The Geometry of Moduli Spaces of Sheaves

Home , Coherent sheaf, Euler sequence, Sheaf of algebras

Daniel Huybrechts and Manfred Lehn

UniversitÈatGH Essen UniversitÈatBielefeld Fachbereich 6 Mathematik SFB 343 FakultÈatfÈur Mathematik UniversitÈatsstraûe 3 Postfach 100131 D-45117 Essen D-33501 Bielefeld Germany Germany [email protected] [email protected]

Preface

The topic of this book is the theory of semistable coherent sheaves on a smooth algebraic surface and of moduli spaces of such sheaves. The content ranges from the de®nition of a semistable sheaf and its basic properties over the construction of moduli spaces to the birational geometry of these moduli spaces. The book is intended for readers with some background in Algebraic Geometry, as for example provided by Hartshorne's text book [98]. There are at least three good reasons to study moduli spaces of sheaves on surfaces. Firstly, they provide examples of higher dimensional algebraic varieties with a rich and interesting geometry. In fact, in some regions in the classi®cation of higher dimensional varieties the only known examples are moduli spaces of sheaves on a surface. The study of moduli spaces therefore sheds light on some aspects of higher dimensional algebraic geometry. Secondly, moduli spaces are varieties naturally attached to any surface. The understanding of their properties gives answers to problems concerning the geometry of the surface, e.g. Chow group, linear systems, etc. From the mid-eighties till the mid-nineties most of the work on moduli spaces of sheaves on a surface was motivated by Donaldson's ground breaking results on the relation between certain intersection numbers on the moduli spaces and the differentiable structure of the four-manifold underlying the surface. Although the interest in this relation has subsided since the introduction of the extremely powerful Seiberg-Witten invariants in 1994, Donaldson's results linger as a third major motivation in the background; they throw a bridge from algebraic geometry to gauge theory and differential geometry. Part I of this book gives an introduction to the general theory of semistable sheaves on varieties of arbitrary dimension. We tried to keep this part to a large extent self-contained. In Part II, which deals almost exclusively with sheaves on algebraic surfaces, we occasionally sketch or even omit proofs. This area of research is still developing and we feel that some of the results are not yet in their ®nal form. Some topics are only touched upon. Many interesting results are missing, e.g. the Fourier- Mukai transformation, Picard groups of moduli spaces, bundles on the projective plane (or more generally on projective spaces, see [228]), computation of Donaldson polynomials on algebraic surfaces, gauge theoretical aspects of moduli spaces (see the book of Friedman and Morgan [71]). We also wish to draw the readers attention to the forthcomingbook of R. Friedman [69]. Usually, we give references and sometimes historical remarks in the Comments at the end of each chapter. If not stated otherwise, all results should be attributed to others. We apologize for omissions and inaccuracies that we may have incorporated in presenting their work. These notes grew out of lectures delivered by the authors at a summer school at Humboldt- UniversitÈatzu Berlin in September 1995. Every lecture was centered around one topic. In vi Preface writing up these notes we tried to maintain this structure. By adding the necessary background to the orally presented material, some chapters have grown out of size and the global structure of the book has become rather non-linear. This has two effects. It should be possible to read some chapters of Part II without going through all the general theory presented in Part I. On the other hand, some results had to be referred to before they were actually introduced. We wish to thank H. Kurke for the invitation to Berlin and I. Quandt for the organization of the summer school. We are grateful to F. Hirzebruch for his encouragement to publish these notes in the MPI-subseries of the Aspects of Mathematics. We also owe many thanks to S. Bauer and the SFB 343 at Bielefeld, who supported the preparation of the manuscript. Many people have read portions and preliminary versions of the text. We are grateful for their comments and criticism. In particular, we express our gratitude to: S. Bauer, V. Brinzanescu, R. Brussee, H. Esnault, L. GÈottsche, G. Hein, L. Hille, S. Kleiman, A. King, J. Klein, J. Li, S. MÈuller-Stach, and K. O'Grady. While working on these notes the ®rst author was supported by the Max-Planck-Institut fÈur Mathematik (Bonn), the Institute for Advanced Study (Princeton), the Institut des Hautes Etudes Scienti®ques (Bures-sur-Yvette),the UniverstitÈatEssen and byagrantfromtheDFG. The second author was supported by the SFB 343 `Diskrete Strukturen in der Mathematik' at the UniversitÈatBielefeld.

Bielefeld, December 1996 Daniel Huybrechts, Manfred Lehn vii

Contents

Introduction x

I General Theory 1

1 Preliminaries 3

1.1 Some Homological Algebra 3

1.2 Semistable Sheaves 9

1.3 The Harder-Narasimhan Filtration 14

1.4 An Example 19

1.5 Jordan-HÈolder Filtration and S-Equivalence 22

1.6 -Semistability 24

1.7 Boundedness I 27

2 Families of Sheaves 32

2.1 Flat Families and Determinants 32

2.2 Grothendieck's Quot-Scheme 38

2.3 The Relative Harder-Narasimhan Filtration 45

Appendix

2.A Flag-Schemes and Deformation Theory 48

2.B A Result of Langton 55

3 The Grauert-MulichÈ Theorem 57

3.1 Statement and Proof 58

3.2 Finite Coverings and Tensor Products 62

3.3 Boundedness II 68

3.4 The Bogomolov Inequality 71

Appendix

3.A e-Stability and Some Estimates 74

4 Moduli Spaces 79

4.1 The Moduli Functor 80

4.2 Group Actions 81

4.3 The Construction Ð Results 88

4.4 The Construction Ð Proofs 93

4.5 Local Properties and Dimension Estimates 101

4.6 Universal Families 105 viii Contents

Appendix

4.A Gieseker's Construction 109

4.B Decorated Sheaves 110

4.C Change of Polarization 114

II Sheaves on Surfaces 119

5 Construction Methods 121

5.1 The Serre Correspondence 123

5.2 Elementary Transformations 129

5.3 Examples of Moduli Spaces 131

6 Moduli Spaces on K3 Surfaces 141

6.1 Low-Dimensional... 141

6.2 ... and Higher-Dimensional Moduli Spaces 150

Appendix

6.A The Irreducibility of the Quot-scheme 157

7 Restriction of Sheaves to Curves 160

7.1 Flenner's Theorem 160

7.2 The Theorems of Mehta and Ramanathan 164

7.3 Bogomolov's Theorems 170

8 Line Bundles on the Moduli Space 178

8.1 Construction of Determinant Line Bundles 178

8.2 A Moduli Space for -Semistable Sheaves 185

8.3 The Canonical Class of the Moduli Space 195

9 Irreducibility and Smoothness 199

9.1 Preparations 199

9.2 TheBoundary 201

9.3 Generic Smoothness 202

9.4 Irreducibility 203

9.5 Proof of Theorem9.2.2 204

9.6 Proof of Theorem9.3.2 210

10 Symplectic Structures 215

10.1 Trace Map, Atiyah Class and Kodaira-Spencer Map 215

10.2 The Tangent Bundle 222

10.3 Forms on the Moduli Space 223

10.4 Non-Degeneracy of Two-Forms 226 ix

11 Birational properties 230

11.1 Kodaira Dimension of Moduli Spaces 230

11.2 More Results 235

11.3 Examples 238

Bibliography 244

Index 261

Glossary of Notations 265 x

Introduction

It is one of the deep problems in algebraic geometry to determine which cohomology

classes on a projective variety can be realized as Chern classes of vector bundles. In low

X c H X Z dimensions the answer is known. On a curve any class can be realized as

the ®rst Chern class of a vector bundle of prescribed rank r . In dimension two the existence

of bundles is settled by Schwarzenberger's result, which says that for given cohomology

c H X Z H X c H X Z X Z

classes and on a complex surface there

c c exists a vector bundle of prescribed rank with ®rst and second Chern class and , respectively. The next step in the classi®cation of bundles aims at a deeper understanding of the set of all bundles with ®xed rank and Chern classes. This naturally leads to the concept of moduli

spaces.

The case r is a model for the theory. By means of the exponential sequence, the set

Pic c X

of all line bundles with ®xed ®rst Chern class can be identi®ed, although not

c H X O H X Z X

canonically for , with the abelian variety . Furthermore, over

Pic X X

the product there exists a `universal line bundle' with the property that its

L X L X restriction to is isomorphic to the line bundle on . The following features are noteworthy here: Firstly, the set of all line bundles with ®xed Chern class carries a natural

scheme structure, such that there exists a universal line bundle over the product with X . c

This is roughly what is called a moduli space. Secondly, if is in the Neron-Severi group

X Z H X H , the moduli space is a nonempty projective scheme. Thirdly, the moduli space is irreducible and smooth. And, last but not least, the moduli space has a distinguished geometric structure: it is an abelian variety. This book is devoted to the analogous questions for bundles of rank greater than one. Although none of these features generalizes literally to the higher rank situation, they serve as a guideline for the investigation of the

intricate structures encountered there.

For r one has to restrict oneself to semistable bundles in order to get a separated ®nite type scheme structure for the moduli space. Pursuing the natural desire to work with complete spaces, one compacti®es moduli spaces of bundles by adding semistable non-locally

free sheaves. The existence of semistable sheaves on a surface, i.e. the non-emptiness of the

c r c

moduli spaces, can be ensured for large while and are ®xed. Under the same assumptions, the moduli spaces turn out to be irreducible. Moduli spaces of sheaves of rank on a surface are not smooth, unless we consider sheaves with special invariants on special surfaces. Nevertheless, something is known about the type of singularities they can attain.

Concerning the geometry of moduli spaces of sheaves of higher rank, there are two guiding

principles for the investigation. Firstly, the geometric structure of sheaves of rank r

c c reveals itself only for large second Chern number while stays ®xed. In other words, xi

only high dimensional moduli spaces display the properties one expects them to have. Sec-

r c Pic X ondly, contrary to the case , , where is always an abelian variety no

matter whether X is ruled, abelian, or of general type, moduli spaces of sheaves of higher rank are expected to inherit geometric properties from the underlying surface. In particular, the position of the surface in the Enriques classi®cation is of uttermost importance for the geometry of the moduli spaces of sheaves on it. Much can be said about the geometry, but at least as much has yet to be explored. The variety of geometric structures exposed by moduli spaces, which in general are far from being `just' abelian, makes the subject highly attractive to algebraic geometers.

