Top with pushforwards for proper morphisms, i.e. for a proper morphism f W X ! Y the pushforward f

f

f W K prop .C ! Top=X / ! K prop .C ! Top=Y / defined by p

f ıp

f .ŒV ! X / D ŒV ! Y is covariantly functorial. f

(ii) K prop .C ! Top=X / has a cross product structure on the category Top: f

f

f

! K prop .C ! Top=X Y / K prop .C ! Top=X / K prop .C ! Top=Y / is defined by p

pk

k

ŒV ! X ŒW ! X D ŒV W ! X Y : (iii) Let ˛ be an additive homology class defined on the category C . Then there exists a unique natural transformation on the category Top of topological spaces: f

˛ W K prop .C ! Top=/ ! H ./ satisfying that for X 2 Obj.C / idX

˛ .ŒX ! X / D ˛.X /: (iv) Let ˛ be an additive-multiplicative homology class defined on the category C . Then the above natural transformation f

˛ W K prop .C ! Top=/ ! H ./ commutes with the cross product, i.e. the following diagram commutes: f

f

˛ ˛

K prop .C ! Top=X / K prop .C ! Top=Y / f

/ H .X / H .Y /

K prop .C ! Top=X Y /

˛

/ H .X Y /:

Remark 3.5. Depending on the additive homology class ˛, the above Grothendieck– f

Thom covariant functor K prop .C ! Top=/ can be made into a much finer one. Here are typical examples. (i) Suppose that the additive homology class ˛ on C is bordism invariant, i.e. if X and Y are bordant, i.e. if there exists W such that @W D X t Y then we

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have ˛.X / D ˛.Y /. (Here we are allowed to be sloppy and we do not consider orientation; if we consider orientation, we should put the sign.) In this case we f

can divide K prop .C ! Top=/ out by the bordism relation, i.e. f

f

prop .C ! Top=X / D K prop .C ! Top=X /= ; where the equivalence relation is defined by h0

h

ŒV ! X ŒV 0 ! X () there exists H W W ! X such that (a) @W D V t V 0 ; (b) H j@W D h t h0 : (ii) If it satisfies the “Grothendieck additivity” ˛.V / D ˛.V n S/ C ˛.S/

for S V;

f

then we can devide K prop .C ! Top=/ out by this strong additivity relation: prop

f

f

K0 .C ! Top=X / D K prop .C ! Top=X /= ; where the equivalence relation is defined by h

hjV nS

hjS

ŒV ! X ŒV n S ! X C ŒS ! X for S V: Remark 3.6. It should be emphasized that even though we consider such a finer category C for a source space V the map h W V ! X of course has to be considered f

Top=X / (with C being the catein the crude category Top. The above prop .C ! gory of closed oriented manifolds) is the so-called bordism group ./, which is a generalized homology theory, in particular ./ is a covariant functor W Top ! AB; where AB is the category of abelian groups. Clearly we can consider this covariant functor on a different category finer than the category Top of topological spaces, e.g. consider the category VC of complex algebraic varieties. Namely we consider continuous maps h W M ! V from closed oriented manifolds M to a complex algebraic variety V . We still get a covariant functor W VC ! AB: In this set-up the following three different categories are involved: • coC 1 of closed oriented manifolds, • Top of topological spaces, • VC of complex algebraic varieties.


More precisely, we have the following forgetful functors fs W coC 1 ! Top and f t W VC ! Top (here “s” and “t” mean “source object” and “target object”): fs

ft

coC 1 ! Top VC : And the commutative triangle MA AA AA AA A h AA A

V

/ M0 } } }} } }} 0 }} h } ~ }

really means fs ./ / fs .M 0 / fs .M /G GG ww GG ww w GG w G ww h GGG ww h0 w G# {ww f t .V /:

As suggested by this situation in [65] we deal with a more general situation of cospan of categories Cs ; C t ; B: S

T

Cs ! B C t : S

T

From this cospan Cs ! B C t we get the canonical generalized .S; T /-reS

lative Grothendieck groups K.Cs ! B=T .// and also from the following commutative diagrams of categories and functors S T /Bo Ct Cs A AA } } AA }} AA }} ˆ A } 0 AA } 0 S A ~}}} T B0

we obtain a categorification of an additive function ˛.X / on objects Obj.Cs / with values ˛.X / 2 T 0 .X /: S

˛ W K.Cs ! B=T .// ! T 0 ./: In particular, for the following commutative diagram S S /Bo Cs A Cs AA } } AA }} AA }} ˆ A } } 0 S 0 AA A ~}}} S B0

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with S W C ! B being a full functor, then the natural transformation S

˛ W K.Cs ! B=T .// ! S 0 ./ satisfying the condition that ˛ .Œ.V; V; idV // D ˛.V / 2 S 0 .V / for V 2 Obj.Cs / is unique. For more details, see [65]. Remark 3.7. (i) If ˛.M / D cl .M / D cl.M / \ ŒM is the Poincaré dual of any characteristic cohomology class cl, then the above natural transformation ˛ is denoted f

by cl W K prop .C ! Top=/ ! H ./ and this can be considered as a very general theory of characteristic homology classes of topological spaces. f

(ii) If we consider cl W K prop .C ! Top=/ ! H ./ on a category B geometrically finer than the category Top of topological spaces with the forgetful functor f W C ! B (e.g. an inclusion functor), then we get a general theory of characteristic homology classes on the subcategory B; for example, a general theory of characteristic homology classes on pseudo-manifolds, on complex algebraic varieties, on real algebraic varieties, etc. f

Top=/ cannot in genThe Grothendieck–Thom covariant functor K prop .C ! eral become a contravariant functor with a reasonable pullback. It is because in the following fiber square M0

f0

/M

h0

h

X

f

/ Y:

with M 2 Obj.C / the fiber product M 0 D X Y M does not necessarily belong to the category C . If it does, we can define the pullback homomorphism f

f

f W K prop .C ! Top=Y / ! K prop .C ! Top=X / by h

h0

Y / D ŒM 0 ! X f .ŒM ! f

and with this K prop .C ! Top=/ becomes a contravariant functor. Lemma 3.8. Let us consider the category VC of complex algebraic varieties, instead of the category Top, and consider the subcategory V C of smooth varieties as the source category C , thus f W V C ! VC is the inclusion functor. Then the functor

Bivariant motivic Hirzebruch class and a zeta function of motivic Hirzebruch class 297 f

K prop .V C ! VC =/ also becomes a contravariant functor for smooth morphisms on the category VC , where for a smooth morphism f W X ! Y the pullback homomorphism f

f

f W K prop .V C ! VC =Y / ! K prop .V C ! VC =X / is defined by h0

h

f .ŒM ! Y / D ŒM 0 ! X : Theorem 3.9 (Verdier-type Riemann–Roch, see [65]). Let the situation be as described in Lemma 3.8. Let cl be any multiplicative characteristic cohomology class of f

complex vector bundles. Then the above natural transformation cl W K prop .V C ! VC =/ ! H ./ on the category VC satisfies the following Verdier-type Riemann– Roch formula: For a smooth morphism f W X ! Y the following diagram commutes: f

K prop .V C ! VC =Y /

cl

cl.Tf /\f

f

K

prop

/ H .Y /

f

.V C ! VC =X /

cl

/ H .X /:

In the case when we consider the identity functor i W C ! C for an object i

X 2 Obj.C /, the functor K prop .C ! C =X / is simply denoted by K prop .C =X /.

Theorem 3.10 (SGA-6-type Riemann–Roch [65]). Consider the category V C of smooth complex varieties. Let K prop.sm .V C =X / be the Grothendieck group of the monoid of the isomorphism classes of proper smooth morphisms h W V ! X. On the category V C let us define Tcl W K prop.sm .V C =X / ! H .X / by h X / D PDX 1 h .cl.Th / \ ŒV / : Tcl .ŒV ! Here PDX W H .X / ! H .X / is the Poincaré duality isomorphism given by taking the cup product with the fundamental class. Then the following diagram is commutative for a proper smooth morphism f W X ! Y :

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Shoji Yokura

Tcl

K prop.sm .V C =X /

/ H .X / fŠ .cl.Tf /[

f

K prop.sm .V C =Y /

Tcl

/

/ H .Y /:

Here the Gysin homomorphism fŠ W H .X / ! H .Y / is defined by fŠ D PDY 1 ı f ı PDX : Remark 3.11. One would not be able to expect such a SGA-6-type Riemann–Roch theorem on the category Top of topological spaces.

3.3 Examples Example 3.12 (The case of fundamental class). Let us consider taking the fundamental class Œ on the category C 1 of differentiable manifolds. The fundamental class Œ is certainly an additive-multiplicative homology class and we have the unique natural transformation on the category Top of topological spaces: Œ

f

K prop .C 1 ! Top=/ ! H ./:

W

Then the classical Steenrod’s realization problem can be interpreted as the problem of asking for the surjectivity of the homomorphism Œ

W

f

K prop .C 1 ! Top=X / ! H .X /

for a topological space. The following results are known (see [44]). • ([51] and [44], Chapter IV, Theorem 7.37) Œ

W

f

K prop .C 1 ! Top=X / !

M

Hi .X /

0i6

is surjective. • ([33]) Let C Poincaré be the category of Poincaré complexes, i.e. topological spaces which satisfies the Poincaré duality. Then the following is surjective: M f Hi .X /: Œ W K prop .C Poincaré ! Top=X / ! i6D3


• ([49] and [44], Chapter VIII, Example 1.25(a)) Let C pseudo be the category of pseudo-manifolds. Then the following is surjective: Œ

W

f

K prop .C pseudo ! Top=X / ! H .X /:

Example 3.13 (The case of Stiefel–Whitney class). Let C D C 1 the category of C 1 manifolds. Let V be a differentiable manifold and let P .w/ .V / 2 H .V; Z2 / be the Poincaré dual P .w/ \ ŒV of a polynomial P .w/ D P .w1 ; w2 ; : : : / of Stiefel– Whitney classes w .T V / 2 H .V; Z2 /. P .w/ .V / is clearly an additive homology class. Then we have a unique natural transformation on the category Top of topological spaces f

P .w/ W K prop .C 1 ! Top=/ ! H .; Z2 / such that for a differentiable manifold X we have idX

P .w/ ŒX ! X / D P .w/ .X /: In particular the Stiefel–Whitney class w is a typical one. If we restrict ourselves to the subcategory VR of real algebraic varieties, instead of Top and we let V R be its subcategory of smooth real algebraic varieties, then we have a finer natural transformation on the category VR f

P .w/ W K prop .V R ! VR =/ ! H .; Z2 /: In the case when P .w/ D w, we have the following more geometric “realization” on the category VR through constructible functions: f

/ F .X / K prop .V R ! VR =X / NNN y y NNN yy NNN y NN yy w NNN yy w y NNN y N' |yy H .X; Z2 / : const

Example 3.14 (The case of Pontryagin class). Let C D C 1 the category of C 1 manifolds. Let V be a differentiable manifold and let P .p/ .V / 2 H .V; Z/ be the Poincaré dual of a rational-coefficient polynomial P .p/ D P .p1 ; p2 ; : : : / of Pontryagin classes p .T V / 2 H .V; Q/. Then P .p/ .V / is clearly an additive homology class with Q-coefficients. Then we have a unique natural transformation on the category Top f

P .p/ W K prop .C 1 ! Top=/ ! H .; Q/

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Shoji Yokura

such that for a differentiable manifold X we have idX

P .p/ ŒX ! X / D P .p/ .X /: Here of course we can consider a Z-coefficient polynomial. If we restrict ourselves to the subcategory VR instead of Top and we let V R be its subcategory of smooth real algebraic varieties, then we have a finer natural transformation on the category VR f