Let us now brie¯y describe the contents of each single chapter of this book. We start out in Chapter 1 by providing the basic concepts in the ®eld. Stability, as it was ®rst introduced for bundles on curves by Mumford and later generalized to sheaves on higher dimensional varieties by Takemoto, Gieseker, Maruyama, and Simpson, is the topic of Section 1.2. This notion is natural from an algebraic as well as from a gauge theoretical point of view, for there is a deep relation between stability of bundles and existence of Hermite-Einstein metrics. This relation, known as the Kobayashi-Hitchin correspondence, was established by the work of Narasimhan-Seshadri, Donaldson and Uhlenbeck-Yau. We will elaborate on the algebraic aspects of stability, but refer to Kobayashi's book [127] for the analytic side (see also [157]). Vector bundles are best understood on the projective line where they always split into the direct sum of line bundles due to a result usually attributed to Grothendieck (1.3.1). In the general situation, this splitting is replaced by the Harder-Narasimhan ®ltration, a ®ltration with semistable factors (Section 1.3). If the sheaf is already semistable, then the Jordan-HÈolder ®ltration ®lters it further, so that the factors become stable. Following Se- shadri, the associated graded object is used to de®ne S-equivalence of semistable sheaves (Section 1.5). Stability in higher dimensions can be introduced in various ways, all gen- eralizing Mumford's original concept. In Section 1.6 we provide a framework to compare the different possibilities. The Mumford-Castelnuovo regularity and Kleiman's boundedness results, which are stated without proof in Section 1.7, are fundamental for the construction of the moduli space. They are needed to ensure that the set of semistable sheaves is small enough to be parametrized by a scheme of ®nite type. Another important ingredient is Grothendieck's Lemma (1.7.9) on the boundedness of subsheaves. Moduli spaces are not just sets of objects; they can be endowed with a scheme structure. The notion of families of sheaves gives a precise meaning to the intuition of what this structure should be. Chapter 2 is devoted to some aspects related to families of sheaves. In Sec- tion 2.1 we ®rst construct the ¯attening strati®cation for any sheaf and then consider ¯at families of sheaves and some of their properties. The Grothendieck Quot-scheme, one of the fundamental objects in modern algebraic geometry, together with its local description will be discussed in Section 2.2 and Appendix 2.A. In this context we also recall the notion of corepresentable functors which will be important for the de®nition of moduli spaces as well. As a consequence of the existence of the Quot-scheme, a relative version of the Harder- xii Introduction

Narasimhan ®ltration is constructed. This and the openness of stability, due to Maruyama, will be presented in Section 2.3. In Appendix 2.A we introduce ¯ag-schemes, a generalization of the Quot-scheme, and sketch some aspects of the deformation theory of sheaves, quotient sheaves and, more general, ¯ags of sheaves. In Appendix 2.B we present a result of Langton showing that the moduli space of semistable sheaves is, a priori, complete. Chapter 3 establishes the boundedness of the set of semistable sheaves. The main tool here is a result known as the Grauert-MÈulich Theorem. Barth, Spindler, Maruyama, Hirscho- witz, Forster, and Schneider have contributed to it in its present form. A complete proof is given in Section 3.1. At ®rst sight this result looks rather technical, butitturnsouttobepow- erful in controlling the behaviour of stability under basic operations like tensor products or restrictions to hypersurfaces. We explain results of Gieseker, Maruyama and Takemoto related to tensor products and pull-backs under ®nite morphisms in Section3.2.Intheproofof boundedness (Section 3.3), we essentially follow arguments of Simpson and Le Potier. The theory would not be complete without mentioning the famous Bogomolov Inequality. We reproduce its by now standard proof in Section 3.4 and give an alternative one later (Sec- tion 7.3). The Appendix to Chapter 3 uses the aforementioned boundedness results to prove a technical proposition due to O'Grady which comes in handy in Chapter 9. The actual construction of the moduli space takes up all of Chapter 4. The ®rst construction, due to Gieseker and Maruyama, differs from the one found by Simpson some ten years later in the choice of a projective embedding of the Quot-scheme. We present Simpson's approach (Sections 4.3 and 4.4) as well as a sketch of the original construction(Appendix4.A). Both will be needed later. We hope that Section 4.2, where we recall some results concerning group actions and quotients, makes the construction accessible even for the reader not familiar with the full machinery of Geometric Invariant Theory. In Section 4.5 deformation theory is used to obtain an in®nitesimal description of the moduli space, including bounds for its dimension and a formula for the expected dimension in the surface case. In particular, we prove the smoothness of the Hilbert scheme of zero-dimensional subschemes of a smooth projective surface, which is originally due to Fogarty. In contrast to the rank one case, a universal sheaf on the product of the moduli space and the variety does not always exist. Conditions for the existence of a (quasi)-universal family are discussed in Section 4.6. In Appendix 4.B moduli spaces of sheaves with an additional structure, e.g. a global section, are discussed. As an application we construct a `quasi-universal family' over a projective birational model of the moduli space of semistable sheaves. This will be useful for later arguments. The dependence of stability on the ®xed ample line bundle on the variety was neglected for many years. Only in connection with the Donaldson invariants was its signi®- cance recognized. Friedman and Qin studied the question from various angles and revealed interesting phenomena. We only touch upon this in Appendix 4.C, where it is shown that for two ®xed polarizations on a surface the corresponding moduli spaces are birational for large second Chern number. Other aspects concerning ®bred surfaces will be discussed in Section 5.3. From Chapter 5 on we mainly focus on sheaves on surfaces. Chapter 5 deals with the exis- xiii

tence of stable bundles on surfaces. The main techniques are Serre's construction (Section 5.1) and Maruyama's elementary transformations (Section 5.2). With these techniques at hand, one produces stable bundles with prescribed invariants like rank, determinant, Chern classes, Albanese classes, etc. Sometimes, on special surfaces, the same methods can in fact be used to describe the geometry of the moduli spaces. Bundles on elliptic surfaces were quite intensively studied by Friedman. Only a faint shadow of his results can be found in Sections 5.3, where we treat ®bred surfaces in some generality and two examples for K3 surface. We continue to consider special surfaces in Chapter 6. Mukai's beautiful results concerning two-dimensional moduli spaces on K3 surfaces are presented in Section 6.1.Someofthe results, due to Beauville, GÈottsche-Huybrechts, O'Grady, concerning higher dimensional moduli spaces will be mentioned in Section 6.2. In the course of this chapter we occasionally make use of the irreducibility of the Quot-scheme of all zero-dimensional quotients of a locally free sheaf on a surface. This is a result originally due to Li and Li-Gieseker. We present a short algebraic proof due to Ellingsrud and Lehn in Appendix 6.A. As a sequel to the Grauert-MÈulich theorem we discuss other restriction theorems in Chap-

ter 7. Flenner's theorem (Section 7.1) is an essential improvement of the former and allows one to predict the -semistability of the restriction of a -semistable sheaf to hyperplane sections. The techniques of Mehta-Ramanathan (Section 7.2) are completely different and

allow one to treat the -stable case as well. Bogomolov exploited his inequality to prove

the rather surprising result that the restriction of a -stable vector bundle on a surface to any curve of high degree is stable (Section 7.3). In Chapter 8 we strive for an understanding of line bundles on moduli spaces. Line bundles of geometric signi®cance can be constructed using the technique of determinant bundles (Section 8.1). Unfortunately, Li's description of the full Picard group is beyond the scope of these notes, for it uses gauge theory in an essential way. We only state a special case of his result (8.1.6) which can be formulated in our framework. Section 8.2 is devoted to the study of a particular ample line bundle on the moduli space and a comparison between the algebraic and the analytic (Donaldson-Uhlenbeck) compacti®cation of the moduli space of stable bundles. We build upon work of Le Potier and Li. As a result we con-

struct algebraically a moduli space of -semistable sheaves on a surface. By work of Li and Huybrechts, the canonical class of the moduli space can be determined for a large class of surfaces (Section 8.3). Chapter 9 is almost entirely a presentation of O'Grady's work on the irreducibility and generic smoothness of moduli spaces. Similar results were obtained by Gieseker and Li. Their techniques are completely different and are based on a detailed study of bundles on ruled surfaces. The main result roughly says that for large second Chern number the moduli space of semistable sheaves is irreducible and the bad locus of sheaves, which are not

-stable or which correspond to singular points in the moduli space, has arbitrary high codimension. In Chapter 10 we show how one constructs holomorphic one- and two-formson the mod- xiv Introduction

uli space starting with such forms on the surface. This re¯ects rather nicely the general philosophy that moduli spaces inherit properties from the underlying surface. We provide the necessary background like Atiyah class, trace map, cup product, Kodaira-Spencer map, etc., in Section 10.1. In Section 10.2 we describe the tangent bundle of the smooth part of the moduli space in terms of a universal family. In fact, this result has been used already in earlier chapters. The actual construction of the forms is given in Section 10.3 where we also prove their closedness. The most famous result concerning forms on the moduli space is Mukai's theorem on the existence of a non-degenerate symplectic structure on the moduli space of stable sheaves on K3 surfaces (Section 10.4). O'Grady pursued this question for surfaces of general type. Chapter 11 combines the results of Chapter 8 and 10 and shows that moduli spaces of semistable sheaves on surfaces of general type are of general type as well. We start with a proof of this result for the case of rank one sheaves, i.e. the Hilbert scheme. Our presentation of the higher rank case deviates slightly from Li's original proof. Other results on the birational type of moduli spaces are listed in Section 11.2. We conclude this chapter with two rather general examples where the birational type of moduli spaces of sheaves on (certain) K3 surfaces can be determined. Part I

General Theory

1 Preliminaries

This chapter provides the basic de®nitions of the theory. After introducing pure sheaves and their homological aspects we discuss the notion of reduced Hilbert polynomials in terms of which the stability condition is formulated. Harder-Narasimhan and Jordan-HÈolder ®ltrations are de®ned in Section 1.3 and 1.5, respectively. Their formal aspects are discussed in Section 1.6. In Section 1.7 we recall the notion of bounded families and the Mumford- Castelnuovo regularity. The results of this section will be applied later (cf. 3.3) to show the boundedness of the family of semistable sheaves. This chapter is slightly technical at times. The reader may just skim through the basic de®nitions at ®rst reading and come back to the more technical parts whenever needed.

1.1 Some Homological Algebra

CohX

Let X be a Noetherian scheme. By we denote the category of coherent sheaves on

E ObCohX X

X . For , i.e. a coherent sheaf on , one de®nes:

E SuppE fx X jE g

De®nition 1.1.1 Ð The support of is the closed set x . Its

dimE

dimension is called the dimension of the sheaf E and is denoted by .

E O E ndE

The annihilator ideal sheaf of , i.e. the kernel of X , de®nes a subscheme

structure on Supp .

d dimF d

De®nition 1.1.2 Ð E is pure of dimension if for all non-trivial coherent

subsheaves F . E Equivalently, E is pure if and only if all associated points of (cf. [172] p. 49) have the

same dimension.

O Y X

Example 1.1.3 Ð The structure sheaf Y of a closed subscheme is of dimension

Y Y dimY dim .Itispureif has no components of dimension less than and no embedded points.

De®nition 1.1.4 Ð The torsion ®ltration of a coherent sheaf E is the unique ®ltration

T E T E E

d dim E T E E i where and i is the maximal subsheaf of of dimension . 4 1 Preliminaries

The existence of the torsion ®ltration is due to the fact that the sum of two subsheaves

F G E i i T E T E

of dimension has also dimension . Note that by de®nition i

E d

is zero or pure of dimension i. In particular, is a pure sheaf of dimension if and only if

T E

d .

X x X

Recall that a coherent sheaf E on an integral scheme is torsion free if for each

s O n fg s E E

x x

and X x multiplication by is an injective homomorphism . Using

T E T E

the torsion ®ltration, this is equivalent to dim . Thus, the property E

of a d-dimensional sheaf to be pure is a generalization of the property to be torsion free.

E F

De®nition 1.1.5 Ð The saturation of a subsheaf F is the minimal subsheaf con-

E F d dim E taining F such that is pure of dimension or zero.