P .p/ W K prop .V R ! VR =/ ! H .; Q/: If we further restrict ourselves to the subcategory VC instead of Top and we let V C be its subcategory of smooth complex algebraic varieties, then we have a finer natural transformation on the category VC f

P .p/ W K prop .V C ! VC =/ ! H .; Q/: In the case when P .p/ D L is a Hirzebruch’s L-class, then we have the following more geometric “realization” on the category VC through Cappell–Shaneson– Youssin–Balmer’s cobordism groups: f

/ .X / K prop .V C !O VC =X / OOO y y OOO y yy OOO y y OOO yy L L OOO y y OO' |yy H .X; Q/ : cobordism

f

VC =X / by the finer relative It follows from [8] that we can replace K prop .V C ! Grothendieck group K0 .VC =X /: / .X / K0 .VC =X / JJ x JJ x x JJ x x JJ J xx L JJJ xx L x JJ % {xx H .X; Q/ : cobordism

This will be recalled in the following section on motivic characteristic classes. We have the following natural transformation f

P .p/ W prop .C 1 ! Top=/ ! H .; Q/: Which is due to the fact that the Pontryagin classes are bordism invariant. We would like to speculate that for any multiplicative sequence P .p/ of Pontryagin classes we


have the following commutative diagram f cobordism / .X / prop .V C ! Top=X / NNN yy NNN yy NNN y NN yy yy P .p/ P .p/ NNN y NNN y ' |yy H .X; Q/ :

In our paper [8] (see [9] also) we could deal with the only L-class, but not with any other multiplicative sequence, because we require a stronger additivity on a given H -class ˛.X /, i.e. the following additivity (provisionally called “Grothendieck additivity”): ˛.X / D ˛.X n Y / C ˛.Y / for a closed subvariety Y X. Example 3.15 (The case of Chern classes). Let AC be the category of almost complex manifolds. Let V be an almost complex manifold and let P .c/ .V / 2 H .V; Z/ be the Poincaré dual of a Z-coefficient polynomial P .c/ D P .c1 ; c2 ; : : : / of Chern classes c .T V / 2 H .V; Z/. P .c/ .V / is clearly an additive H -class with H D H .; Z/. Then we have a unique natural transformation on the category Top f

P .c/ W K prop .AC ! Top=/ ! H .; Z/ such that for an almost complex manifold X we have idX

P .c/ ŒX ! X / D P .c/ .X /: If we further restrict ourselves to the subcategory VC instead of Top and we let V C be its subcategory of smooth complex algebraic varieties, then we have a finer natural transformation on the category VC f

P .c/ W K prop .V C ! VC =/ ! H .; Z/: In the case when P .c/ D c is the Chern class, then we have the following more geometric “realization” on the category VC through constructible functions via MacPherson’s theorem: / F .X / K prop .V CL=X / LLL x x x LLL x x LL xx c LLL xx cMac LLL x & {xx H .X; Z/ : const

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Shoji Yokura f

It follows from [8] that we can replace K prop .V C ! VC =X / by the finer relative Grothendieck group K0 .VC =X /: / F .X / K0 .VC =X / JJ JJ xx JJ xx x JJ J xx c JJJ xx cMac x JJ $ {xx H .X; Z/ : const

More on this will be discussed in the following section. Example 3.16 (The case of Chern classes of other types). Let EMV C be the subcategory of complex algebraic varieties embeddable into smooth varieties and let cFJ (resp., cFJ ) be Fulton–Johnson’s Chern class (resp., Fulton’s canonical class) defined on a scheme embeddable into a nonsingular scheme. Namely, let X be a scheme embeddable into a nonsingular scheme M. cFJ .X / ([19], Example 4.2.6(c)) is defined by cFJ .X / D c.TM jX / \ s.NX M /; where TM is the tangent bundle of M and s.NX M / is the Segre class of the conormal sheaf NX M of X in M [19], §4.2. Fulton’s canonical class cF .X / ([19], Example 4.2.6(a)) is defined by cF .X / D c.TM jX / \ s.X; M /; where s.X; M / is the relative Segre class [19], §4.2. The local complete intersection variety X defines a normal bundle NX in M , from which we can define the virtual tangent bundle TX of X by TX D TM jX NX M which is a well-defined element of the Grothendieck group K 0 .X /. As shown in [19], Example 4.2.6, for a local complete intersection variety X in a non-singular variety M these two classes are both equal to cFJ .X / D cF .X / D c.TX / \ ŒX : Then there exists the following unique natural transformations on the category Top f

cF W K prop .EMV C ! Top=/ ! H .; Z/ idX

satisfying that cF .ŒX ! X / D cF .X / for X 2 Obj.EMV C /, and f

cFJ W K prop .EMV C ! Top=/ ! H ./

Bivariant motivic Hirzebruch class and a zeta function of motivic Hirzebruch class 303 idX

satisfying that cFJ .ŒX ! X / D cFJ .X /, and f

cM W K prop .EMV C ! Top=/ ! H ./ idX

satisfying that cM .ŒX ! X / D cM .X /, the Chern–Mather class. Since we have the canonical map f

ˆ W K prop .EMV C ! VC =/ ! K0 .VC =/; we can consider the natural transformation f

Ty ı ˆ W K prop .EMV C ! VC =/ ! H ./ ˝ QŒy; which is simply denoted by f

Ty W K prop .EMV C ! VC =/ ! H ./ ˝ QŒy: The natural transformation Ty W K0 .VC =/ ! H ./ ˝ QŒy will be described in the following §4. Then we can consider the following three natural transformations: f

VC =/ ! H ./; MF D cF T1 W K prop .EMV C ! f

MFJ D cFJ T1 W K prop .EMV C ! VC =/ ! H ./; f

MM D cM T1 W K prop .EMV C ! VC =/ ! H ./: A geometric realization problem for these natural transformations is, e.g. if or how one can factorize them through the covariant functor of constructible functions (which is supposed to be adapted or admissible with these distinguished Chern homology classes, i.e. cl D c the Chern class): f

K prop .EMV C ! VC =/ OOO q q OOO q q q OOMF constqqq OOO; MF J; MM q OOO q q OOO qq q q O' xq / F .X / H .X / ˝ Q : ‹

f

Here const W K prop .EMV C ! VC =/ ! F .X / is defined by p

! X / D p 11V : const.ŒV

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4 Motivic characteristic classes: the most sophisticated categorification of additive-multiplicative homology classes As mentioned in the previous section, in this section we recall the three well-known theories of characteristic classes of singular varieties, which are the most sophisticated categorification of additive-multiplicative homology classes: • MacPherson’s Chern class transformation [36]: cMac W F .X / ! H .X /; which is a unique natural transformation satisfying the normalization condition that for a smooth variety X the value of the characteristic function is the Poincaré dual of the total Chern cohomology class: cMac .11X / D c.TX / \ ŒX , • Baum–Fulton–MacPherson’s Todd class or Riemann–Roch [4]: W G0 .X / ! H .X / ˝ Q; tdBMF which is a unique natural transformation satisfying the normalization condition that for a smooth variety X the value of the structure sheaf OX is the Poincaré dual of the total Todd cohomology class: tdBMF .OX / D td.TX / \ ŒX , • Goresky–MacPherson’s homology L-class [22], which is extended as a natural transformation by Sylvain Cappell and Julius Shaneson [12] (also see [60]): LCS W .X / ! H .X / ˝ Q; which is a unique natural transformation satisfying the normalization condition that for a smooth variety X the value of the constant sheaf QX is the Poincaré dual of the total Hirzebruch–Thom L-class: LCS .QX / D L.TX / \ ŒX . Here the homology theory H .X / is the Borel–Moore homology theory. From now on the category VC shall be simply denoted by V without the suffix C. In [8] (cf. [9], [48], and [62]) we introduced the motivic Hirzebruch class Ty W K0 .V =X / ! H .X / ˝ QŒy; which is a unique natural transformation satisfying the normalization condition that idX

for a smooth variety X the value of the isomorphism class ŒX ! X of the identity map idX is the Poincaré dual of the total Hirzebruch cohomology class: idX

Ty .ŒX ! X / D td.y/ .TX / \ ŒX :


Here the Hirzebruch class td.y/ .E/ of the complex vector bundle E (see [26] and [27]) is defined to be td.y/ .E/ D

rank YE iD1

˛i .1 C y/ ˛ y 2 H .X / ˝ QŒy; i 1 e ˛i .1Cy/

where ˛i is the Chern root of E, i.e. c.E/ D

rank YE

.1 C ˛i /: Note that

iD1

• td.1/ .E/ D c.E/ the Chern class, • td.0/ .E/ D td.E/ the Todd class and • td.1/ .E/ D L.E/ the Thom–Hirzebruch L-class. The motivic Hirzeruch class Ty W K0 .V =X / ! H .X / ˝ QŒy “unifies” the ; LCS above three characteristic classes cMac ; tdBMF (also see §3) in the sense that the following diagrams commute: K0 .V =XJ / JJ y JJ yy JJT1 yyy JJ JJ yy y JJ y J% |yy / H .X / ˝ Q; F .X / Mac c

K0 .V =X / JJ x JJ xx JJ T0 xxx JJ JJ x JJ xx x JJ x {x % / H .X / ˝ Q; G0 .X / BMF td

and K0 .V =XJ / JJ y JJ yy JJT1 ! yyy JJ y JJ y y JJ y J% |yy / H .X / ˝ Q: .X / CS L

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Shoji Yokura

This “unification” could be considered as a positive answer to the following remark in MacPherson’s survey article [37]: “It remains to be seen whether there is a unified theory of characteristic classes of singular varieties like the classical one outlined above.”1 In the rest of the paper we consider the following two problems on the motivic Hirzebruch class Ty : (i) A bivariant-theoretic analogue of the motivic Hirzebruch class Ty ; (ii) A zeta function of the motivic Hirzebruch class Ty .

5 A bivariant-theoretic analogue of the motivic Hirzebruch class Ty In early 1980’s Fulton and MacPherson introduced Bivariant Theory as a categorical framework for the study of singular spaces [21] (see also Fulton’s book [19]): A bivariant theory is defined on morphisms, instead of objects, and unifies both a covariant functor and a contravariant functor. Important objects to be investigated in bivariant theories are Grothendieck transformations between given two bivariant theories; a Grothendieck transformation is a bivariant version of a natural transformation.

5.1 Fulton–MacPherson’s bivariant theory We quickly recall some basic ingredients of Fulton–MacPherson’s bivariant theory; see [21]. Let V be a category which has a final object pt and on which the fiber product or fiber square is well-defined. Let us suppose that V has a class of maps, called confined maps (e.g. proper maps), which are closed under composition and base change and contain all the identity maps, and a class of fiber squares, called independent squares (e.g. Tor-independent), which satisfy the following conditions. (i) If the two inside squares in X 00

h0

f 00

Y 00

/ X0

g0

f0

h

/ Y0

/X f

g

/X

1At that time Goresky–MacPherson’s homology L-class was not available yet and it was defined only after the theory of Intersection Homology was invented by Mark Goresky and Robert MacPherson in 1980.


or h00

X0

/X

f0

f

Y0

/ X0

h0

g0

g

Z0

/Z

h

are independent, then the outside square is also independent, (ii) Any square of the following forms is independent: X

idX

f

/X

X

f

Y

idX

f

/Y

idX

/X

X

idY

f

/Y

where f W X ! Y is any morphism. A bivariant theory B on a category V with values in the category of graded abelian groups is an assignment to each morphism f

X ! Y in the category V a graded abelian group f

B.X ! Y / which is equipped with the following three basic operations. The i-th component of f

f

B.X ! Y /, i 2 Z, is denoted by Bi .X ! Y /.