Clearly, the saturation of F is the kernel of the surjection

E E F E F T E F

Next, we brie¯y recall the notions of depth and homological dimension. Let M bea mod-

a m A M

ule over a local ring A. Recall that an element in the maximal ideal of is called -

M M

regular, if the multiplication by a de®nes an injective homomorphism . A sequence

a a m M a M a a M i

i i

is an -regular sequence if is -regular for all . The M

maximal length of an M -regular sequence is called the depth of . On the other hand the

homological dimension, denoted by dh , is de®ned as the minimal length of a projec- A tive resolution of M . If is a regular ring, these two notions are related by the Auslander-

Buchsbaum formula:

M depthM dim A

dh (1.1)

E X dhE maxfdhE j x X g X

For a coherent sheaf on one de®nes x . If is not X

regular, the homological dimension of E might be in®nite. For regular it is bounded by

X dh E dimX

dim and for a torsion free sheaf. Both statements follow from

X dhk x dim X

(1.1). Also note that for a regular closed point x , one has and

E F G F dhE

for a short exact sequence with locally free one has

f dhG g max . In the sequel we discuss some more homological algebra. In particular, we will study the

restriction of pure (torsion free, re¯exive, ) sheaves to hypersurfaces. The reader interested in vector bundles or sheaves on surfaces exclusively might want to skip the next part and to go directly to 1.1.16 or even to the next section. For the sake of completeness and in

order to avoid many ad hoc arguments later on we explain this part in broader generality.

n k

Let X be a smooth projective variety of dimension over a ®eld . Consider a coherent

d E c n d

sheaf E of dimension . The codimension of is by de®nition . The following

S k

generalizes Serre's conditions k ( ):

S depthE minfk dimO cg for all x Supp E

k c x X x

1.1 Some Homological Algebra 5

S S E S

c c c

The condition is vacuous. Condition is equivalent to the purity of . Indeed,

x SuppE dimO c depthE x

is equivalent to the following: if with X x , then .

depth E k x O m E x

X x x x

But x if and only if does not embed into , i.e. is not

E E S E c c

an associated point of . Hence satis®es if and only if is pure. Note, for the

S fx X j dhE g

c x

condition implies that the set of singular points has codimension

Supp E S E c

. More generally, if is normal, then implies that is locally free on an

E SuppE

open subset of Supp whose complement in has at least codimension two. S

The conditions k c can conveniently be expressed in terms of the dimension of certain

local Ext-sheaves.

d c n d Proposition 1.1.6 Ð Let E be a coherent sheaf of dimension and codimension

on a smooth projective variety X .

q q

E xt E SuppE E xt E X

i) The sheaves X are supported on and for

X X

q c co dimE xt E q q c

all . Moreover, X for .

E S co dimE xt E q k X

ii) satis®es the condition k c if and only if for all

c q .

Proof. The ®rst statement in i) is trivial. For the second one takes m large enough such that

q q

Ext E m E m E O m H X E xt H X E xt

X X

X and

X X

Ext E m E xt E H X E m

uses Serre duality X to conclude

n q d M

for . For ii) we apply (1.1) and the fact that for a ®nite module over a regular

dhM maxfq j Ext M A g

ring A one has . Then

depthE min fk dim O cg

x Xx

maxfdim O k cg dhE maxfq jExt E O g

X x x x X x

Ext E O q maxfdim O k cg

x X x X x

q c x X

For all and the following holds:

Ext E O E xt E dim O q k

x X x X x X x

E n H omE O

For a sheaf of dimension , the dual X is a non-trivial torsion free sheaf. If n the dimension of E is less than , then, with this de®nition, the dual is always trivial. Thus

a modi®cation for sheaves of smaller dimension is in order.

d c n d

De®nition 1.1.7 Ð Let E be a coherent sheaf of dimension and let be its

D c

E E xt E

codimension. The dual sheaf is de®ned as X .

c E

If , then differs from the usual de®nition by the twist with the line bundle X ,

E E

i.e. X . The de®nition of the dual in this form has the advantage of being

Y i X Y

independent of the ambient space. Namely, if X and are smooth, is a closed

D D

i E E X i E

embedding and is a sheaf on , then . In particular, this property can be used to de®ne the dual of a sheaf even if the ambient space is not smooth. 6 1 Preliminaries

Lemma 1.1.8 Ð There is a spectral sequence

pq p q

E E xt E xt E E

X X

E E E

In particular, there is a natural homomorphism E .

Proof. The existence of the spectral sequence is standard: take a locally free resolution

L E I X

and an injective resolution and compare the two possible ®ltrations

H omH omL I X

of the total complex associated to the double complex . Note that

q pq

co dimE xt E q E p q

one has X and therefore if . Hence the only non-

E p q p dimX

vanishing -terms lie within the triangle cut out by the conditions ,

cc cc

cc cc DD

q c E E E E E E

and . Moreover, and thus E is

naturally de®ned. 2

pp p p

E E xt E xt E X

The spectral sequence also shows that X is pure of

X X

p E xt E

codimension or trivial. Indeed, one ®rst shows that X is pure of codimension

c c

c E xt E xt E X

. Then the assertion for X follows directly. In fact, we show that

X X

c pc

E xt E S co dim E p E p c

X even satis®es : Since and for , the

exact sequences

pc prcr

E E E

r r

show

pc pc pc

dimE maxfdimE dimE g

prcr

max fdimE g

E p p c

Hence co dim for .

E c

De®nition 1.1.9 Ð A coherent sheaf of codimension is called re¯exive if E is an iso-

DD E morphism. E is called the re¯exive hull of .

We summarize the results: c Proposition 1.1.10 Ð Let E be a coherent sheaf of codimension on a smooth projective

variety X . Then the following conditions are equivalent:

1) E is pure

co dimE xt E q q c

2) X for all

E S c

3) satis®es

4) E is injective.

Similarly, the following conditions are equivalent:

1') is re¯exive, i.e. E is an isomorphism c

2') E is the dual of a coherent sheaf of codimension

co dimE xt E q q c

3') X for all

E S c 4') satis®es .

1.1 Some Homological Algebra 7

Proof. i) have been shown above. If E is injective, then is a subsheaf

c c

E E xt E xt E E X

of the pure sheaf X . Hence is pure as well. If is pure, then

X X

q p q

co dimE xt E q q c E xt E xt E p

X X

X for . Hence for

X X X

q q

q E q c

. In particular, for and, therefore, E is injective.

ii) follows from 1.1.6. Also is obvious. Now assume that con-

co dimE xt E q q c

dition 3') holds true, i.e. that we have X for . Then

q p pq

E E xt E xt p q E p q c X

X for . Hence for .

X X

cc q q

E E q c E

This shows and for . Hence E is an isomorphism, i.e. 1')

holds. It remains to show , but this was explained after the proof of the previous

lemma. 2 DD

Note that the proposition justi®es the term re¯exive hull for E . A familiar example

of a re¯exive sheaf is the following: if Y is a proper normal projective subvariety of

X O dimY X S

, then Y is a re¯exive sheaf of dimension on . Indeed, Serre's condition

S c co dimY c is equivalent to where

The interpretation of homological properties of a coherent sheaf E in terms of local Ext-

E j H

sheaves enables us to control whether the restriction H to a hypersurface shares these properties. Roughly, the properties discussed above are preserved under restriction to hypersurfaces which are regular with respect to the sheaf. Both concepts generalize naturally

to sheaves as follows:

E X

De®nition 1.1.11 Ð Let X be a Noetherian scheme, let be a coherent sheaf on and

X s H X L E

let L be a line bundle on . A section is called -regular if and only if

E L E s s H X L E s

is injective. A sequence is called -regular if is

E s s E L i

-regular for all .

H X L E H jLj

Obviously, s is -regular if and only if its zero set contains none

H jLj E

of the associated points of E . We also say that the divisor is -regular if the corre-

H X L E sponding section s is -regular. The existence of regular sections is ensured

k E

Lemma 1.1.12 Ð Assume X is a projective scheme de®ned over an in®nite ®eld . Let

X E

be a coherent sheaf and let L be a globally generated line bundle on . Then the -regular

divisors in the linear system j form a dense open subscheme.

x x E I

N X

Proof. Let denote the associated points of , and let be the ideal sheaves

X fx g H jLj x H

i i

of the reduced closed subschemes i . Then contains if and only if

Lj jLj P jI L X

is contained in the linear subspace i . Since is globally generated,

L h X L h X I P jLj i

X , so that the linear subspaces are proper subspaces in i

and their complement is open and dense. 2

H jLj Lemma 1.1.13 Ð Let X be a smooth projective variety and .

8 1 Preliminaries

E c S k

i) If is a coherent sheaf of codimension satisfying k c for some integer and

H E E j X S

k c

is -regular, then H , considered as a sheaf on , satis®es .

q q

H E xt E q E xt E j

H X

ii) If in addition is X -regular for all , then

X X

E xt E Lj E S E j S

X H H k c

. In particular, if satis®es k c , then satis®es .

E L E E j

Proof. By assumption we have an exact sequence H . The

associated long exact sequence

q q q

E xt E L E xt E j E xt E

X H X X

X X X

gives

q q q

co dimE xt E j minfco dimE xt E L co dim E xt E g

H X X X

X X X

xt The second regularity assumption implies that the above complex of E -groups splits

up into short exact sequences

q q q

E xt E E xt E L E xt E O

X X H X

X X X

This gives the second assertion. 2

H jLj

Corollary 1.1.14 Ð Let X be a smooth projective variety and .

E c H E E j

i) If is a re¯exive sheaf of codimension and is -regular then H is pure of

codimension c .

E H E E xt E q

ii) If is pure (re¯exive) and is -regular and X -regular for all

E j c

then H is pure (re¯exive) of codimension .

P k

Corollary 1.1.15 Ð Let X be a normal closed subscheme in and an in®nite ®eld.

H jO j H X

Then there is a dense open subset U of hyperplanes such that intersects

properly and such that X is again normal.

X H S

Proof. One must show that is regular in codimension one and satis®es property .

O P O

X H

By assumption X is a re¯exive sheaf on . Hence Corollary 1.1.14 implies that is

jO j X X

re¯exive again for all H in a dense open subset of . Let be the set of singular

X co dim X H X co dim X

X H

points of . Then X . If intersects properly, then

H , too. Hence it is enough to show that a general hyperplane intersects the regular

X X 2 part r eg of transversely, but this is the content of the Bertini Theorem. 1.2 Semistable Sheaves 9

Example 1.1.16 Ð For later use we bring the results down to earth and specify them in the

case of projective curves and surfaces. E

First, let X be a smooth curve. Then a coherent sheaf might be zero or one-dimensional.

E SuppE E T E

If dim , then is a ®nite collection of points. In general,

E T E E E T , where is locally free. Indeed, a sheaf on a smooth curve is torsion free

if and only if it is locally free. E

If X is a smooth surface, then a sheaf of dimension two is re¯exive if and only if it is

E E E

locally free. Any torsion free sheaf E embeds into its re¯exive hull such that

I M

has dimension zero. In particular, a torsion free sheaf of rank one is of the form Z ,

M I

where is a line bundle and Z is the ideal sheaf of a codimension two subscheme. Note

dhE E E

that a for torsion free sheaf E on a surface . The support of is called the

set of singular points of the torsion free sheaf E . We will also use the fact that if a locally

F E T E F

free sheaf is a subsheaf of a torsion free sheaf , then . The restriction

results are quite elementary on a surface: if E is of dimension two and re¯exive, i.e. locally

free, then the restriction to any curve is locally free. If E is purely two-dimensional, i.e. torsion free, then the restriction to any curve avoiding the ®nitely many singular points of

E is locally free.

1.1.17 Determinant bundles Ð Recall the de®nition of the determinant of a coherent

s detE E sheaf. If E is locally free of rank , then is by de®nition the line bundle .

More generally, let E be a coherent sheaf that admits a ®nite locally free resolution

E E E E

n n

det E detE X

De®ne i . The de®nition does not depend on the resolution. If is

a smooth variety, every coherent sheaf admits a ®nite locally free resolution. See exc. III 6.8

dimE dim X det E O

and 6.9 in [98] for the non-projective case. If , then X .

1.2 Semistable Sheaves k

Let X be a projective scheme over a ®eld . Recall that the Euler characteristic of a coherent

i i i i

E E h X E h X E dim H X E

sheaf is , where k . If we ®x an

X P E

ample line bundle O on , then the Hilbert polynomial is given by

m E O m

E d H H jO j d Lemma 1.2.1 Ð Let be a coherent sheaf of dimension and let

be an E -regular sequence. Then

m i

E j P E m E O m

j i

10 1 Preliminaries

Proof. We proceed by induction. If d the assertion is trivial. Assume that d

and that the assertion of the lemma has been proved for all sheaves of dimension . Let

H H

and consider the short exact sequence

E m E m E mj H

Then by the induction hypothesis

m i

E j E m E m E mj

j i

m E m

This means that if f denotes the difference of and the term on the right hand

m f m f f side in the lemma, then f . But clearly , so that vanishes

identically. 2

In particular, P can be uniquely written in the form

dimE

E P E m

E i dimE E

with integral coef®cients i ( ). Furthermore, if the leading

E O

E dimX

coef®cient dim , called the multiplicity, is always positive. Note that

is the degree of X with respect to .

d dimX

De®nition 1.2.2 Ð If E is a coherent sheaf of dimension , then

rkE

d X

is called the rank of E .

d d E

On an integral scheme X of dimension there exists for any -dimensional sheaf an

U X E j rkE

open dense subset such that U is locally free. Then is the rank of the

E j rkE X

vector bundle U . In general, need not be integral, and if is reducible it might

even depend on the polarization.

E E De®nition 1.2.3 Ð The reduced Hilbert polynomial p of a coherent sheaf of dimen-

sion d is de®ned by

P E m

pE m

E d

Recall that there is a natural ordering of polynomials given by the lexicographic order of

g f m g m m

their coef®cients. Explicitly, f if and only if for . Analogously,

g f m g m m f if and only if for . We are now prepared for the de®nition of stability.

1.2 Semistable Sheaves 11

d E

De®nition 1.2.4 Ð A coherent sheaf E of dimension is semistable if is pure and for

E pF pE E E

any proper subsheaf F one has . is called stable if is semistable

F pE F E and the inequality is strict, i.e. p for any proper subsheaf .

We want to emphasize that the notion of stability depends on the ®xed ample line bundle

O O m on X . However, replacing by has no effect. We come back to this problem in 4.C. Notation 1.2.5 Ð In order to avoid case considerations for stable and semistable sheaves

we will occasionally employ the following short-hand notation: if in a statement the word

ª(semi)stableº appears together with relation signs ª º or ª º, the statement encodes

in fact two assertions: one about semistable sheaves and relation signs ªº and ª º, re-

spectively, and one about stable sheaves and relation signs ªº and ª º, respectively. For

pF pE

example, we could say that E is (semi)stable if and only if it is pure and for

E every proper subsheaf F .

An alternative de®nition of stability would have been the following: a coherent sheaf E

d E P F F P E d

of dimension is (semi)stable if d for all proper subsheaves

F . This is obviously the same de®nition except that it does not require explicitly that

E F T E T E

d d

is pure. But applying the inequality to d and using we

P T E T E E

get d . This immediately implies , i.e. is pure.

d E Proposition 1.2.6 Ð Let E be a coherent sheaf of dimension and assume is pure. Then the following conditions are equivalent:

i) E is (semi)stable.

E pF pE

ii) For all proper saturated subsheaves F one has .

E G G pE pG

iii) For all proper quotient sheaves with d one has .

E G pE pG

iv) For all proper purely d-dimensional quotient sheaves one has .

ii iii iv

Proof. The implications i and are obvious. Consider an exact sequence

F E G

E F G P E P F P G F pF

d d d

Using d and , we get

pE G pE pG G d F

d . Since is pure and -dimensional if and only if is

i iii ii iv ii i F

saturated, this yields and . Finally, follows from d

F P F P F F F E 2

d and , where is the saturation of in .

G d

Proposition 1.2.7 Ð Let F and be semistable purely -dimensional coherent sheaves. If

F pG HomF G pF pG f F G f

p , then . If and is non-trivial then

F G pF pG F G d

is injective if is stable and surjective if is stable. If and d

F G F G then any non-trivial homomorphism f is an isomorphism provided or is stable.

12 1 Preliminaries

F G

Proof. Let f be a non-trivial homomorphism of semistable sheaves with

F pG E f pF pE pG

p . Let be the image of . Then . This contradicts

F pG pF pG

immediately the assumption p . If it contradicts the assumption

F E G

that F is stable unless is an isomorphism, and the assumption that is stable unless

E G F G F pF

is an isomorphism. If and have the same Hilbert polynomial d

G pG f F G f

d , then any homomorphism is an isomorphism if and only if is

injective or surjective. 2

EndE

Corollary 1.2.8 Ð If E is a stable sheaf, then is a ®nite dimensional division al-

k k k E EndE gebra over . In particular, if is algebraically closed, then , i.e. is a simple

sheaf. E Proof. If E is stable then according to the proposition any endomorphism of is either

or invertible. The last statement follows from the general fact that any ®nite dimensional

x D n k

division algebra D over an algebraically closed ®eld is trivial: any element would

D 2

generate a ®nite dimensional and hence algebraic commutative ®eld extension of k in .

E EndE k The converse of the assertion in the corollary is not true: if is simple, i.e. ,

then E need not be stable. An example will be given in in 1.2.10.

De®nition 1.2.9 Ð A coherent sheaf E is geometrically stable if for any base ®eld extension

X X Sp ecK X E K

k k K the pull-back is stable.

A stable sheaf need not be geometrically stable. An example will be given in 1.3.9. But note that a stable sheaf on a variety over an algebraically closed ®eld is also geometrically stable (cf. 1.5.11). The corresponding notion of geometrically semistable sheaves does not differ from the ordinary semistability due to the uniqueness of the Harder-Narasimhan ®l- tration (cf. 1.3.7). Historically, the notion of stability for coherent sheaves ®rst appeared in the context of

vector bundles on curves [190]: let X be a smooth projective curve over an algebraically

E r closed ®eld k , and let be a locally free sheaf of rank . The Riemann-Roch Theorem for

curves says

E deg E r g X

where g is the genus of . Accordingly, the Hilbert polynomial is

P E m r deg X m deg E r g deg X m E g r

E deg E r E E

where is called the slope of . Then is said to be (semi)stable, if for

E rkF rkE F E

all subsheaves F with one has . Note that this is

F pE equivalent to our stability condition p . 1.2 Semistable Sheaves 13

Example 1.2.10 Ð Examples of stable or semistable bundles are easily available: any line

L F L

bundle is stable. Furthermore, if is a non-trivial extension with

L L F

line bundles and of degree and , respectively, then is stable: since the degree

F F M F

is additive, we have deg and . Let be an arbitrary subsheaf.

M F M M

If rk , then is a sheaf of dimension zero of length, say , and

F F rkM M L

. If consider the composition . This is either zero

M L M L

or injective. In the ®rst case and therefore . In the

M L M L M

second case and therefore . If , then necessarily

M L M

and would provide a splitting of the extension in contrast to the assumption.

M L L Hence again . On the other hand, a direct sum of line bundles of different degree is not even semistable. By a similar technique, one can also construct

semistable bundles which are not stable, but simple: let X be a projective curve of genus

g k E E

over an algebraically closed ®eld and let and be two non-isomorphic stable

r r E E HomE E

vector bundles of rank and , respectively, with . Then

Ext E E by Proposition 1.2.7. Hence the dimension of can be computed using the

Riemann-Roch formula:

dimExt E E E E r r g

E E E E

Therefore, there are non-trivial extensions . Of course, is semi-

E E

stable, but not stable. We show that E is simple: Suppose is a non-trivial endo-

E E E E E E

morphism. Then the composition must vanish, hence .

id E j k id

E E E

Since is simple, for some scalar . Consider Then

E E E

is trivial when restricted to and hence factorizes through a homomorphism

E E E E E

. If the composition were non-zero, it would be an

isomorphism and hence a multiple of the identity and would provide a splitting of the se-

E E E

quence de®ning . Hence factorizes through some homomorphism . But since

HomE E , one concludes .

If we pass from sheaves on curves to higher dimensional sheaves the notion of stability can be generalized in different ways. One, using the reduced Hilbert polynomial, was presented above. This version of stability is sometimes called Gieseker-stability. Another possible generalization uses the slope of a sheaf. The resulting stability condition is called

Mumford-Takemoto-stability or -stability. Compared with the notion of Gieseker-stability

-stability behaves better with respect to standard operations like tensor products, restric-

tions to hypersurfaces, pull-backs, etc., which are important technical tools. We want to give

d dimX the de®nition of -stability in the case of a sheaf of dimension . For a com-

pletely general treatment compare Section 1.6

d dimX De®nition 1.2.11 Ð Let E be a coherent sheaf of dimension . The degree of

E is de®ned by

degE E rkE O

d d X 14 1 Preliminaries

and its slope by

degE

rkE

On a smooth projective variety the Hirzebruch-Riemann-Roch formula shows deg

H degE degdet E c E H

, where is the ample divisor. In particular, . If we

H deg E E

want to emphasize the dependence on the ample divisor we write and H .

d d

deg E n deg E E n E H

Obviously, and nH .

nH H

d dim X

De®nition 1.2.12 Ð A coherent sheaf E of dimension is -(semi)stable if

T E T E F E F E rkF

d and for all subsheaves with

E rk .

The condition on the torsion ®ltration just says that any torsion subsheaf of E has codi-

E dim X

mension at least two. Observe, that a coherent sheaf of dimension dim is

rkE deg F rkF deg E F E

-(semi)stable if and only if for all subsheaves

F rkE

with rk (compare the arguments after 1.2.4). One easily proves

d dimX Lemma 1.2.13 Ð If E is a pure coherent sheaf of dimension , then one has

the following chain of implications

E is stable E is stable E is semistable E is semistable 2

For later use, we also formulate the following easy observation.

E d dimX

Lemma 1.2.14 Ð Let X be integral. If a coherent sheaf of dimension is

rkE deg E E

-semistable and and are coprime, then is -stable.

F E rkF

Proof. If E is not -stable, then there exists a subsheaf with

E degF rkE degE rkF

rk and . This clearly contradicts the assumption

cdrkE deg E 2 g .

1.3 The Harder-Narasimhan Filtration

Before we state the general theorem, let us consider the special situation of vector bundles

on P over a ®eld .

r P

Theorem 1.3.1 Ð Let E be a vector bundle of rank on . There is a uniquely determined

a a a E O a O a

r r decreasing sequence of integers such that .

1.3 The Harder-Narasimhan Filtration 15

Proof. The theorem is clear for r . Assume that the theorem holds for all vector

r E r

bundles of rank and that is a vector bundle of rank . Then there is a line bundle

a E

O such that the quotient is again a vector bundle: simply take the saturation of

E a O a

any rank 1 subsheaf of . Let be maximal with this property, and let i be a

E O a

decomposition of the quotient . Consider the twisted extension:

O a a O E a

E a O a E

Any section of would induce a non-trivial homomorphism ,

a H E a H O

contradicting the maximality of . Hence . Since

H O a a i a a a

we have also i for all . This implies , so that

a a r

. It remains to show that the sequence splits. But clearly

M M

Ext O a O a HomO a O a

i i

a a a

i i since . We can rephrase the existence part of the theorem as follows:

There is an isomorphism

E V O a

a k

aZ V

for ®nite dimensional vector spaces a , almost all of which vanish. To prove uniqueness

E dimV

amounts to showing that determines the dimensions a . b

We de®ne a ®ltration of E in the following way: for every integer let

H P E b O b E

E E b

denote the canonical evaluation map and b its image. Since has no global sections for b

very negative b and is globally generated for very large , we get a ®nite increasing ®ltration

E E E E

L L

V O a V O a E E

a k a k

Moreover, it is clear that, if , then b . This

ab a

dim V rkE E 2

a a shows: a .