Product operations. For morphisms f W X ! Y and g W Y ! Z, the product operation f

g

gf

W Bi .X ! Y / ˝ Bj .Y ! Z/ ! BiCj .X ! Z/ is defined.

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Shoji Yokura

Pushforward operations. For morphisms f W X ! Y and g W Y ! Z with f confined, the pushforward operation gf

g

f W Bi .X ! Z/ ! Bi .Y ! Z/ is defined. Pullback operations. For an independent square X0

g0

/X

f0

f

Y0

g

/ Y;

the pullback operation f

f0

g W Bi .X ! Y / ! Bi .X 0 ! Y 0 / is defined. And these three operations are required to satisfy the seven compatibility axioms (see [21], Part I, §2.2, for details): (B-1) product is associative, (B-2) pushforward is functorial, (B-3) pullback is functorial, (B-4) product and pushforward commute, (B-5) product and pullback commute, (B-6) pushforward and pullback commute, and (B-7) projection formula. idX

We also assume that B has units, i.e. there is an element 1X 2 B0 .X ! X / such that ˛ 1X D ˛ for all morphisms W ! X, all ˛ 2 B.W ! X /; such that 1X ˇ D ˇ for all morphisms X ! Y , all ˇ 2 B.X ! Y /; and such that g 1X D 1X 0 for all g W X 0 ! X. Let B; B0 be two bivariant theories on a category V . Then a Grothendieck transformation from B to B0

W B ! B0 is a collection of homomorphisms B.X ! Y / ! B0 .X ! Y /


for a morphism X ! Y in the category V , which preserves the above three basic operations: (i)

.˛ B ˇ/ D .˛/ B0 .ˇ/,

(ii)

.f ˛/ D f .˛/, and

(iii)

.g ˛/ D g .˛/.

A bivariant theory B is called commutative (see [21], §2.2), if whenever both W

g0

/X

f0

Y

W

f

/Z

g

f0

/Y

g0

g

X

/Z

f

f

g

are independent squares, then for ˛ 2 B.X ! Z/ and ˇ 2 B.Y ! Z/ g ˛ ˇ D f ˇ ˛: If g ˛ ˇ D .1/deg.˛/ deg.ˇ / f ˇ ˛; then it is called skew-commutative. B .X / D B.X ! pt/ becomes a covariant functor for confined morphisms and id

B .X / D B.X ! X / becomes a contravariant functor for any morphisms. As to the

id

grading, Bi .X / D Bi .X ! pt/ and Bj .X / D Bj .X ! X /. Definition 5.1 ([21], Definition 2.6.2, Part I). Let be a class of maps in V , which is closed under compositions and containing all identity maps. Suppose that to each f

! Y / satisfying that f W X ! Y in there is assigned an element .f / 2 B.X (i) .g ı f / D .f / .g/ for all f W X ! Y , g W Y ! Z 2 and idX

(ii) .idX / D 1X for all X with 1X 2 B .X / D B.X ! X / the unit element. Then .f / is called a canonical orientation of f. If we need to refer to which bivariant theory we consider, we denote the bivariant theory B by B .f /. A canonical orientation makes B a contravariant functor and B a covariant functor with the corresponding Gysin homomorphisms: f

g

Proposition 5.2. For the composite X !Y ! Z, if f 2 has a canonical orientation B .f /, then we have the Gysin homomorphism defined by f Š .˛/ D .f / ˛: g

gf

! Z/ ! B.X ! Z/; f Š W B.Y

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which is functorial, i.e. .gf /Š D f Š g Š : In particular, when Z D pt, we have the Gysin homomorphism: f Š W B .Y / ! B .X /: Proposition 5.3. For an independent square X0

g0

/X

f0

Y0

f

g

/ Y:

if g 2 C \ and g has a canonical orientation B .g/, then we have the Gysin homomorphism defined by gŠ .˛/ D g0 .˛ .g//: f0

f

gŠ W B.X 0 ! Y 0 / ! B.X ! Y /; which is functorial, i.e. .gf /Š D gŠ fŠ : In particular, for an independent square f

X ! ? ? idX y

Y ? ?id y Y

X ! Y; f

with f 2 C \ , we have the Gysin homomorphism: fŠ W B .X / ! B .Y /: The symbols f Š and gŠ should carry the information of and the canonical orientation , but it will be usually omitted unless some confusion is possible. Let W B ! B0 be a Grothendieck transformation of two bivariant theories B and idX

B0 and let us assume that there is a bivariant element uf 2 B .X / D B0 .X ! X / such that

.B .f // D uf B0 .f /: This formula is called Riemann–Roch formula (see [21]). The Grothendieck transformation W B ! B0 induces natural transformations

W B ! B0 and W B ! B0 .


For any morphism f W X ! Y we have the commutative diagram

B .Y /

/ B0 .Y /

f

f

B .X /

/ B0 .X /:

For a confined morphism f W X ! Y we have the commutative diagram

B .X /

/ B0 .X /

f

f

B .Y /

/ B0 .Y /:

Furthermore the above Riemann–Roch formula .B .f // D uf B0 .f / gives rise to the following commutative diagrams for the above Gysin homomorphisms fŠ ; f Š : B .X /

fŠ .

fŠ

B .Y /

/ B0 .X /

B .Y / uf /

/ B0 .Y /

/ B0 .Y / uf f Š

fŠ

B .X /

/ B0 .X /:

5.2 A pre-motivic bivariant theory on the category of complex algebraic varieties Let V be the category of complex algebraic varieties, let Prop be the class of proper morphisms, Sm be the class of smooth morphisms, and let any fiber square be an independent square. Theorem 5.4. We define Prop

f

! Y/ MSm .V =X to be the free abelian group generated by the set of isomorphism classes of proper morphisms h W W ! X such that the composite of h and f is a smooth morphism: h 2 Prop and f ı h W W ! Y 2 Sm;

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Shoji Yokura

in other words, h W V ! X is “a left quotient” of a smooth morphism s W V ! Y divided by the given morphism f : s ; f

f ı h D s or h D

h

V@ @@ @@ @@ s @@ @@

Y:

/X ~ ~ ~~ ~~ ~ ~~ f ~~~

Prop

Then the association MSm is a bivariant theory if the three operations are defined as follows. • Product operations. For morphisms f W X ! Y and g W Y ! Z, the product operation f

Prop

g

Prop

gf

Prop

W MSm .V =X ! Y / ˝ MSm .V =Y ! Z/ ! MSm .V =X ! Z/ p

f

Prop

k

is defined for ŒV ! X 2 MSm .V =X ! Y / and ŒW ! Y 2 Prop

g

MSm .V =Y ! Z/, by p

pık 00

k

ŒV ! X ŒW ! Y D ŒV 0 ! X ; and bilinearly extended. Here we consider the following fiber squares p0

V0

/ X0

k 00

f0

k0

k

V

/W

p

/X

f

/Y

g

/ Z:

• Pushforward operations. For morphisms f W X ! Y and g W Y ! Z with f 2 Prop, the pushforward operation Prop

gf

Prop

p

f ıp

g

f W MSm .V =X ! Z/ ! MSm .V =Y ! Z/ is defined by f .ŒV ! X / D ŒV ! Y : and linearly extended.


• Pullback operations. For an independent square X0

g0

/X

f0

Y0

f

g

/ Y;

the pullback operation Prop

f

Prop

p

p0

f0

g W MSm .V =X ! Y / ! MSm .V =X 0 ! Y 0 / is defined by g .ŒV ! X / D ŒV 0 ! X 0 ; and linearly extended. Here we consider the following fiber squares: V0

g 00

p0

X0

p g0

f0

Y

/V

/X f

g

/ Y:

Prop

The bivariant theory MSm shall be called a pre-motivic bivariant relative Grothendieck group on the category of complex algebraic varieties. Theorem 5.5. Let cl be a multiplicative characteristic cohomology class of complex vector bundles, i.e. for a complex vector bundle E over X we have cl.E/ 2 H .X /˝R with a certain commutative ring R and we have cl.E ˚ F / D cl.E/cl.F /. Then there exists a unique Grothendieck transformation from the pre-motivic bivariant relative Prop Grothendieck group MSm to the Fulton–MacPherson bivariant homology theory H Prop

f

f

cl W MSm .V =X ! Y / ! H.X ! Y / ˝ R satisfying the normalization condition that for a smooth morphism f W X ! Y idX

f

cl .ŒX ! X / D cl.Tf / Uf 2 H.X ! Y / ˝ R: f

Here Uf 2 H.X ! Y / is the canonical orientation of the smooth morphism f.

314

Shoji Yokura f

Prop

f

Remark 5.6. (i) cl W MSm .V =X ! Y / ! H.X ! Y / ˝ R can be called a general bivariant theory of pre-motivic characteristic classes. When Y is a point pt, Prop

cl W MSm .V =X ! pt/ ! H.X ! pt/ D HBM .X / ˝ R is a unique natural transformation satisfying the normalization condition that for a smooth variety idX

cl .ŒX ! X / D cl.TX / \ ŒX : In other words, this gives rise to a pre-motivic characteristic classes for singular varieties. In a sense, this could be also a very general solution or answer to the aforementioned MacPherson’s question about the existence of a unified theory of characteristic classes for singular varieties. We emphasize that when it comes to the pre-motivic characteristic class for singular varieties we do not have to require the characteristic class cl to be multiplicative, but it can be any characteristic class. (ii) In particular, we have the following commutative diagrams: Prop Prop Here we set MSm .V =X / D MSm .V =X ! pt/. Prop

MSm .V =X / II w II ww II c w w w II II ww w II w w I$ {w w / F .X / H .X /: Mac c

h

Here .ŒV ! X / D h 11V . Prop

MSm .V =X / LLL v LLL vv v v LLLtd v v LLL v v LLL vv v z % v / H .X / ˝ Q: G0 .X / BMF td

h

Here .ŒV ! X / D h OV . Prop

MSm .V =X / LLL w LLL ww w LLLL ! ww w LLL ww w LLL w {ww % / .X / H .X / ˝ Q: CS L

h

X / D h QV . Here !.ŒV !


(iii) In fact it follows from Hironaka’s resolution of singularities that there exists a surjection Prop MSm .V =X / ! K0 .V =X /: And it turns out that if we require the normalization condition and another extra condition that the degree of the 0-dimensional component of the class cl .CP n / equals Prop 1y Cy 2 C: : : .y/n , the natural transformation cl W MSm .V =X / ! H .X /˝R can be pushed down to the relative Grothendieck group K0 .V =X /, the multiplicative characteristic class has to be the Hirzebruch class and that is the only case; i.e. the following diagram commutes: Prop

MSm .X / KKK vv KKK td v v q vv KKK.y/ v KKK v v KKK vv v z % v / H .X / ˝ QŒy: K0 .V =X / Ty

And one of the main results of our previous paper [8] claims that the above three diaProp grams also commute with Mm .V =X / being replaced by the finer group K0 .V =X /. Thus we are led to the following natural problem. Problem 5.7. Formulate a reasonable bivariant-theoretic analogue f

K0 .V =X ! Y / of the relative Grothendieck group K0 .V =X / so that the following hold. (i) K0 .V =X ! pt/ D K0 .V =X /. f

Prop

f

(ii) Bq W MSm .V =X ! Y / ! K0 .V =X ! Y / is a certain quotient map which Prop specializes to the quotient map q W MSm .V =X / ! K0 .V =X / when Y is a point. f

f

(iii) Ty W K0 .V =X ! Y / ! H.X ! Y / ˝ QŒy is a bivariant-theoretic analogue so that when Y is a point it specializes to the original motivic Hirzebruch class transformation Ty W K0 .V =X / ! H .X / ˝ QŒy, and (iv) the following diagram commutes: f

Prop

MSm .V =X ! Y / PPP p PPP ppp p PPPtd.y/ p Bq ppp PPP p p PPP p p p PP' w p p f / H.X f! Y / ˝ QŒy: K0 .V =X ! Y / Ty

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Shoji Yokura f

Remark 5.8. If we can get a reasonable one K0 .V =X ! Y /, then its associated idX

contravariant functor K 0 .V =X / D K0 .V =X ! X / can be considered as the contravariant counterpart of the relative Grothendieck group K0 .V =X /.