In fact, we proved more than the theorem required, namely the existence of a certain unique split ®ltration, though the splitting homomorphisms are not unique. In general, we still have a ®ltration for a given coherent sheaf with similar properties as above but which is non-split. The following de®nition and theorem give a ®rst justi®cation for the notion of a semistable sheaf: we can think of semistable sheaves as building blocks for arbitrary pure di-

mensional sheaves. Let X be a projective scheme with a ®xed ample line bundle.

16 1 Preliminaries d De®nition 1.3.2 Ð Let E be a non-trivialpuresheafof dimension . A Harder-Narasimhan

®ltration for E is an increasing ®ltration

HN E HN E HN E E

g r HN E HN E i

such that the factors i for are semistable sheaves

d p

of dimension with reduced Hilbert polynomials i satisfying

p E p p p E

max min

E E p E p E min

Obviously, is semistable if and only if is pure and max . Apriori,the E de®nition of the maximal and minimal p of a sheaf depends on the ®ltration. We will see in the next theorem, that the Harder-Narasimhan ®ltration is uniquely determined, so that there is no ambiguity in the notation. For the following lemma, however, we ®x Harder-

Narasimhan ®ltrations for both sheaves:

F G d p F p G max

Lemma 1.3.3 Ð If and are pure sheaves of dimension with min ,

F G

then Hom .

F G i HN F

Proof. Suppose is non-trivial. Let be minimal with i

j HN F HN G j

and let be minimal with i . Then there is a non-trivial ho-

HN HN HN

g r F g r G pg r F p F

momorphism . By assumption min

i j i

p G pg r G 2

max . This contradicts Proposition 1.2.7. j

Theorem 1.3.4 Ð Every pure sheaf E has a unique Harder-Narasimhan ®ltration.

We will prove the theorem in a number of steps:

d F E

Lemma 1.3.5 Ð Let E be a purely -dimensional sheaf. Then there is a subsheaf

E pF pG F G such that for all subsheaves G one has , and in case of equality .

Moreover, F is uniquely determined and semistable. E De®nition 1.3.6 Ð F is called the maximal destabilizing subsheaf of .

Proof. Clearly, the last two assertions follow directly from the ®rst.

E F F

Let us de®ne an order relation on the set of non-trivial subsheaves of by if

F F pF pF

and only if and . Since any ascending chain of subsheaves ter-

E F F E

minates, we have for every subsheaf F a subsheaf which is maximal

F E F

with respect to . Let be -maximal with minimal multiplicity d among all

maximal subsheaves. We claim that F has the asserted properties.

E pG pF

Suppose there exists G with . First, we show that we can assume

F G G F G F F F G

G by replacing by . Indeed, if , then is a proper subsheaf of

F pF G

and hence p . Using the exact sequence

F G F G F G

1.3 The Harder-Narasimhan Filtration 17

P F P G P F G P F G P F G F G d

one ®nds and d

F GpG pF G F G F G F G

d d d

d . Hence,

F GpF G pF G F GpF pG

d d

d . Together with

F pG pF pF G pF pG

the two inequalities p and this shows

G F pG pF F F G

p . Next, ®x with which is maximal in with respect to .

G E pF pG pG

Then let G contain and be -maximal in . In particular, . By the

G F G F G F d

maximality of and we know , since otherwise d contradicting

F F F G pF

the minimality of d . Hence, is a proper subsheaf of . Therefore,

F G pF pG pF pF G pF

p . As before the inequalities and imply

G F G F G 2 pG pG G . Since , this contradicts the assumption on .

The lemma allows to prove the existence part of the theorem: let E be a pure sheaf of

d E

dimension and let be the maximal destabilizing subsheaf. By induction we can assume

E E G G G E E

that has a Harder-Narasimhan ®ltration . If

E E G pE pE E

i denotes the pre-image of , all that is left is to show that .

pE pE E

But if this were false, we would have contradicting the maximality of .

E E

For the uniqueness part assume that and are two Harder-Narasimhan ®ltrations.

pE j E pE E j

Without loss of generality . Let be minimal with . Then the

E E E E

j j

composition j is a non-trivial homomorphism of semistable sheaves.

pE E pE pE pE E

j j j

This implies j by Proposition 1.2.7. Hence,

E pE j E pE

equality holds everywhere, implying so that . But then

E E

because of the semistability of , and one can repeat the argument with the rÃoles of

E E E

and reversed. This shows: . By induction we can assume that uniqueness holds

E E E E E E

for the Harder-Narasimhan ®ltrations of . This shows and ®nishes i

the proof of the uniqueness part of the theorem. 2

d K Theorem 1.3.7 Ð Let E be a pure sheaf of dimension and let be a ®eld extension of

k . Then

HN E K HN E K

k k

i.e. the Harder-Narasimhan ®ltration is stable under base ®eld extension.

E F K E K

Proof. If F is a destabilizing subsheaf then so is . Hence if

K E

E is semistable, then is also semistable. It therefore suf®ces to prove: there exists a

E E HN E K E K HN E K

i i i

®ltration of such that . The sheaves are ®nitely

k L K

presented and hence de®ned over some ®eld L, , which is ®nitely generated

L K k x

over k . Filtering by appropriate sub®elds we can reduce to the case that for

K some single element x and that either

1. K k is purely transcendental or separable, or

2. K k is purely inseparable.

18 1 Preliminaries

G GalK k

In the ®rst case, k is the ®xed ®eld under the action of . In general any

N E K N N K N E

K k k

submodule K is of the form for some submodule

N G N K

if and only if K is invariant under the induced action of on . This applies to all

g G g HN E K

members of the Harder-Narasimhan ®ltration: For any , is again an

HN E K

HN-®ltration, and hence coincides with .

A Der K E K N E K K

In the second case, the algebra k acts on , and can

N N K N E N N A

k k K K

be written as K for some if and only if for all

F HN E K

(Jacobson descent). Let i and consider the composition

F E K E K E K F

Though certainly is not -linear, the composition is:

f f f f mo d F

F F 2 Lemma 1.3.3 imlies . This means , we are done.

A special case of the theorem is the following:

E K k E K

Corollary 1.3.8 Ð If is a semistable sheaf and is a ®eld extension of , then k

is semistable as well. 2

Example 1.3.9 Ð Here we provide an example of a stable sheaf which is not geometrically

X Pro jRx x x x x x H

stable. Let and let be the skew ®eld of real

K I J K J I

quaternions, i.e. the real algebra with generators I J and and relations

J K

and I . De®ne a homomorphism

H O H O

R X R X

H O I x J x K x

of R -left bimodules as right-multiplication by the element .

H O F coker R X

The R -structure inherited by induces an -algebra homomorphism

H End F H

X , which is injective as is a skew ®eld. Complexifying, we get identi®ca-

tions

i P Pro jC u v X Sp ecC i O O

C C

via

u v x uv x u v x

H C M C

and R with

i i

K J I

i i

With respect to these identi®cations, C is right multiplication by

1.4 An Example 19

u uv i u

v i u

v i v uv i

so that C factors as follows

v i u

v i

M C O C O M C O

P P P

C C C

C O i F i F

C C

From this we get P . Since is locally free and semistable by 1.3.8,

C F

the same holdsfor F , but obviously is not geometrically stable. Moreover, by the ¯at base

O dim End F dim End H

X C

extension theorem, R which implies

P P

C C

F End F

X . We claim that is stable. For otherwise there would exist a short exact sequence

L F L L L

with line bundles and of the same degree. Comparison with

L F L L

the complexi®ed situation implies that which leads to the contradiction

End F 2 M R H

X .

1.4 An Example

Here we want to show that the cotangent bundle of the projective space is stable and at the same time supply ourselves with some detailed information which will be needed later in the proof of Flenner's Restriction Theorem 7.1.1. At one point in the proof we will use the

existence and the uniqueness of the Harder-Narasimhan ®ltration.

n V

Let k be algebraically closed and of characteristic . Let be an integer and a

n PV

k -vector space of dimension . We want to study a sequence of vector bundles on

V related to the cotangent bundle P . It is well known that the cotangent bundle is

given by the Euler sequence

V O O P

P (1.2)

V O O P

Here the homomorphism X is the evaluation map for the global sections

of P . Symmetrizing sequence (1.2) we get exact sequences

d d d

S S V O S V O P

P (1.3)

W V

where the map d at a closed point corresponding to a hyperplane is given by

v v v v mo d W v v

i d i d

The assumption that the characteristic of be zero is necessary for the surjectivity of d . More general, we consider the epimorphisms

20 1 Preliminaries

i d di

i S V O S V O i

di d

id i id

di di

where is short for O , and we agree that and

d d

i i

i d K ker i d

for . In particular, d . The subbundles , , form a

d d d

®ltration

d d

K K K K S V O

(1.4)

d d d d

with factors of the same nature:

i j d

Lemma 1.4.1 Ð For there are natural short exact sequences

j j i

K K K i

d di

If j , the sequence is non-split.

j j i

i i

Proof. The ®rst claim follows from the identity . In particular, for

d di

j one gets:

i di

K K K i S O i

d di d If the corresponding short exact sequence were split, there would be a non-trivial homomor-

phism

di d

S K i S V O i

S V O i

On the other hand, applying Hom to the short exact sequence (1.3)

d i

(with d replaced by ), one gets the exact sequence

di d di d

HomS V O S V O i HomS S V O i

di d

Ext S V O S V O i

where the exterior terms vanish, and hence the one in the middle as well. 2 i

Lemma 1.4.2 Ð The slopes of the sheaves K satisfy the following relations:

i S

ii K K K

d d d

Proof. From the exact sequence (1.3) we deduce:

dim S V d

nd nd

d d

dim S V dim S V n

n n Therefore, the slopes

1.4 An Example 21

d d i

i di

i K K S O i i

n n n

K S V

are strictly increasing with i. Since the last term of the sequence is

O , the lemma is proved.

V PV O S V O

The group SL acts naturally on . The sheaves , and also carry

a natural SL -action with respect to which the homomorphisms are equivariant.

Lemma 1.4.3 Ð The vector bundles S have no proper invariant subsheaves.

SLV

Proof. Any invariant subsheaf G must necessarily be a subbundle, since acts tran-

V x PV W V

sitively on P . Let be a closed point corresponding to a hyperplane .

SLV SLV GLW x

The isotropy subgroup x acts via the canonical surjection on the

d d d

x S W G Gx S W

®bre S . For any invariant subbundle the ®bre is an

W S W Gx

GL -subrepresentation. But is an irreducible representation, so that or

d d

S W G G S 2

, which means or .

i d

S V O

Lemma 1.4.4 Ð The bundles K are the only invariant subsheaves of .

d d

Proof. We proceed by induction on d. The case is trivial. Hence, assume that

d d G S V O

and that the assertion is true for all . Let be a proper invariant

i i di

K G G K G G G S O i

i i i

subbundle. Then i and are

d d

i G G G S

i i

also invariant subbundles. Let be minimal with i . Then

O because of 1.4.3. But this isomorphism provides a splitting of the exact sequence

i di

K K S O i

i K S

According to Lemma 1.4.1 this is impossible unless . Since is irre-

G K G K G

ducible, . Therefore, let be the maximal index such that . If

d d

G G GG S V O K

we are done. If not, is a proper invariant subbundle of

V O

S . By the induction hypothesis

G K K K K G G G

d d d

G K 2

so that contradicting the maximality of .

Lemma 1.4.5 Ð The vector bundles K are semistable. Moreover, is -stable,

hence stable.

SLV

Proof. The Harder-Narasimhan ®ltration of K is invariant under the action of be- d

cause of its uniqueness. By the previous lemma, all subsheaves of the Harder-Narasimhan

®ltration also appear in the ®ltration (1.4). But according to Lemma 1.4.2 one has

K i

for all j . Hence, none of these bundles can have a bigger reduced Hilbert poly-

i i

K n

nomial than K , i.e. is semistable. The last assertion follows from ,

d d

2 since -semistability implies -stability whenever degree and rank are coprime (1.2.14). 22 1 Preliminaries

1.5 Jordan-HolderÈ Filtration and S-Equivalence

Just as the Harder-Narasimhan ®ltration splits every sheaf in semistable factors the Jordan-

HÈolder ®ltration splits a semistable sheaf in its stable components. More precisely, d De®nition 1.5.1 Ð Let E be a semistable sheaf of dimension . A Jordan-HÈolder ®ltration

of E is a ®ltration

E E E E

g r E E E pE

i i

such that the factors i are stable with reduced Hilbert polynomial .

E i pE

Note that the sheaves i , , are also semistable with Hilbert polynomial . Taking the direct sum of two line bundles of the same degree one immediately ®nds that a Jordan-

HÈolder ®ltration need not be unique.

Proposition 1.5.2 Ð Jordan-HÈolder ®ltrations always exist. The graded object g r

g r E

i does not depend on the choice of the Jordan-HÈolder ®ltration.

Proof. Any ®ltration of E by semistable sheaves with reduced Hilbert polynomial E

has a maximal re®nement, whose factors are necessarily stable. Now, suppose that and

E are two Jordan-HÈolder ®ltrations of length and , respectively, and assume that the

g r F F F E d d

uniqueness of has been proved for all with d , where is the dimen-

E i E E

sion of and d is the multiplicity. Let be minimal with . Then the composite

E E E E E

map is non-trivial and therefore an isomorphism, for both and

i i i

E E E E pE pE E E

are stable and . Hence , so that there is a

i i i i i i

short exact sequence

E E E E E

i i

F E E F E E

j j

The sheaf inherits two Jordan-HÈolder ®ltrations: ®rstly, let for

F E j i F

j . And secondly, let for and let be the preimage

j j j

E j i F

of E for . The induction hypothesis applied to gives and

j i

M M

E E E E

j j

j j i

E E E 2

Since , we are done.

i i

E E

De®nition 1.5.3 Ð Two semistable sheaves and with the same reduced Hilbert poly-

g r E g r E

nomial are called S-equivalent if .

The importance of this de®nition will become clear in Section 4. Roughly, the moduli space of semistable sheaves parametrizes only S-equivalence classes of semistable sheaves. We conclude this section by introducing the concepts of polystable sheaves and of the socle and the extended socle of a semistable sheaf.

1.5 Jordan-HÈolder Filtration and S-Equivalence 23 E De®nition 1.5.4 Ð A semistable sheaf E is called polystable if is the direct sum of stable sheaves.

As we saw above, every S-equivalence class of semistable sheaves contains exactly one polystable sheaf up to isomorphism. Thus, the moduli space of semistable sheaves in fact parametrizes polystable sheaves.

Lemma 1.5.5 Ð Every semistable sheaf E contains a unique non-trivial maximal polysta-

ble subsheaf of the same reduced Hilbert polynomial. This sheaf is called the socle of E .

Proof. Any semistable sheaf E admits a Jordan-HÈolder ®ltration. Thus there always ex-

E ists a non-trivial stable subsheaf with Hilbert polynomial p . If there were two maximal polystable subsheaves, then, similarly to the proof of 1.5.2, one inductively proves that ev-

ery direct summand of the ®rst also appears in the second. 2

De®nition 1.5.6 Ð The extended socle of a semistable sheaf E is the maximal subsheaf

E pF pE g r F F with and such that all direct summands of are direct summands

of the socle. E

Lemma 1.5.7 Ð Let F be the extended socle of a semistable sheaf . Then there are no

E F HomF E F

non-trivial homomorphisms form F to , i.e. .

E F F E F G

Proof. If G is the image of a non-trivial homomorphism and

G F g r G

denotes its pre-image in E , then contains properly and the direct summands of

F 2

and g r coincide. This contradicts the maximality of the extended socle.

X L E L

Example 1.5.8 Ð Let be a curve and let be a non-trivial

E L

extension of two line bundles of the same degree. The socle of is . The extended socle

L E E L L

of is itself if and it is otherwise.

Lemma 1.5.9 Ð The socle and the extended socle of a semistable sheaf E are invariant

E E under automorphisms of X and . Moreover, if is simple, semistable, and equals its ex-

tended socle, then E is stable.

E F

Proof. The ®rst assertion is clear. Suppose that E is not stable. If equals its socle ,

F E

then E is not simple. Suppose . Since the last factor of a Jordan-HÈolder ®ltration of

F E F F

E F is isomorphicto a submodulein there is a non-trivialhomomorphism , 2

inducing a non-trivial nilpotent endomorphism of E .

E E Lemma 1.5.10 Ð If E is a simple sheaf, then is stable if and only if is geometrically stable.

24 1 Preliminaries K

Proof. Assume E is simple and stable but not geometrically stable. Let be a ®eld ex-

E K E tension of k . According to the previous lemma, the extended socle of is a proper

submodule. The extended socle is invariant under all automorphisms of K k and satis®es

E K E E E k the condition Hom . Thus is already de®ned over . (Compare the

arguments in the proof of Theorem 1.3.7.) 2

Combined with 1.2.8 this lemma shows:

E E Corollary 1.5.11 Ð If k is algebraically closed and is a stable sheaf, then is also ge-

metrically stable. 2

p CohX

Remark 1.5.12 Ð Consider the full subcategory C of consisting of all semi-

p C p stable sheaves E with reduced Hilbert polynomial . Then is an abelian category in which all objects are Noetherian and Artinian. All de®nitions and statements made in this section are just specializations of corresponding de®nitions and statements within this more

general framework. Our stable and polystable sheaves are the simple and semisimple objects

p in C . Be aware of the very different meanings that the word ºsimpleº assumes in these contexts.

1.6 -Semistability

We have encountered already two different stability concepts; using the Hilbert polynomial and the slope, respectively. In fact there are others. We present an approach which allows

one to deal with the different stability de®nitions in a uniform manner. In particular, for - stability it takes care of things happening in codimension two which do not effect the stability condition. As it turns out, almost everything we have said about Harder-Narasimhan and Jordan-HÈolder ®ltrations remains valid in the more general framework.

Let us ®rst introduce the appropriate categories.

Coh X CohX

De®nition 1.6.1 Ð d is the full subcategory of whose objects are sheaves d

of dimension .

X d d dim X Coh

For two integers the category d is a full subcategory of

X Coh X Coh d d . In fact, is a Serre subcategory, i.e. it is closed with respect to subob-

jects, quotients objects and extensions. Therefore, we can form the quotient category.

X Coh Coh X Coh X

d d

De®nition 1.6.2 Ð dd is the quotient category .

Coh X Coh X f F G d

Recall that dd has the same objects as . A morphism

X Coh F G G

in dd is an equivalence class of diagrams of morphisms in

Coh X kers cokers d G F

d such that and are at most -dimensional. and are

1.6 -Semistability 25

X Coh d

isomorphic in dd if they are isomorphic in dimension . Moreover, we say that

X X E ObCoh T E Coh T E T E

d dd d d

dd is pure, if in , i.e. ,

X F E E F Coh

and that is saturated, if is pure in dd .

Q T fP Q T j deg P dg Q T Q T

d d

Similarly, if we let d , then is

Q T Q T Q T

d d a linear subspace, and the quotient space dd inherits a natural

ordering. There is a well de®ned map

X Q T Coh P

dd dd dd

F d

given by taking the residue class of the Hilbert polynomial. For if E and are -dimensional

X Coh P E m P F m

sheaves which are isomorphic as objects in dd then modulo

E X d P E Coh

terms of degree . In particular, dd if and only if in . The

reduced Hilbert polynomials dd are de®ned analogously.

X Coh

We can now introduce a notion of stability in the categories dd which general-

izes the notion given in Section 1.2:

X E ObCoh E

De®nition 1.6.3 Ð dd is (semi)stable , if and only if is pure in

E X F p Coh F p

dd dd dd and if for all proper non-trivial subsheaves one has .

Lemma 1.2.13 immediately generalizes to the following

d j i

Lemma 1.6.4 Ð If E is a pure sheaf of dimension and , then one has:

E Coh X E Coh X dj

is stable in di is stable in

Coh X E Coh X E dj

is semistable in di is semistable in

Coh X Coh X P P d

d d

Example 1.6.5 Ð By de®nition d and . In the case

d one has

d d

T T

P E E E

dd d d

d d

Q T

in dd and hence

d d

T T

p E E E

dd d d

d d

dimX E d

Hence, for d and a sheaf of dimension the (semi)stability in the category

Coh X

dd is equivalent to the -(semi)stability in the sense of 1.2.12.

The veri®cation of the following meta-theorem is left to the reader.

Theorem 1.6.6 Ð All the statements of the previous sections remain true for the categories

X Coh 2

dd if appropriately adopted. The proofs carry over literally.

Two results, however, shall be mentioned explicitly.

26 1 Preliminaries d

Theorem 1.6.7 Ð i) If E is a sheaf of dimension and pure as an object of the category

X X Coh Coh dd dd , then there exists a unique ®ltration in (the Harder-Narasimhan

®ltration)

E E E E

E E Coh X

i dd

such that the factors i are semistable in and their reduced Hilbert poly-

E E E p p

nomials satisfy dd .

X X E ObCoh Coh dd ii) If dd is semistable, then there exists a ®ltration in (the

Jordan-HÈolder ®ltration)

E E E E

E X E E p E E ObCoh p

i i dd i dd dd

such that the factors i and . The

X g r E Coh

graded sheaf of the ®ltration is uniquely determined as an object in dd .

Note that for a pure sheaf the Harder-Narasimhan ®ltration with respect to ordinary stabil-

X Coh

ity is a re®nement of the Harder-Narasimhan ®ltration in dd , whereas the Jordan-

X Coh

HÈolder ®ltration in dd is a re®nement of the standard Jordan-HÈolder ®ltration pro-

vided the sheaf E is semistable.

Example 1.6.5 suggests to extend the de®nition of -stability to sheaves of dimension less

X than dim . We ®rst introduce a modi®ed slope which comes in handy at various places

later on.

E d E E

De®nition 1.6.8 Ð Let be a coherent sheaf of dimension . Then d is de-

E P d P

d d

noted by . Fora polynomial i of degree we write

When working with Hilbert polynomials is the more natural slope, but for historical

E deg E rkE dimX

reasons for a sheaf of dimension will be used whenever

dim X E E

possible. Note that for d the usual slope differs from by the constant

O O E O E

X d X d X

factor d and the constant term . More precisely,

d .