5.3 A bivariant-theoretic relative Grothendieck group f

K0 .V =X ! Y / First we recall the following result of Franziska Bittner [6]. Prop

Theorem 5.9. The group K0 .V =X / is isomorphic to MSm .V =X / modulo the blowup relation Œ; ! X D 0 and ŒBlY X 0 ! X ŒE ! X D ŒX 0 ! X ŒY ! X ;

(bl)

for any following Cartesian diagram (which shall be called the blow-up diagram from here on) E

i0

q0

/ BlY X 0 q

Y

i

/ X0

f

/ X:

with i a closed embedding of smooth (pure dimensional) spaces and f W X 0 ! X proper. Here BlY X 0 ! X 0 is the blow-up of X 0 along Y with exceptional divisor E. Note that all these spaces over X are also smooth (and pure dimensional and/or quasi-projective). We want a bivariant-theoretic analogue of the above “blow-up relation”. Lemma 5.10. Let h W X 0 ! X be a smooth morphism, i W S ! X 0 be a closed embedding such that the composite h ı i W Z ! X is also a smooth morphism. E

i0

q0

S

/ BlS X 0 q

i

/ X0

h

/ X:

In the above diagram, let q W BlS X 0 ! X 0 be the blow-up of X 0 along S and let q 0 W E ! S be the exceptional divisor map of the blow up q W BlS X 0 ! X 0 . Then it follows that h ı q W BlS X 0 ! X is a smooth morphism and h ı q ı i 0 W E ! X is a smooth morphism.


Definition 5.11. For a morphism f W X ! Y we consider the above diagram E

/ BlS X 0

i

q0

q

S

/ X0

i

/X

h

f

/ X:

such that (i) h W X 0 ! X is proper, (ii) f ı h W X 0 ! Y is smooth, (iii) f ı h ı i W Z ! Y is smooth, (iv) q W BlS X 0 ! X 0 is the blow-up of X 0 along S and q 0 W E ! S is the exceptional divisor map of the blow up q W BlS X 0 ! X 0 . f

f

Prop

! Y / be the free abelian subgroup of MSm .V =X ! Y / generated Let BL.V =X by ŒBlS X 0 ! X ŒE ! X ŒX 0 ! X C ŒS ! X and we set

f

Prop

f

K0 .V =X ! Y / D p

MSm .V =X ! Y / f

:

BL.V =X ! Y /

f

p

The class ŒV ! X C BL.V =X ! Y / represented by ŒV ! X shall be p

denoted by ŒV ! X . f

Remark 5.12. When Y D pt, the blowup diagram defining BL.V =X ! pt/ is nothing but the following: E

i0

q0

S

/ BlS X 0 q

i

/ X0

h

/ X:

such that (i) S and X 0 are nonsingular, (ii) h W X 0 ! X is proper, (iii) q W BlS X 0 ! X 0 is the blow-up of X 0 along S, q 0 W E ! S is the exceptional divisor map of the blow up q W BlS X 0 ! X 0 :

318

Shoji Yokura f

Hence BL.V =X ! pt/ is nothing but BL.V =X /, i.e. we have K0 .V =X ! pt/ D K0 .V =X /: f

Theorem 5.13. K0 .V =X ! Y / becomes a bivariant theory with the following three operations induced by the corresponding ones defined in Theorem 5.4. • Product operations. For morphisms f W X ! Y and g W Y ! Z, the product operation f

gf

g

? W K0 .V =X ! Y / ˝ K0 .V =Y ! Z/ ! K0 .V =X ! Z/ is defined by p

k

h

k

ŒV ! X ?ŒW ! Y D ŒV ! X ŒW ! Y : and bilinearly extended. • Pushforward operations. For morphisms f W X ! Y and g W Y ! Z with f 2 Prop, the pushforward operation gf

g

f W K0 .V =X ! Z/ ! K0 .V =Y ! Z/ is defined by

p p f ŒV ! X D f .ŒV ! X /

and linearly extended. Pullback operations. For an independent square X0 f0

g0

Y0

/X

g

f

/ Y;

the pullback operation f

f0

g W K0 .V =X ! Y / ! K0 .V =X 0 ! Y 0 / is defined by

p p g ŒV ! X D g .ŒV ! X /

and linearly extended.


5.4 A bivariant-theoretic motivic Hirzebruch class Theorem 5.14 (A bivariant-theoretic motivic Hirzebruch class). The transformation Ty W K0 .V = / ! H. / ˝ QŒy defined by

p p Ty ŒV ! X D Ty .ŒV ! X /

is a unique natural transformation satisfying the normalization condition that for a smooth morphism f W X ! Y idX Ty ŒX ! X D td.y/ .Tf / Uf : Corollary 5.15. For a morphism X ! pt to a point, the above Grothendieck transformation td.y/ W K0 .V =X ! pt/ ! H.X ! pt/ ˝ QŒy is equal to the motivic Hirzebruch class Ty W K0 .V =X / ! H .X / ˝ QŒy. f

Let Kalg .X ! Y / be Fulton–MacPherson’s bivariant K-theory, introduced in [21]. In the case when y D 0, we have the following corollary. Corollary 5.16. The following diagram commutes f

K0 .V =X ! Y/ MMM ss MMM s s Kalg sss MMMtd s MMM ss s MMM s y s s & f f / H.X Kalg .X ! Y/ ! Y / ˝ Q: BMF td

Let f

f

cBr W F .X ! Y / ! H.X ! Y / be a bivariant Chern class constructed by J.-P. Brasselet [7], i.e. a bivariant-theoretic version of the MacPherson’s Chern class transformation c W F .X / ! H .X /. This is a Grothendieck transformation satisfying the normalization condition (called “weak” normalization condition) that for a smooth variety X

cBr .11X / D c.TX / \ ŒX idX

where 11X 2 F .X ! pt/, c.TX / 2 H.X ! X /, and ŒX 2 H.X ! pt/:

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Shoji Yokura

Conjecture 5.17. The Brasselet bivariant Chern class cBr satisfies the normalization condition (called “strong” normalization condition) that for a smooth mophism fWX !Y

cBr .11f / D c.Tf / Uf f

f

idX

where 11f D 11X 2 F .X ! Y /, c.Tf / 2 H.X ! X / and Uf 2 H.X ! Y /: Remark 5.18. The “strong” normalization condition implies the “weak” normalization condition. Corollary 5.19. If Conjecture 5.17 is correct, then the following diagram commutes: f

K0 .V =X ! Y/ MMM r r M r MMM rr MMMc F rrr r MMM r r r MM& r x r r f / H.X f! Y /: F .X ! Y / Br c

f

Problem 5.20. Define a bivariant-theoretic analogue B.X ! Y / of the cobordism group .X / in such a way that the following diagram commutes: f

! Y/ K0 .V =X KK KK ss s KK ss KK L B sss KK s KK s KK ss s ys % f f / B.X ! Y/ H.X ! Y /: BL

6 A zeta function of the motivic Hirzebruch class Now in the rest of the paper we will discuss a generating function of the motivic Hirzebruch class. We simply call it the “zeta function” of the motivic Hirzebruch class. We take a general look at the zeta functions of covariant functors and natural transformations.


6.1 Some zeta functions Given a sequence fcn gnD1 nD0 , the formal power series 1 X

cn t n

nD0

is called an ordinary generating function (of the sequence), or simply a generating function without qualifier ordinary. There are other types of generating functions such as 1 X tn cn ; nŠ nD0 called exponential generating function and 1 X

cn e n

nD0

tn ; nŠ

called Poisson generating function (e.g. see [55]). If each cn is some invariant .X .n/ / of the n-th symmetric product X .n/ D X n =Sn of some geometric object X, then the generating function 1 X

cn t n

nD0

is called a zeta function of X , and denoted by .X /.t/: Example 6.1 (The case when X is a finite set). Let X be a finite set of cardinality m, i.e. ].X / D m. Let cn be the cardinality of the n-th symmetric product of X, i.e. cn D ].X .n/ / D mHn D m1Cn Cn : Then we have the following ] .X /.t/ D D

1 X nD0 1 X nD0

cn t n ! m1Cn n t n

1 .1 t/m 1 D .1 t/].X/ D

Thus the zeta function ] .X /.t/ of a finite set X is a rational Here we note P function. n c t is expressed as a that for a given sequence fcn g the generating function 1 nD0 n rational function if and only if the sequence is a linear recursive sequence or LRS.

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Shoji Yokura

Example 6.2 (The case when X is a compact topological space). Let X be a compact topological space and let the invariant be the Euler–Poincaré characteristic . Then we have, as mentioned in Introduction, the following formula due to I. Macdonald: .X /.t/ D

1 : .1 t/.X/

This is clearly a generalization of the above formula. Example 6.3 (Hasse–Weil zeta function and Weil conjecture). Let X be an algebraic variety defined over the finite field Fp . Consider the following Hasse–Weil zeta function: 1 X ] X.Fp /.n/ t n : ] .X.Fp //.t/ D nD0

The celebrated Weil conjecture is about the rationality of this zeta function of a nonsingular projective algebraic variety; to be more precise, it is P1 .t/P3 .t/ : : : P2N 1 .t/ , where N D dim X and Pi .t/ is an P2 .t/P4 .t/ : : : P2N .t/ integral polynomial whose degree deg Pi .t/ D dim H i .X / for the cohomology H .X / and

• ] .X.Fp //.t/ D

i

• the absolute value of the roots of Pi .t/ is equal to p 2 . This conjecture was solved by Pierre Deligne in [16] and [17]. Example 6.4 (The Kapranov motivic zeta function). Kapranov introduced the following motivic zeta function: Kap .X /.t/ D

1 X

ŒX .n/ t n 2 K0 .V /ŒŒt:

nD0

Kapranov [29] showed that for a non-singular projective curve the motivic zeta function Kap .X /.t/ is a rational function. But Larsen and Lunts [32] showed that for a surface X the motivic zeta function Kap .X /.t/ is not necessarily a rational function, in fact they showed that it is a rational function if and only if the Kodaira dimension of it is negative. However, when it comes to the Grothendieck ring of Chow motives, which is finer that the Grothendeick ring of algebraic varieties, Y. André [2] showed that if the Chow motive of X is Kimura-finite (see [30]) then the Chow motivic zeta function Chow .X /.t/ D

1 X

ŒCh.X .n/ /t n 2 K0 .C M/ŒŒt

nD0

is a rational function. Here C M denotes the category of Chow motives.