De®nition and Corollary 1.6.9 Ð A coherent sheaf E of dimension is called -(semi)-

Coh X E

stable if it is (semi)stable as an object in dd . Then, is -(semi)stable if and

T E T E F E F E Coh X

d dd

only if d and for all in .

dim X

If d the Harder-Narasimhan and Jordan-HÈolder ®ltration of a torsion free

Coh X

sheaf considered as an object in dd are also called -Harder-Narasimhan and -

Jordan-HÈolder ®ltration, respectively. In this case, if E is torsion free and we require that in

the Harder-Narasimhan ®ltration all factors are torsion free, then the ®ltration is unique in

X g r E Coh . On the other hand, for a torsion free -semistable sheaf the graded sheaf is uniquely de®ned only in codimension one. Since two re¯exive sheaves which are isomorphic in codimension one are isomorphic, we have

1.7 Boundedness I 27

d dimX

Corollary 1.6.10 Ð If E is a -semistable torsion free sheaf of dimension ,

then the re¯exive hull g r of the graded sheaf is independent of the choice of the

Jordan-HÈolder ®ltration. 2

Coh X

The concept of polystability also naturally generalizes to objects in dd : a sheaf

X X E ObCoh E Coh E E

dd dd i

is polystable if i in , where the sheaves are

E X E p Coh p d d E

i dd dd stable in dd and . Again, for such a sheaf is

called -polystable. Since a saturated sheaf of a locally free sheaf is re¯exive and a direct

summand of a locally free sheaf is locally free, one has

E X Coh X

Corollary 1.6.11 Ð A locally free sheaf on is polystable in dd if and only if

E CohX E E E

i i

i in , where the sheaves are -stable locally free sheaves with

F E F E E

. In this case any saturated non-trivial subsheaf with is a direct 2 summand of E .

1.7 Boundedness I

In order to construct moduli spaces one ®rst has to ensure that the set of sheaves one wants to parametrize is not too big. In fact, this is one of the two reasons why one restricts attention to semistable sheaves. As we eventually will show in Section 3.3 the family of semistable sheaves is bounded, i.e. it is reasonably small. This problem is rather intriguing. Here, we give the basic de®nitions, discuss some fundamental results and prove the boundedness of

semistable sheaves on a smooth projective curve.

k O

Let X be a projective scheme over a ®eld and let be a very ample line bundle.

F m

De®nition 1.7.1 Ð Let m be an integer. A coherent sheaf is said to be -regular, if

X F m i i H for all

For the proof of the next lemma we refer the reader to [191] or [124]. m

Lemma 1.7.2 Ð If F is -regular, then the following holds:

m m m

i) F is -regular for all integers .

ii) F is globally generated.

iii) For all n the natural homomorphisms

X F m H X O n H X F m n

H are surjective.

m F Because of Serre's vanishing theorem, for any sheaf F there is an integer such that is

m-regular. And because of i) the following de®nition makes sense: 28 1 Preliminaries

De®nition 1.7.3 Ð The Mumford-Castelnuovo regularity of a coherent sheaf F is the num-

F inf fm Zj m g

ber reg F is -regular .

F F The regularity is reg if and only if is -dimensional.

The following important proposition allows to estimate the regularity of a sheaf F in terms

of its Hilbert polynomial and the number of global sections of the restriction of F to a se-

quence of iterated hyperplane sections. For the proof we again refer to [124].

P Q T T

Proposition 1.7.4 Ð There are universal polynomials i such that the

F d H H d

following holds: Let be a coherent sheaf of dimension and let be an

a b F F j h F j

i i H H

-regular sequence of hyperplane sections. If and

j j

j i j i

then

regF P a b a b a b

d d d 2

De®nition 1.7.5 Ð A family of isomorphism classes of coherent sheaves on X is bounded

k S O F

if there is a -scheme of ®nite type and a coherent S -sheaf such that the given

fF j js S g

k sX family is contained in the set Sp ec a closed point in .

Note that later we use the word family of sheaves in a different setting (cf. Chapter 2.)

Here it still has its set-theoretical meaning.

fF g

I Lemma 1.7.6 Ð The following properties of a family of sheaves are equivalent:

i) The family is bounded.

fP F g

ii) The set of Hilbert polynomials is ®nite and there is a uniform bound

regF I

for all .

fP F g F

iii) The set of Hilbert polynomials is ®nite and there is a coherent sheaf

F F F 2

such that all admit surjective homomorphisms .

As an example consider the family of locally free sheaves on P with Hilbert polynomial

m m

P , that is, bundles of rank 2 and degree 0. We know that any such sheaf is

F O a O a a regF a a isomorphic to a for some . And it is clear that . In particular, this family cannot be bounded, since the regularity can get arbitrarily large. The lemma already suf®ces to prove the boundedness of semistable sheaves on curves:

Corollary 1.7.7 Ð The family of semistable sheaves with ®xed Hilbert polynomial P on a smooth projective curve is bounded. 1.7 Boundedness I 29

Proof. The family of zero-dimensional sheaves with ®xed Hilbert polynomial, i.e. of ®xed length, is certainly bounded. Any integer can be taken as a uniform regularity. For one-

dimensional semistable sheaves one applies Serre duality

H X E m HomE m

The latter space vanishes due to the semistability of E if

g X dr

deg O

r P r deg O m g d 2 where d and are given by .

Combining Lemma 1.7.6 with Proposition 1.7.4 we get the following crucial bounded-

ness criterion:

fF g X

Theorem 1.7.8 (Kleiman Criterion) Ð Let be a family of coherent sheaves on

with the same Hilbert polynomial P . Then this family is bounded if and only if there are

C i d degP F F

constants i , such that for every there exists an -regular

C 2 H H h F j

i d

sequence of hyperplane sections , such that H

j i

Next, we prove a useful boundedness result for quotient sheaves of a given sheaf.

Lemma 1.7.9 (Grothendieck) Ð Let P be a polynomial and an integer. Then there is a

P X k

constant C depending only on and such that the following holds: if is a projective -

E d

scheme with a very ample line bundle O , is a -dimensional sheaf with Hilbert polyno-

regE F d

mial P and Mumford-Castelnuovo regularity and if is a purely -dimensional

F C d

quotient sheaf of E then . Moreover, the family of purely -dimensional quotients

F with bounded from above is bounded.

j X P

Proof. We can assume that X is a projective space: choose an embedding

E j E L P N

and replace by . Then we can choose a linear subspace in of dimension

N d

Supp E P L P

d disjoint from . The linear projection induces a ®nite

Supp E P O d O G

SuppE

map with P . If is a coherent sheaf on

SuppE G G G d

, then is also coherent, and if is purely -dimensional, then the same

is truefor G , which in this case is the same as saying that is torsion free. Moreover, using

the projection formula, we see that G and have the same Hilbert polynomial, regularity

E E E

and . But this implies, that we can safely replace by and hence assume that is a

d P coherent sheaf of dimension d on . The assumption on the regularity allows to write down

a surjective homomorphism

G V O d E P

30 1 Preliminaries

P G P

where V is a vector space of dimension . Note that the bundle depends on and

G E G

only. Any quotient of E is a quotient of as well, and we may therefore replace by .

G F

Let q be a surjective homomorphism onto a torsion free coherent sheaf of rank

s rkG P q

. Then induces a generically surjective homomorphism

s s s

O d deg F q G V O d s detF

P P

deg F s F O

This shows that , and hence P is uniformly bounded.

This proves the ®rst part of the theorem. Now ®x C . In order to prove the second assertion it

s rkG P

is enough to show that the family of pure quotient sheaves F of rank

deg F s C O d q G F

and with P is bounded. For a given quotient

F rkF s

with deg and consider the induced homomorphism

detq

s s

G G G O

and the adjoint homomorphism

G O G

U P F ker j

Let denote the dense open subscheme where is locally free. Then U

kerq j G G

U . Since the quotients of corresponding to these two subsheaves of are torsion

ker

free and since they coincide on a dense open subscheme of P , we must have

kerq F im everywhere, i.e. . Now, the family of such image sheaves certainly is

bounded. 2

Remark 1.7.10 Ð Note that in particular the set of Hilbert polynomials of pure quotients

F with ®xed is ®nite.

Comments: Ð The presentation of the homological algebra in Section 1.1 is inspired by Le Potiers's article [147]. The reader may also consult the books of Okonek, Schneider, Spindler [211] and of Kobayashi [127]. For the details concerning the de®nition of the determinant 1.1.17 of a coherent sheaf see the article of Knudson and Mumford [126]. Ð The concept of stable vector bundles on curves goes back to Mumford [190] and was later gen-

eralized by Takemoto [242] to -stable vector bundles on higher dimensional varieties. The notion of stability using the Hilbert polynomial appears ®rst in Gieseker's paper [77] for sheaves on surfaces

and in Maruyama's paper[162] for sheaves on varieties of arbitrary dimension. Later Simpson introduced pure sheaves and their stability ([238], also [145]). This led him to consider the multiplicity

of a coherent sheaf instead of the slope . Ð In modern language Theorem 1.3.1 was proved by Grothendieck in [92]. Ð The Harder-Narasamhan ®ltration, as the name suggest, was introduced by Harder and Nara- simhan in [95]. For generalizations see articles by Maruyama or Shatz [164], [236]. In particular, 1.3.4 in the general form was proved in [236]. Another important notion is the notion of the Harder- Narasimhan polygon which can also be found in Shatz' paper [236]. 1.7 Boundedness I 31

Ð The example in Section 1.4 is due to Flenner [63]. For other results concerning bundles on projective spaces see [211]. Ð S-equivalence was again ®rst de®ned for bundles on curves by Seshadri [233]. There, two S- equivalent sheaves are called strongly equivalent. Ð Langton de®ned the socle and the extended socle in [135]. For another reference see the paper of Mehta and Ramanathan [176]. Ð De®nitions 1.7.1, 1.7.3 and Lemma 1.7.2 can be found in Mumford's book [191]. Proposition 1.7.4 is proved in [191] for the special case of ideal sheaves and in general in Kleiman's exposÂein [124]. Lemmas 1.7.6 and 1.7.9 are taken from [93]. 32

2 Families of Sheaves

In the ®rst chapter we proved some elementary properties of coherent sheaves related to semistability. The main topic of this chapter is the question how these properties vary in algebraic families. A major technical tool in the investigations here is Grothendieck's Quot- scheme. We give a complete existence proof in Section 2.2 and discuss its in®nitesimal structure. As an application of this construction we show that the property of being semistable is open in ¯at families and that for ¯at families the Harder-Narasimhan ®ltrations of the members of the family form again ¯at families, at least generically. In the appendix the notion of the Quot-scheme is slightly generalized to Flag-schemes. We sketch some parts of deformation theory of sheaves and derive important dimension estimates for Flag-schemes that will be used in Chapter 4 to get similar a priori estimates for the dimension of the moduli space of semistable sheaves. In the second appendix to this chapter we prove a theorem due to Langton, which roughly says that the moduli functor of semistable sheaves is proper (cf. Chapter 4 and Section 8.2).