6.2 Covariant functors and their zeta functions Let C be a category satisfying that terminal objects and the limit exist, hence, in particular, for any object X 2 C the product X n D X X X exists in C . Let pt denote a terminal object. Let A be a category of Q-vector spaces, or one can think of a category of abelian groups and then tensor each object with Q. Or more generally one could think of a tensor category with coefficients in a field of characteristic zero. Let F W C ! A be a covariant functor which carries the structure of cross products F W F .X / F .Y / ! F .X Y /: In the rest of the paper, for any functor F W C ! A and for any object X the abelian group F .X / is considered as an F .pt/-module by the cross product F W F .pt/ F .X / ! F .pt X / D F .X /: From now on, unless some confusion is possible, we omit the subscript F from the cross product F . Let F ; H W C ! A be two covariant functors carrying cross products and let T W F ! H be a natural transformation which is compatible with the structures of cross products: / F .X Y / F .X / F .Y / T

H .X / H .Y /

T

/ H .X Y /:

For a covariant functor K W C ! A, the Cartesian product T n W C ! A is simply defined by K n .X / D K.X n /: And for a morphism f W X ! Y , fn W K n .X / ! K n .Y / is of course defined by fn D .f f f / : Then it is obvious that K n is a covariant functor. When n D 0, we understand that for any object X K 0 .X / D K.pt/ and for any morphism f W X ! Y K 0 .f / D idK.pt/ W K.pt/ ! K.pt/: The formal power series of the covariant functor is naturally defined: 1 X nD0

K n t n W C ! AŒŒt D

1 X nD0

At n

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Shoji Yokura

is defined by: for any object X 1 X

n n

K t

.X / D

nD0

1 X

K n .X /t n :

nD0

A natural transformation T W F ! H gives rise to the obvious natural transformation: T W

1 X

F n t n !

nD0

1 X

H nt n:

nD0

For ˛ 2 K.X / we define n

‚ …„ ƒ W K.X / ! K .X / D K.X /I .˛/ D ˛ D ˛ K K ˛ : n

n

n

n

n

Obviously n W K ! K n is a natural transformation and in the case when n D 0 0 W K ! K 0 D K.pt/ is but the pushforward to a terminal object pt. For any power series f .t/ D Pnothing 1 n nD0 an t with an 2 Q we define f ./ W K !

1 X

K nt n

nD0

by, for an element ˛ 2 K.X /, f ./.˛/ D

1 X

an ˛ n t n 2

nD0

1 X

K n .X /t n :

nD0

For a natural transformation T W F ! H the composite T ı n W F ! F n ! H n is denoted by T n W F ! H n : because the diagram F .X / F .X /

T T

/ F .X n / T

H .X / H .X /

commutes we have T n .˛/ D T .˛ n /:

/ H .X n /


Then the composite T ı f ./ W F !

1 X

F n .X /t n !

nD0

1 X

H n .X /t n

nD0

is simply denoted by f .T /. Remark 6.5. Since the Cartesian product T n is defined by cross products, the above simple Cartesian product K n can be replaced by the image of the homomorphism of cross products K K W K.X / K.X / ! K.X n /: If we use this finer one, we denote the finer covariant functor by

K n .X / D Image .K K W K.X / K.X / ! K.X n // :

For just now let us assume furthermore that the category C satisfies the following conditions: (A) for any object X the symmetric product X .n/ D X n =Sn exists in C ; (B) for any object X the projection n W X n ! X .n/ exists in C . Then similarly we can define a finer version of the above. For a covariant functor K W C ! A, the symmetric product K .n/ W C ! A is simply defined by K .n/ .X / D K.X .n/ /: For a natural transformation T W F ! H , we define the symmetric product T .n/

T

.n/

W F ! H .n/

by, for any object X and for any ˛ 2 F .X / T

.n/

.˛/ D n .T .˛/ T .˛// 2 H .n/ .X / D H .X .n/ /:

Here n W X n ! X .n/ is the canonical projection to the quotient space. Then T .n/ is a natural transformation. Since the diagram F .X / F .X /

T T

/ F .X n / T

H .X / H .X /

/ H .X n /

commutes, we have T where ˛ .n/ D n .˛ n /:

n

.n/

.˛/ D T .˛ .n/ /;

/ F .X .n/ / T

n

/ H .X .n/ /

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Shoji Yokura

In this paper we deal with a more general category for which the above additional conditions (A) and (B) might not be satisfied. So we consider the following Sn -invariant one. For a covariant functor K W C ! A, the “Sn -invariant product” K Sn W C ! A is defined as the invariant subgroup under the action of Sn : K Sn .X / D K.X n /Sn D fx 2 K.X n / j g x D x; far all g 2 Sn g: Remark 6.6. As above, we can consider the following finer one: S K Sn .X / D K n .X / n D fx 2 K n .X / j g x D x; for all g 2 Sn g: For a partition … PD .k1 ; k2 ; : : : ; kn / of n,i.e. a collection of non-negative integers ki such that niD1 iki D n, the number ].…/ of permutation 2 Sn of cycle-type … is given by ].…/ D

nŠ : k1 Šk2 Š : : : kn Š1k1 2k2 : : : nkn

For ˛i 2 H ri .X / D H .X ri / with r1 C r2 C C rm D n, the symmetrized cross product of ˛i ’s , denoted by ˛1 ˛2 ˛m , is defined by 1 X ˛1 ˛2 ˛m D .˛1 ˛2 ˛m /: nŠ 2Sn

Note that the operator n D

1 X nŠ 2Sn

is usually called symmetrizer in representation theory. In passing, the following operator 1 X An D sign./ nŠ 2S n

is called alternizer. Definition 6.7. Let … D .k1 ; k2 ; : : : ; kn / be a partition and let ˛ 2 F .X /, and let ‰ D f n j n 2 F .pt/ .n D 1; 2; 3; : : : /g be a sequence of elements of F .pt/. … (i) The ‰-weighted …-cross products of ˛, denoted by ˛‰ , is defined by … D. ˛‰

k1 1 ˛/

.

2 k2 2 .˛//

.

n kn n .˛//

Here m W X ! X m is the diagonal morphism and m m m W F .X / ! F .X / D F .X /:

2 F n .X /:


(ii) The ‰-weighted symmetrized …-cross products of ˛, is defined by … … ˛‰ D n .˛‰ / 2 F Sn .X / D F .X n /Sn ;

that is to say … ˛‰ D.

k1 1 ˛/

.

2 k2 2 .˛//

n kn n .˛// :

.

(iii) The average of all ‰-weighted symmetrized …-cross products of ˛ with all cycle Sn types …’s, denoted by ˛‰ , is defined by Sn ˛‰ D

1 X … ].…/˛‰ 2 F Sn .X /; nŠ …

which is called the total ‰-weighted symmetrized cross products of ˛. Sn as coefficients (iv) The following formal power series of ˛ with ˛‰

ZS ‰ .˛/.t/ D 1 C

1 X

Sn n ˛‰ t 21C

nD1

1 X

F Sn .X /t n

nD1

shall be called the (pre- or stacky) ‰-weighted zeta function of ˛. If we need to refer to which covariant functor we treat, we put the suffix F as ZS F ;‰ .˛/.t/, otherwise we omit the suffix F to avoid messy symbols. When the sequence ‰ is unital, i.e. when n D 1 for all n, then ZS ‰ .˛/.t/ is S simply denoted by Z .˛/.t/ and called the zeta function of ˛. Proposition 6.8. The ‰-weighted zeta function ZS ‰ .˛/.t/ of ˛ is exponential: ZS ‰ .˛/.t/ D exp

1 r X t rD1

r

r r .˛/ :

and we have S S ZS ‰ .˛ C ˇ/.t/ D Z‰ .˛/.t/ Z‰ .ˇ/.t/:

Proof. The proof is standard: ZS ‰ .˛/.t/

D1C

1 X nD1

Sn n ˛‰ t

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Shoji Yokura

D

1 X

X

tn

P

nD0

D

i

1 X nD0

iki Dn

X P

iki Dn

i

].…/ . nŠ

k1 1 ˛/

1 X nD0

D

X P

nD0

iki Dn

i

1 X

X P

iki Dn

i

.

n kn n .˛//

t k1 C2k2 C:::nkn k1 Š : : : kn Š1k1 : : : nkn k1 1 ˛/

. D

2 k2 2 .˛//

.

2 k2 2 .˛//

.

n Y t rkr . k Šr kr rD1 r

r kr r .˛//

n Y 1 tr k Š r rD1 r

r r .˛/

1 X 1 Y 1 tr D kr Š r rD1

.

n kn n .˛//

Q ( with respect to )

kr

kr

r r .˛/

kr D1

D

1 Y

exp

tr

rD1

D exp

r r .˛/

r

1 r X t rD1

r

r r .˛/

:

Remark 6.9. Let T be an indeterminate. Since we have ˛

log.1 T /

D ˛ log.1 T / D ˛

1 X Tr rD1

r

;

we get the following equality: 1 X Tr

exp ˛

rD1

r

D exp.log.1 T /˛ / D .1 T /˛ :

So, in the case when the sequence ‰ is unital, if we understand that .t /r D t r r

and

1 r X t rD1

r

r ˛ D

1 r X t rD1

then we can express the above formula ZS .˛/.t/ D exp

1 r X t rD1

r

r ˛

r

r ˛


symbolically as follows: ZS .˛/.t/ D .1 t /˛ : Definition 6.10. Suppose that conditions (A) and (B) are satisfied. The pushforward of the total symmetrized cross products ˛ Sn of ˛ under the projection n W X n ! X .n/ is denoted by Sn Sn ˛‰ D n ˛‰ 2 F .X .n/ /: We set ZS ‰ .˛/.t/ D 1 C

1 X

Sn n ˛‰ t 21C

nD1

1 X

F .X .n/ /t n :

nD1

Corollary 6.11. If conditions (A) and (B) are satisfied, the ‰-weighted zeta function ZS ‰ .˛/.t/ of ˛ is exponential exactly the same as above and we have S S ZS ‰ .˛ C ˇ/.t/ D Z‰ .˛/.t/ Z‰ .ˇ/.t/:

Definition 6.12. Suppose that the conditions (A) and (B) are satisfied. If there is an assignment fX 2 F .X / for each object X such that fX .n/ D .fX /Sn 2 F .X .n/ / then the assignment f is called a symmetrically distinguished assignment and the element fX is called a symmetrically distinguished element of F .X /. Let V be the category of complex algebraic varieties. Then conditions (A) and (B) are satisfied and we have the following proposition, the proof of which is given in T. Ohmoto [43] and thus is omitted. Proposition 6.13. The characteristic function 11X 2 F .X / is a symmetrically distinguished element, namely we have 11X .n/ D .11X /Sn 2 F .X .n/ /: Remark 6.14. Note that for any integer d such that d 6D 0; 1 the constructible function d 11X 2 F .X / is not a symmetrically distinguished element. Corollary 6.15. We have ZS .11X /.t/ D 1 C

1 X nD1

11X .n/ t n 2 1 C

1 X

F .X .n/ /t n :

nD1

For the constructible function covariant functor F (note that we considered the Q-tensored one F .X / ˝ Q), the pushforward for a mapping X W X ! pt to a point X W F .X / ! F .pt/ D Q

330

Shoji Yokura

is nothing but taking the Euler characteristic , in particular X .11W / D .W / for a subvariety W X. So, suggested by this, for a covariant functor F and a morphism X W X ! pt, where pt is a terminal object, the pushforward X W F .X / ! F .pt/ is called the F -characteristic and denoted by F , mimicking the Euler characteristic . If F is the usual Z-homology theory H , then H W H .X / ! H .pt/ D Z R is the integration X or taking the degree of the 0-dimensional component of any homology class. For a symmetrically distinguished element fX of F .X /, F .X / D F .fX / shall be called the F -characteristic of X. The function F .˛/.t/ D F .ZS .˛/.t// is called the zeta function of the F -characteristic of ˛ and in particular, for a symmetrically distinguished element fX of F .X /, F .X/ .t/ D F .fX /.t/ is called the zeta function of the F -characteristic of X. We set

F S .X /ŒŒt D 1 C

1 X

F Sn .X /t n ;

nD1

which is called the symmetrized F -valued formal power series of X. Proposition 6.16. (i) Let ‰ D f n j n 2 F .pt/ .n D 1; 2; 3; : : : /g be a sequence. For a covariant functor F W C ! A, the correspondence S ZS F ;‰ .t/ W F ! F .X /ŒŒt

defined by S ZS F ;‰ .t/.˛/ D ZF ;‰ .˛/.t/


is a natural transformation and the following diagram commutes: F S C.X /ŒŒt ZS F ;‰ .t/ d dt j tD0 / F .X / F .X /

(ii) F .Z

S

1

.˛/.t// D exp

1 r X t

F

/ F .pt/ŒŒt

d dt j tD0

/ F .pt/:

F

F .˛/ D .1 t/F .˛/ :

r (iii) Suppose that the above conditions (A) and (B) are satisfied and there exist symmetrically distinguished elements fX in F .X / for each object X. Then we have rD1

F .X/ .t/ D .1 t/F .X/ : S The above natural transformation ZS F ;‰ .t/ W F ! F ./ŒŒt shall be called the ‰-weighted zeta function of natural transformation, abusing words.

Remark 6.17. Of course we defined the above ‰-weighted zeta function ZS ‰ .˛/.t/ backward from the expected formula: ZS ‰ .˛/.t/ D exp

1 r X t rD1

r

r r .˛/

:

A typical model of this is of course the constructible function functor F .X / and the Euler characteristic. In this sense the ‰-weighted zeta natural transformation ZS ‰ .t/ could be called a ‰-weighted zeta function of natural transformation of Euler-type. Theorem 6.18. Let ‰ D f n j n 2 K0 .V =pt/ D K0 .V / .n D 1; 2; 3; : : : /g be a sequence of “motivic” classes of complex algebraic varieties. Then the ‰-weighted zeta function of natural transformation S ZS ‰ .t/ W K0 .V =/ ! K0 .V =/ ŒŒt Kap

is a generalization of the Kapranov motivic zeta function ŒM .t/ in the following sense:

332

Shoji Yokura

K0 Z

S

k

.t/.ŒM ! X /.t/ D exp

1 r X t rD1

D exp

r

1 r X t rD1

r

k

K0 .ŒM ! X /

ŒM

D .1 t/ŒM Kap

D ŒM .t/: Remark 6.19. (i) In the case of the constructible function functor F .X /, our ‰-weighS ted zeta function of natural transformation ZS ‰ .t/ W F .X / ! F .X /ŒŒt is described as follows, using the expression of Ohmoto’s paper [43], Remark 3.8. If we let '.t/ D 1 C

1 X

n n .˛/

2 F .X / ˝ QŒŒt;

nD1

where the diagonal of X n is identified with X, ZS ‰ .˛/.t/ D T .'.t// with the correP1 sponding operator T D nD0 Tn W F .X / ˝ QŒŒt ! FX;sy m ŒŒt: (ii) In the case of K0 .V =X /, K .ZS ‰ .ŒM ! X /.t/ is almost the same as 0 the “motivic power series” A.t/ŒM introduced by Gusein-Zade, Luengo and MelliP1 n Hernández [23], where A.t/ D 1 C nD1 n t 2 K0 .V /ŒŒt. More precisely, if n is the first zero term, then ŒM .mod t n /: K0 .ZS ‰ .ŒM ! X /.t/ A.t/

In particular, if the sequence ‰ is unital, then we have K0 .ZS .ŒM ! X /.t/ D

1 X

tn

ŒM

D .1 t/ŒM :

nD0

6.3 Natural transformations and their zeta functions Definition 6.20. Let F ; H W C ! A be covariant functors. Let ‰ D f n j n 2 F .pt/ .n D 1; 2; 3; : : : /g be a sequence of elements of F .pt/. For a natural transformation T W F ! H , we define the Sn -invariant product T Sn : T‰Sn W F ! H Sn by, for any object X and for any ˛ 2 F .X / Sn /: T‰Sn .˛/ D T .˛‰

Lemma 6.21. T‰Sn is a natural transformation.


Proposition 6.22. Suppose that the above conditions (A) and (B) are satisfied. Then for any natural transformation T W F ! H , we have the canonical natural transformation n W H Sn ! H .n/ and the following diagram commutes: kk5 kkk S T‰ n kkkk kkk kkk k k k kkk kkk F SSS SSS SSS SSS SSS SSS .n/ SSS T‰ SSS S)

H Sn H n

n

H .n/ :

Definition 6.23. If the above conditions (A) and (B) are satisfied, then we set H .1/ .X /ŒŒt D

1 X

H .n/ .X /t n :

nD0

Here we set that H .0/ .X / D Q. Definition 6.24. Let T W F ! H be a natural transformation and let ‰ D f F .pt/ .n D 1; 2; 3; : : : /g be a sequence of elements of F .pt/.

nj

n

2

(i) The ‰-weighted zeta function of the natural transformation T is defined by ZS T ;‰ .t/ D

1 X

T‰Sn t n W F ./ ! H S ./ŒŒt:

nD0

Here we set that T‰S0 .X / D 1 for any object X. (ii) If the above conditions (A) and (B) are satisfied, then the ‰-weighted zeta function of the natural transformation T is defined by ZT ;‰ .t/ D

1 X

n T‰Sn t n W F ./ ! H .1/ ./ŒŒt:

nD0

Here we set that T‰.0/ .X / D 1 for any object X.

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Shoji Yokura

In other words, since the natural transformation T W F ! H induces the canonical natural transformation T W F S ./ŒŒt ! H S ./ŒŒt; the above ‰-weighted zeta function of the natural transformation T is nothing but S ZS T ;‰ .t/ D T ı ZF ;‰ .t/:

Corollary 6.25. (1) The ‰-weighted zeta functions ZS T ;‰ .t/ and ZT ;‰ .t/ are both natural transformations. (2) The ‰-weighted zeta functions ZS T ;‰ .t/ and ZT ;‰ .t/ are both exponential, i.e. we have S S ZS T ;‰ .˛ C ˇ/.t/ D ZT ;‰ .˛/.t/ ZT ;‰ .ˇ/.t/;

ZT ;‰ .˛ C ˇ/.t/ D ZT ;‰ .˛/.t/ ZT ;‰ .ˇ/.t/ and explicitly we have ZS T ;‰ .t/

D ZT ;‰ .t/ D exp

1 r X t rD1

r

r r T

:

In particular, in the case when the sequence ‰ is unital, then we have ZS T .t/ D ZT .t/ D exp

1 r X t rD1

r

r T

D .1 t /T :

(3) The following diagram commutes: jj5 jjjj j ZS .t/ .or ZT .t// j j T jjjj jjjj j j j j jjjj F ./ TT TTTT TTTT TTTT TTTT TTTT T TTTT *

H S ./ŒŒt .or H .1/ .ŒŒt /

d dt j tD0

H ./:


For a natural transformation T W F ! H , we let us suppose that F .pt/ D H .pt/ and T .pt/ W F .pt/ ! H .pt/ is the identity. Then we have the following commutative diagram: H S .X /ŒŒt @ ZS .t/ T d dt j tD0 T / H .X / F .X / >> >> >> >> >> >> H F >> >> >> >> >> F .pt/ D H .pt/:

H

/ H .pt/ŒŒt

d dt j tD0

H

/ H .pt/

S The above naïve zeta function of natural transformation ZS T .t/ W F ! H ./ŒŒt of a given natural transformation T W F ! H could be called a zeta function of natural transformation of Chern class-type in the sense that if T W F ! H is the Chern–MacPherson class transformation c W F ! H then ZT .t/ is nothing but the generating series of the Chern–MacPherson class constructed by T. Ohmoto [43].

6.4 A zeta function of the motivic Hirzebruch class For other well-studied characteristic classes such as Baum–Fulton–MacPherson’s Todd class or Riemann–Roch [4] and Goresky–MacPherson’s homology L-class [22] and Cappell–Shaneson’s homology L-class [12], the canonical generating series of them, i.e. the zeta function cannot be described as above. In a sense the above ‰-weighted zeta function of natural transformation is more or less a correct one, but we need to interpret the “multiplication by r ” differently and also we need to Sn /, as done in the following section. For that modify the symmetrized product T .˛‰ purpose the description of the motivic characteristic class or the motivic Hirzebruch characteristic class Ty W K0 .V =X / ! H .X / ˝ QŒy is essential and crucial. In other words, for a general category or in a general set-up, there is no reason to think of such a modification or trick. For the sake of presentation given below, from here we consider Ty instead of Ty , in other words we change our parameter y by y. Thus we have that set-

336

Shoji Yokura

ting y D 1, y D 0, and y D 1 produce the Chern class, Todd class and L-class, respectively. In this section we consider another kind of ‰-weighted zeta natural transformation for the above motivic Hirzebruch characteristic class Ty , but with a completely different meaning of the weight, i.e. the multiplication by each weight r is understood as an Adams operator together with another extra “deformation”. Definition 6.26. We set ‰ZS Ty .t/ D with

1 X

S n n ‰Ty S t W K0 .V =/ ! H .I QŒy/ŒŒt;

nD0

‰Ty S0 .X /

D 1 for any object X, and for any ˛ 2 K0 .V =X /, X 1 ‰Ty Sn .˛/ D ].…/‰Ty .˛ … / 2 H Sn .X / ˝ QŒy nŠ …D.k1 ;:::;kn /

and ‰Ty .˛ … / D

k1 1 Ty .˛/

:::

kr r r Ty r . .˛//

kn n n Ty n . .˛//;

thus by a linear extension we have Sn n ‰Ty S /; .˛/ D ‰Ty .˛

where the operator

is defined by

r

kr r Ty r .˛/ D

1 X 1 kr r T .˛/ ; y 2m rm mD0

where A2m denotes the degree-2m part of the total homology class A. (In this case only even-degree homology classes appear.) To emphasize that “‰” functions as an operator, not simply as a multiplication, we n shall call ‰Ty S .˛/ a ‰-operated symmetrized product of Ty .˛/. Remark 6.27. (i) Note that the above ‰-operated symmetrized product n ‰Ty S .˛/ has a “deformation” in each “r-th component” unlike the previous formal ‰-weighted symmetrized product, which would be simply k1 1 Ty .˛/

:::

r kr r Ty . .˛//

n kn n Ty . .˛//:

Namely, in each “r-th component” r Ty r kr .r .˛//, the original natural transformation Ty is replaced by the deformed one Ty r with the different parameter y r . k (ii) By the definition Ty r kr .r .˛// D Ty r .r .˛// r . Thus it is clear that by the definition of the operator r we have kr kr r r : r Ty r . .˛// D r Ty r . .˛//


(iii) As to the above ‰-operated symmetrized product, in fact one can also define such a ‰-operated symmetrized product (but without the above deformation) for any natural transformation T W F ! H as long as the Q-vector space H .X / has a Zgrading. Namely, for a natural transformation T W F ! H , we can define the “simple” ‰-operated symmetrized product of T .˛/ as follows: 1T

k1

where the operator

.˛/ : : : r

rT

kr

.r .˛//

nT

kn

.n .˛//;

is defined by r T .˛/ D

1 X

1 p m .T .˛//m : . r/ mD0

We make such a definition so that it is compatible with the above special case. Otherwise, if we do not take care of such a compatibility, putting aside the issue of geometric meaning, we could simply define it as follows: r T .˛/ D

1 X 1 .T .˛//m : rm mD0

00 From here on we simply denote ZS Ty .t/ without ‰, since the operator “‰ is used in a unique way.

Theorem 6.28. (1) The zeta function ZS Ty .t/ is a natural transformation.

ZS Ty .t/

is exponential, i.e. for any object X and for any (2) The zeta function element ˛; ˇ 2 K0 .V =X /, we have S S ZS Ty .˛ C ˇ/.t/ D ZTy .˛/.t/ ZTy .ˇ/.t/

and explicitly we have ZS Ty .t/

D exp

1 r X t rD1

r

r r Ty r

:

More in detail we have 8 1 r X ˆ t r ˆ êxp T D .1 t /T1 ; y D 1; ˆ 1 ˆ r ˆ ˆ rD1 ˆ ˆ ˆ 1 r X ˆ t ˆ r ˆ ˆ y D 0; exp r T0 ; ˆ < r rD1 ZS Ty .t/ D 1 1 X X ˆ tr r tr ˆ r ˆ ˆ T exp T exp 1 r 1 ˆ ˆ r r ˆ rDeven ˆ rDodd ˆ ˆ 1 ˆ X T ˆ tr ˆ 2 2 1 r ˆ 2 D .1 t / exp T ˆ r 1 ; y D 1: : r rDodd

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Shoji Yokura

(3) The following diagram commutes: HS .X I QŒy/ŒŒt p7 ppp ZS Ty .t/ pp d pp dt j tD0 ppp p p ppp / H .X I QŒy/ K0 .V =X / Ty

K0

H

/ QŒyŒŒt d dt j tD0

H

/ QŒy

H

K0 .V /

/ QŒy:

y

The diagram K0 .V =X n /Sn 8 q q qq Sn qqq q n q qqq q q qq / K0 .V =X .n/ / K0 .V =X / Sn

commutes and we have the following result. idX

Proposition 6.29. ŒX ! X is a symmetrically distinguished element of K0 .V =X /. Corollary 6.30. For any complex quasi-projective variety X the following holds: idX

H .ZS ! X /.t//: D 1 C Ty .ŒX

1 X

y .X .n/ /t n :

nD1

In particular, (i) when y D 1, we have idX

! X /.t// D .X/ .t/ D .1 t/.X/ I H .ZS T1 .ŒX (ii) when y D 0, we have idX

H .ZS ! X /.t// D a .X/ .t/ D .1 t/ T0 .ŒX

a .X/

:

Here a .X / is the arithmetic genus, i.e. the degree of the 0-dimensional component of the Todd class T0 .X /.


(iii) when y D 1, we have idX

! X /.t// D .X/ .t/ H .ZS T1 .ŒX D .1 t 2 / D

.X/ 2

1 C t .X/ 2

1t

.1 C t/

.X/ .X/ 2

.1 t/

.X/C .X/ 2

:

Here .X / is the Goresky–MacPherson’s signature, i.e. the degree of the 0-dimensional component of the Goresky–MacPherson’s homology L-class T1 .X /. Corollary 6.31. Theorem 6.28 and Corollary 6.30 also hold with and HS ./ŒŒt being replaced by and H.1/ ./ŒŒt, respectively. Corollary 6.32. For any complex projective variety X the following holds: idX

! X /.t// D exp ZS Ty .ŒX D1C

1 r X t rD1 1 X

r

r r Ty r .X /

Ty .X .n/ /t n :

nD1

More details and more related things will be treated in different papers.

References [1]

D. Abramovich, K. Karu, K. Matsuki, and J. Włodarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), 531–572.

[2]

Y. André, Motif de dimension finie (d’après S.- I. Kimura, P. O’Sullivan), Séminaire Bourbaki, Vol. 2003/2004, Astérisque 299 (2005) Exp. 929, 115–145. 322

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Y. André, An introduction to motivic zeta functions of motives, in Motives, quantum field theory, and pseudodifferential operators. Papers from the conference held at Boston University, Boston, MA, June 2–13, 2008, ed. by A. Carey, D. Ellwood, S. Paycha, and S. Rosenberg, Clay Mathematics Proceedings 12, American Mathematical Society and Clay Mathematics Institute, Providence, RI, and Cambridge, MA, 2010, 3–17

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P. Baum, W. Fulton, and R. MacPherson, Riemann–Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 101–145. 286, 304, 335

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P. Baum, W. Fulton and R. MacPherson, Riemann–Roch and topological K-theory for singular varieties, Acta Math. 143 (1979), 155–192.

[6]

F. Bittner, The universal Euler characteristic for varieties of characteristic zero, Compos. Math. 140 (2004), 1011–1032. 316

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[7]

J.-P. Brasselet, Existence des classes de Chern en théorie bivariante, in Analyse et topologie sur les espaces singuliers II, III, Proceedings of the colloquium held at Luminy, July 6–11, 1981. Astérisque, 101–102 (1981), Société Mathématique de France, Paris, 1983, 7–22. 287, 319

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J.-P. Brasselet, J. Schürmann, and S.Yokura, Hirzebruch classes and motivic Chern classes for singular spaces, J. Topol. Anal. 2 (2010), 1 – 55. 286, 287, 290, 300, 301, 302, 304, 315

[9]

J.-P. Brasselet, J. Schürmann, and S. Yokura, Classes de Hirzebruch et classes de Chern motivique, C. R. Acad. Sci. Paris 342 (2006), 325–328. 287, 301, 304

[10] J.-P. Brasselet, J. Schürmann, and S. Yokura, On the uniqueness of bivariant Chern classes and bivariant Riemann–Roch transformations, Adv. Math. 210 (2007), 797–812. [11] J.-P. Brasselet, J. Schürmann, and S. Yokura, On Grothendieck transformation in Fulton– MacPherson’s bivariant theory, J. Pure and Appl. Algebra 211 (2007), 665–684. [12] S. E. Cappell and J. L. Shaneson, Stratifiable maps and topological invariants, J. Amer. Math. Soc. 4 (1991), 521–551. 286, 304, 335 [13] S. E. Cappell, L. Maxim, J. Schürmann, and J. L. Shaneson, Characteristic classes of complex hypersurfaces, Adv. Math. 225 (2010), 2616–2647. [14] S. E. Cappell, L. Maxim, J. Schürmann, J. L. Shaneson, and S. Yokura, Characteristic classes of symmetric products of complex quasi-projective varieties, preprint 2010 arXiv:1008.4299 [15] V. I. Danilov and A. G. Khovanskii, Newton polyhedra and an algorithm for computing Hodge–Deligne numbers, Math. USSR, Izv. 29 (1987), 279–298, translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50 (1986), 925–945. 290 [16] P. Deligne, La conjecture de Weil I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307. 286, 322 [17] P. Deligne, La conjecture de Weil II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252. 286, 322 [18] D. Eisenbud and J. Harris, The geometry of schemes, Graduate Texts in Mathematics 197, Springer, New York 2000. [19] W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Springer, Berlin 1984. 287, 302, 306 [20] W. Fulton and K. Johnson, Canonical classes on singular varieties, Manuscripta Math. 32 (1980), 381–389. [21] W. Fulton and R. MacPherson, Categorical frameworks for the study of singular spaces, Mem. Amer. Math. Soc. 243, 1981. 287, 306, 308, 309, 310, 319 [22] M. Goresky and R. MacPherson, Intersection homology theory, Topology 149 (1980), 155–162. 304, 335 [23] S. M. Gusein-Zade, I. Luengo and A. Melle-Hernández, A power structure over the Grothendieck ring of varieties, Math. Res. Lett. 11 (2004), 49–57. 332 [24] A. Grothendieck, Éléments de géométrie algébrique IV. Étude locale des schémas et des morphismes de schémas, Inst. Hautes Études Sci. Publ. Math. 32 (1967). [25] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I, II, Ann. Math. (2) 79 (1964), 109–103 (I) and 205–326 (II).

Bivariant motivic Hirzebruch class and a zeta function of motivic Hirzebruch class 341 [26] F. Hirzebruch, Topological methods in algebraic geometry, 3 rd enlarged ed., Die Grundlehren der Mathematischen Wissenschaften 131, Springer Verlag New York, New York, 1966; Neue topologische Methoden in der algebraischen Geometrie, 1st German ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.) 9, Springer, Berlin etc., 1956. 305 [27] F. Hirzebruch, T. Berger, and R. Jung, Manifolds and modular forms, Aspects of Mathematics E20, Friedrich Vieweg & Sohn, Braunschweig 1992. 305 [28] B. Kahn, Motivic zeta functions of motives, preprint 2006. arXiv:math/0606424 [29] M. Kapranov, The elliptic curve in the S-duality theory and Eisenstein series for Kac– Moody groups, preprint 2000. arXiv:math/0001005 288, 322 [30] S.-I. Kimura, Chow groups are finite dimensional, in some sense, Math. Ann. 331 (2005) 173–201. 322 [31] M. Kontsevich, Notes on motives in finite characeteristic. In Algebra, arithmetic, and geometry. In honor of Y. I. Manin on the occasion of his 70th birthday, ed by Y. Tschinkel and Y. G. Zarhin, Progress in Mathematics 270, Birkhäuser, Boston 2007, 213–247. [32] M. Larsen and V. A. Lunts, Rationality criteria for motivic zeta functions, Compos. Math. 140 (2004), 1537–1560. 322 [33] N. Levitt, Applications of Poincaré duality spaces to the topology of manifolds, Topology 11 (1972), 205–221. 298 [34] E. Looijenga, Motivic measures, Séminaire Bourbaki, Vol. 1999/2000. Astérisque 276 (2002), 267–297 [35] I. G. Macdonald, The Poincaré polynomial of a symmetric product, Proc. Cambridge Philos. Soc. 58 (1962), 563–568. 286 [36] R. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423–432. 286, 304 [37] R. MacPherson, Characteristic classes for singular varieties, in Proceedings of the 9 th Brazilian Mathematical Colloquium (Poços de Caldas July 9–28, 1973), Vol. II, Instituto de Matemática Pura e Aplicada, São Paulo 1977, 321–327 306 [38] L. Maxim and J. Schürmann, Twisted genera of symmetric products, Selecta Math. (N.S.) 18 (2012), 283–317. [39] L. Maxim and J. Schürmann, Hirzebruch invariants of symmetric products, in Topology of algebraic varieties and singularities. Papers from the Conference on Topology of Algebraic Varieties, in honor of Anatoly Libgober’s 60 th birthday, held in Jaca, June 22–26, 2009, ed. by J. I. Cogolludo-Agustín and Eriko Hironaka, Contemporary Mathematics 538. American Mathematical Society and Real Sociedad Matemática Española, Providence, R.I., and Madrid 2011, 163–177. [40] L. Maxim, J. Schürmann, and M. Saito, Symmetric products of mixed Hodge modules, J. Math. Pures Appl. (9) 96 (2011), 462–483. [41] B. Moonen, Das Lefschetz–Riemann–Roch-Theorem für singuläre Varietäten, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonner Mathematische Schriften 106, Universität Bonn, Mathematisches Institut, Bonn 1978. 286 [42] T. Ohmoto, Equivariant Chern classes of singular algebraic varieties with group actions, Math. Proc. Cambridge Philos. Soc. 140 (2006), 115–134. 286

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Bivariant motivic Hirzebruch class and a zeta function of motivic Hirzebruch class 343 [59] S. Yokura, A singular Riemann–Roch for Hirzebruch characteristics, in Singularities Symposium–Łojasiewicz 70. Dedicated to Stanisław Łojasiewicz on the occasion of his 70 th birthday. Papers from the Symposium on Singularities held at Jagellonian University, Kraków, September 25–29, 1996 and the Seminar on Singularities and Geometry held in Warsaw, September 30–October 4, 1996, ed. by B. Jakubczyk, W. Pawłucki and J. Stasica, Banach Center Publications 44, Polish Academy of Sciences, Institute of Mathematics, Warsaw 1998, 257–268. [60] S. Yokura, On Cappell–Shaneson’s homology L-class of singular algebraic varieties, Trans. Amer. Math. Soc. 347 (1995), 1005–1012. 286, 304 [61] S. Yokura, Oriented bivariant theory I, Internat. J. Math. 20 (2009), 1305–1334. [62] S. Yokura, Motivic characteristic classes, in Topology of stratified spaces. Proceedings of the workshop held in Berkeley, CA, September 8–12, 2008, ed. by G. Friedman, E. Hunsicker, A. Libgober and L. Maxim, Mathematical Sciences Research Institute Publications 58, Cambridge University Press, Cambridge, 2011, 375–418. 285, 290, 304 [63] S. Yokura, Motivic Milnor classes, J. Singul. 1 (2010), 39–59. [64] S. Yokura, Motivic bivariant characteristic classes, preprint 2009. arXiv:1005.1124 [65] S. Yokura, Generalized relative Grothendieck groups and a categorification of additive functions on objects, preprint 2010. 295, 296, 297 [66] B. Youssin, Witt Groups of Derived Categories, K-Theory 11 (1997), 373–395. [67] D. Zagier, Equivariant Pontrjagin classes and applications to orbit spaces. Applications of the G-signature theorem to transformation groups, symmetric products and number theory, Lecture Notes in Mathematics 290, Springer, Berlin and New York, 1972. 286 [68] J. Zhou, Calculations of the Hirzebruch y genera of symmetric products by the holomorphic Lefschetz formula, preprint 1999. arXiv:math/9910029

Minimality of hyperplane arrangements and basis of local system cohomology Masahiko Yoshinaga Department of Mathematics, Kyoto University Sakyo-ku, Kyoto, 606-8502, Japan email: [email protected]

Abstract. The purpose of this paper is applying minimality of hyperplane arrangements to local system cohomology groups. It is well known that twisted cohomology groups with coefficients in a generic rank one local system vanish except in the top degree, and bounded chambers form a basis of the remaining cohomology group. We determine precisely when this phenomenon happens for two-dimensional arrangements.

1 Introduction The purpose of this paper is applying minimality of hyperplane arrangements to local system cohomology groups. In §1.1 and §1.2, we will recall basic notions and results on these topics. In §1.3, we will give the plan of the paper.

1.1 Minimality of hyperplane arrangements Let A D fH1 ; : : : ; Hn g be a hyperplane arrangement in C` . Namely a finite set of affine hyperplanes. We assume each hyperplane Hi D f˛i D 0g C` is defined by an affine linear equation ˛i . We denote the complement of hyperplanes by M.A/ D C` n

n [

Hi :

iD1

After the discovery of a combinatorial description of the ring H .M.A/; Z/ in [15] and of the K.; 1/-property for simplicial arrangements in [4], it has been found that the complement M.A/ of a hyperplane arrangement A has a very special homotopy type among other complex affine varieties. In particular, the following minimality property seems one of the most peculiar characteristics of M.A/; see [6], [18], [17], and [10]. Work

partially supported by JSPS Grant-in-Aid for Young Scientists (B) 20740038.

346

Masahiko Yoshinaga

Theorem 1.1 (Minimality of arrangements). The complement M.A/ is homotopy equivalent to a finite minimal CW-complex X . Namely, X satisfies the following minimality property: the number of k-dimensional cells ]fk- dim cellsg is equal to the k-th Betti number bk .X /. The minimality is expected to be useful for computations of local system cohomology groups. An immediate corollary is the following upper bounds for dimensions of rank one local system cohomology groups, which were conjectured by Aomoto and first proved in [2] by using another method. Corollary 1.2. Let L be a complex rank one local system on M.A/. Then the dimension of the L-coefficients cohomology group is bounded by the Betti number: dim H k .M.A/; L/ bk .M.A//; for k D 0; 1; : : : ; `. For further applications of the minimality to computations of local system cohomology groups, the description of the minimal CW-complex X, in particular the attaching map of each cell, is needed. However Theorem 1.1 does not tell it. It should be noted that the proof of Theorem 1.1 is based on Morse theoretic arguments. The constructions of cells are relying on a transcendental method, namely using gradient flows of a Morse function. Both of the two recent approaches to the problem of describing attaching maps of minimal cells are: • assuming A is defined over the real numbers R, and • describing attaching maps by using combinatorial structure of chambers. However they used different methods. • In [23], we studied Lefschetz’s hyperplane section theorem for M.A/, and described the attaching maps of the top cells. • In [20], Salvetti and Settepanella developed discrete Morse theory on the Salvetti complex, and then described the minimal cell complex by using discrete Morse flows. See [5] and [7] for subsequent developments. Furthermore, in [11], a 2-dimensional algebraic minimal chain complex is described. The present article can be considered as a counterpart of [11].

1.2 Non-resonant local systems A nonempty intersection of elements of A is called an edge. We denote by L.A/ the set of edges. An edge X 2 L.A/ is called a dense edge if the localization AX D fH 2 A j H X g is indecomposable. We denote by D.A/ L.A/ the set of dense edges.

Minimality of hyperplane arrangements and basis of local system cohomology

347

Let D .1 ; : : : ; n / 2 Cn . Then determines a rank one representation of 1 .M.A// by n Z X 1 .M.A// 3 7! exp i d log ˛i 2 C iD1

and the associated local system L D L . In other words, L is determined by the p local monodromy qi D e 2 1 i 2 C around each hyperplane Hi . For an edge X 2 L.A/, denote Y qi : qX D XHi p 1 i

. We also denote the half twist by qi1=2 D e We can embed the affine space C` in CP ` as C` D CP ` n H1 . We call x j H 2 Ag [ fH1 g A1 D fH the projective closure of A. The monodromy of L around the hyperplane at infinity Q H1 is niD1 qi1 . It is natural to define q1 D

n Y

qi1 :

iD1

The structure of the cohomology group H k .M.A/; L/ with local system coefficients has been well studied; see [1], [9], [13], and [21]. In particular, it is known that if L is generic, then the cohomology vanishes except in k D `. Among others, let us recall two results in this direction; see [8], [14], and [3]. Theorem 1.3 ([8]). Suppose that A is defined over R and the local system L satisfies qX ¤ 1; Then H k .M.A/; L / D

for X 2 D.A1 /:

8 ˆ 0 q . (i) For an arbitrary chamber C , if belonging to chq .A/, then C \ F q F>0 0 0 (ii) For any two X , X 2 L.A/ with dim X D dim X D ` q (i.e. satisfying X \ F q D fptg and X 0 \ F q D fptg), if X 6D X 0 ,

hq .X \ F q / 6D hq .X 0 \ F q /: In the remainder of the paper we fix a generic flag F satisfying the above conditions. And also fix the orientation of F q by the oriented basis .@h1 ; : : : ; @hq / of the tangent space Tx F q . Next we further decompose chq .A/ into two subsets.


351

Definition 2.8. Define subsets bchq .A/ and uchq .A/ of chq .A/ by bchq .A/ D fC 2 chq .A/ j C \ F q is boundedg; uchq .A/ D fC 2 chq .A/ j C \ F q is unboundedg:

We note that bch` .A/ D bch.A/. Example 2.9. Let us consider the arrangement of four lines A D fH1 ; H2 ; H3 ; H4 g with a generic flag F as in Figure 2. C2_ C1_

C3_

F 2 D R2 C0

C1

C3 C2

F0 H1

C4_

C4

F1

H2

H3

H4

Figure 2. bchq .A/ and uchq .A/.

Then we have by definition ch0 .A/ D fC0 g;

ch1 .A/ D fC1 ; C2 ; C3 ; C0_ g;

ch2 .A/ D fC1_ ; C2_ ; C3_ ; C4 g;

bch0 .A/ D fC0 g;

bch1 .A/ D fC1 ; C2 ; C3 g;

bch2 .A/ D fC4 g;

uch0 .A/ D ;;

uch1 .A/ D fC0_ g;

uch2 .A/ D fC1_ ; C2_ ; C3_ g:

Theorem 2.10. The involution induces a bijection

W bchq1 .A/ ! uchq .A/: Proof. Suppose C 2 bchq1 .A/, that is, C \ F q1 is bounded. Then we have C _ \ F q1 D ;. By the assumption on the flag, C \ F q is unbounded. Since F q is generic, cl.F q / intersects cl.C / \ H1 transversally. Hence C _ \ F q ¤ ; and it is unbounded. We have C _ 2 uchq .A/. Conversely if C _ 2 uchq .A/, then C __ D C intersects F q1 . Suppose C \F q1 is unbounded. In this case, C _ also intersects F q1 . This contradicts the hypothesis C _ 2 uchq .A/ chq .A/. Corollary 2.11. ]bchq1 .A/ D ]uchq .A/.

352

Masahiko Yoshinaga

Remark 2.12. (1) Corollary 2.11 together with Proposition 2.6 and ]chq .A/ D ]bchq .A/ C ]uchq .A/, gives a “bijective proof” for Zaslavsky’s formula ]bch.A/ D P` ì bi .M.A//. iD0 .1/ (2) The bijective correspondence (Theorem 2.10) plays a crucial role in §4.

3 Minimal complexes Let A be an essentially real arrangement and F be a generic flag as in the previous section. Set F D F `1 ˝C, the complexification of F `1 . Compare the complexified complement M.A/ with the generic hyperplane section M.A/ \ F. Lefschetz’s hyperplane section theorem [12] tells us that M.A/ is homotopy equivalent to the space obtained from M.A/ \ F by attaching some `-dimensional cells. Namely we have the following homotopy equivalence: [ M.A/ .M.A/ \ F / ['i D`; i

where 'i W @D ` ! M.A/\F is the attaching map. In [23], we described the homotopy type of the attaching maps. The `-dimensional cells are naturally encoded by the set ch` .A/ of chambers which do not intersect F `1 . By using the description of attaching maps, we constructed a cochain complex dL

.CŒchq .A/; dL /`qD0 W : : : ! CŒchq .A/ ! CŒchqC1 .A/ ! : : : which computes local system cohomology groups for arbitrary rank one local system L. Namely, we have H .CŒch .A/; dL / ' H .M.A/; L/. In §3.1, we shall describe the cochain complex .CŒch .A/; dL / based on [23], and in §3.2 we investigate the case ` D 2 closely.

3.1 Minimal complex arising from Lefschetz’s Theorem Definition 3.1 (Separating hyperplanes). Let C1 ; C2 2 ch.A/ be chambers. Set Sep.C1 ; C2 / D fH 2 A j H separates C1 and C2 g and 1=2 D qSep.C 1 ;C2 /

Y

qi1=2 :

Hi 2Sep.C1 ;C2 /

To describe the coboundary map dL W CŒchq .A/ ! CŒchqC1 .A/, we need the notion of degree map deg W chq .A/ chqC1 .A/ ! Z;


353

which we will define below. Suppose C 2 chq .A/ and C 0 2 chqC1 .A/ are given. Let D D D q F q be a q-dimensional ball with sufficiently large radius so that every 0-dimensional edge x 2 L.A \ F q / is in the interior of D q . There exists a tangent vector field U.x/ 2 Tx F q for x 2 D which satisfies the following properties: • if x 2 @D, then U.x/ … Tx .@D/, and U.x/ directs inside of D; smallskip • if x 2 H with H 2 A, then U.x/ … Tx .H \ F q / Tx F q and U.x/ directs the side in which C 0 is contained. From the properties above, we have U.x/ ¤ 0 for x 2 @.cl.C / \ D/, where cl.C / is the closure of C in F q . Roughly speaking, the degree deg.C; C 0 / is defined to be the degree of the Gauss map U W @.cl.C / \ D/ ! S q1 : jU j Definition 3.2. Let C 2 chq .A/ and C 0 2 chqC1 .A/. Fix U as above. Then define deg.C; C 0 / as follows. (1) When q D 0, then deg.C; C 0 / D 1. (1) When q D 1, then cl.C / \ D ' Œ1; 1. In this case S 0 ' f˙1g. The degree of the Gauss maps U gD W f˙1g ! f˙1g jU j is defined by

8 ˆ

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