2.1 Flat Families and Determinants

X S g T S S

Let f be a morphism of ®nite type of Noetherian schemes. If is an -

X T X g X X

S X T

scheme we will use the notation T for the ®bre product , and and

f X T s S f s Sp eck s X

T S

T for the natural projections. For the ®bre

X F O F g F F

X T s

is denoted s . Similarly, if is a coherent -module, we write and

F F s S F j s

X . Often, we will think of as a collection of sheaves parametrized by . The

s F requirement that the sheaves s and their properties should vary `continuously' is made pre-

cise by the following de®nition:

f O

De®nition 2.1.1 Ð A ¯at family of coherent sheaves on the ®bres of is a coherent X - S

module F which is ¯at over .

x X F

Recall that this means that for each point the stalk x is ¯at over the local ring

O F S F T T S F F

Sf . If is -¯at, then is -¯at for any base change . If

O F S F F

is a short exact sequence of coherent X -sheaves and if is -¯at then is

S F S X F S F S

-¯at if and only if is -¯at. If then is -¯at if and only if is locally free. S

A special case that will occur frequently in these notes is the following: k is a ®eld,

Y k X S Y

and are -schemes and k . In this situation the natural projections will almost

X S q X Y always be denoted by p and , and we will say `sheaves on Y' rather 2.1 Flat Families and Determinants 33

than `sheaves on the ®bres of p'.

f X S O

Assume from now on that is a projective morphism and that X is an

f X O X F s

-ample line bundle on , i.e. the restriction of X to any ®bre is ample. Let be O

a coherent X -module. Consider the following assertions: S

1. F is -¯at

m f F m

2. For all suf®ciently large the sheaves are locally free.

P F s S

3. The Hilbert polynomial s is locally constant as a function of .

S Proposition 2.1.2 Ð There are implications 1 2 3. If is reduced then also 3 1.

Proof. Thm. III 9.9 in [98] 2

This provides an important ¯atness criterion. If S is not reduced, it is easy to write down

counterexamples to the implication 3 1. However, in the non-reduced case the following

criteria are often helpful:

S S

Lemma 2.1.3 Ð Let be a closed subscheme de®ned by a nilpotent ideal sheaf

I O F S F S

S . Then is -¯at if and only if is -¯at and the natural multiplication map

F I F I 2

O is an isomorphism.

F F F O

Lemma 2.1.4 Ð Let be a short exact sequence of X -modules.

F F S F S s S F

If is -¯at, then is -¯at if and only if for each the homomorphism s s

is injective. 2

For proofs see Thm. 49 and its Cor. in [172]. The following theorem of Mumford turns out to be extremely useful as it allows us to `¯at-

ten' any coherent sheaf by splitting up the base scheme in an appropriate way.

X S

Theorem 2.1.5 Ð Let f be a projective morphism of Noetherian schemes, let

X S F

O be an invertible sheaf on which is very ample relative , and let be a coherent

O P fP F js S g F s

X -module. Then the set of Hilbert polynomials of the ®bres of is

S S

®nite. Moreover, there are ®nitely many locally closed subschemes P , indexed by the

polynomials P , with the following properties:

S S j

1. The natural morphism P is a bijection.

S S g F S

2. If g is a morphism of Noetherian schemes, then is ¯at over if and

X j

only if g factorizes through . F Such a decomposition is called a ¯attening strati®cation of S for . It is certainly unique. We begin with a weaker version of this due to Grothendieck: 34 2 Families of Sheaves

Lemma 2.1.6 Ð Under the assumptions of the theorem there exist ®nitely many pairwise

S S S F S

S i

disjoint locally closed subschemes i of which cover such that is ¯at over .

U S F U

Proof. It suf®ces to show that there is an open subset such that is ¯at over red .

S S Sp ecA

Moreover, the problem is local in X and . One may therefore assume that

K X Sp ecB

for some Noetherian integral domain A with quotient ®eld , that for some

A B F M

®nitely generated A-algebra, that is injective, and that for some ®nite

M M B

B -module . This module has a ®nite ®ltration by -submodules with factors of the

B p M p B

i i

form i for prime ideals . It suf®ces to consider these factors separately,

B A B

so that we may further reduce to the case that M is integral and injective. By

b b B K B n

Noether's normalization lemma there are elements such that is a

K b b n

®nite module over the polynomialring . `Clearing denominators' we can ®nd

f A M B B A b b

f n

an element such that f is still a ®nite module over .

B A M B A

Replace M , and by , and and apply the same procedure again. By induction

M B B

over the dimension of B we may ®nally reduce the problem to the case that and 2

is a polynomial ring over A, in which case ¯atness is obvious.

S S S i

Proof of the theorem. Let be a decomposition of as in the lemma and let

S S i F

be the natural morphism. Then is ¯at, and since the Hilbert polynomial of a

¯at family is locally constant as a function on the base, we conclude that the set P de®ned

in Theorem 2.1.5 is indeed ®nite.

L L

f F m F m F

For any let . Recall that there is

m m

Z O O X

a functor which converts -graded S -modules into -modules and is inverse to the

F F g S

functor (cf. [98] II.5.). Thus there is a natural isomorphism , and if

g F g F S

is any morphism of Noetherian schemes, then . Moreover, there is an

mg g m mg g F g F

integer , depending on , such that for all we have m

g i

(cf. [98], exc. II 5.9). We apply this to the case and conclude that there is an integer

m m m

such that for all we have

H F m i s S

s for all and for all .

F s F s H F m i F s i s S

m m m

s for all .

g F g F

By Proposition 2.1.2, we see that is ¯at if and only if m is locally free for all

X m

suf®ciently large m. Fixing for a moment, we claim that there are ®nitely many locally S

closed subschemes mr such that

S S j mr

1. m is a bijection,

F j r S

2. m is locally free of rank and

g S S j g F m 3. factors through m if and only if is locally free.

2.1 Flat Families and Determinants 35

S fs S j dim F s r g

mr m

Set-theoretically, this decomposition is given by k . S

We must endow the sets mr with appropriate scheme structures. Because of the universal

s S

property of these sets, this can be done locally: let be a point in mr . Then there is an

U s S F j U

open neighbourhood of in such that m admits a presentation

r r

O F j O

m U

U U

S U U

Let mr be the closed subscheme in which is de®ned by the ideal generated by the

r A

entries of the r -matrix and check that it has the required properties.

F S S g S

Now suppose that g is a morphism such that is -¯at with Hilbert poly-

X g

nomial P . According to what was said before, must factor through the locally closed sub-

S m m

scheme mP for all . We therefore consider the sets

S fs S jP F P g P P S

P s

mP for all (2.1)

S S P By 2.1.6 and the ®rst description of P ,weknowthat is the ®nite union of locally closed

subsets. But then it is evident from the second description and the fact that S is Noetherian,

that the intersection on the right hand side in (2.1) is in fact ®nite, even when considered as S

an intersection of subschemes. Let P be endowed with this subscheme structure and check

S P P 2

that the collection P , , thus de®ned has the properties postulated in the theorem.

F O x X s f x

Lemma 2.1.7 Ð Let be a coherent X -module, a point and . Assume

F O F F

Ss x s x

that x is ¯at over . Then is free if and only if the restriction is free.

k x

Proof. The `only if' direction is trivial. For the `if' direction let r be the -dimension

F x F m F K O F

x x x

of x . Then there is a short exact sequence ,

X x

F K m O F

s Ss x

and x is free if . Let denote the maximal ideal of the local ring . Since

O K m K O F

s s x

is Ss -¯at, is the kernel of the isomorphism . By Nakayama's

X x

2 Lemma K .

Lemma 2.1.8 Ð Let F be a ¯at family of coherent sheaves. Then the set

fs S jF g

s is a locally free sheaf

is an open subset of S .

A fx X jF xg X

Proof. The set x is not locally free at is closed in , and the set de®ned

A f f A 2 in the lemma is the complement of f . Since is projective, is closed.

De®nition 2.1.9 Ð Let P be a property of coherent sheaves on Noetherian schemes. P is

X S

said to be an open property, if for any projective morphism f of Noetherian

f s S

schemes and any ¯at family F of sheaves on the ®bres of the set of points such that

F S F s S

s has P is an open subset in . is said to be a family of sheaves with P, if for all F

the sheaf s has P. 36 2 Families of Sheaves

Examples of open properties are: being locally free (as we just saw), of pure dimension,

semistable, geometrically stable (as will be proved in Section 2.3).

S k f X S

Proposition 2.1.10 Ð Let k be a ®eld, a -scheme of ®nite type and a smooth F projective morphism of relative dimension n. If is a ¯at family of coherent sheaves on the

®bres of f then there is a locally free resolution

F F F F

n n

n i

R f F n R f F i n n

such that is locally free for , for and .

R f F

Moreover, in this case the higher direct image sheaves can be computed as the ho-

n ni n

R f F R f F h R f F

mology of the complex : Namely, .

O f X f

Proof. Let X be an -very ample line bundle on . Since the ®bres of are smooth,

it follows from Serre duality and the Base Change Theorem for cohomology that there is

m m m O R f O m

S X

an integer such that for all the -module is locally free and

R f O m i n S K G

vanishes for all . De®ne -¯at sheaves , for

K F K

inductively as follows: Let , and assume that has been constructed for some

m m K s S m

. For suf®ciently large all ®bres , , are -regular. Hence

f K m G f f K mm

is locally free and there is a natural surjection

i i

K G R f G f K m R f O m

. Then is locally free and by the projection

R f G

formula. In particular, is locally free and the other direct image sheaves vanish.

K G K

Finally, let be the kernel of the map . This procedure yields an (in®nite)

G F G

locally free resolution . Since all sheaves involved are ¯at, it follows that

F s S K

n s

is a locally free resolution of s for all . In particular, is isomorphic to the

G G f

s n s

kernel of n , and as any coherent sheaf on the ®bres of has homological

n K K

s n

dimension , n is locally free. According to Lemma 2.1.7 the sheaf is itself

G F n F n

locally free. Hence we can truncate the resolution at the -th step and de®ne

K F G n

n and for . To prove the last statement split the resolution

F F R f 2

into short exact sequences and apply the functors .

K X K X

Recall the notion of Grothendieck's groups and for a Noetherian scheme

X O

: these are the abelian groups generated by locally free and coherent X -modules, respec-

F F F F F F

tively, with relations for any short exact sequence

K X O

. Moreover, the tensor product turns into a commutative ring with X and

K X K X f X S

gives a module structure over . A projective morphism induces

R f F f K X K S f F

a homomorphism de®ned by .

Corollary 2.1.11 Ð Under the hypotheses of Proposition 2.1.10: if F is an -¯at family of

f F K X f F K S

coherent sheaves on the ®bres of , then and .

P P P

ni n i i i

R f F F R f F F f F 2

i i

Proof. and .

i i i 2.1 Flat Families and Determinants 37

Since the determinant is multiplicative in short exact sequences, it de®nes a homomor-

K X PicX X

phism det for any Noetherian scheme (1.1.17). Applying this ho-

F K X f F K S momorphism to the elements and in the corollary, we get

well de®ned line bundles

det F detF PicX detR f F detf F PicS

and

F F F

More explicitly, if is a ®nite locally free resolution of as in Proposition 2.1.10,

detF detF

then . This construction commutes with base change. For ex-

ample, there is a natural isomorphism

det H F detR f F s

S for each s . We conclude this section with a standard construction of a ¯at family that will be used

frequently in the course of these notes.

F F O k X

Example 2.1.12 Ð Let and be coherent -modules on a projective -scheme

F F E Ext E

and let . Since elements correspond to extensions

F F F

S PE F F

the space parametrizes all non-split extensions of by up to scalars. More-

over, there exists a universal extension

q F p O F q F

X p q S X

on the product S (with projections and to and , respectively), such that for

S F F id

each rational point , the ®bre is isomorphic to . Indeed, the identity gives

F E F E E Ext

a canonical extension class in k . Let denote the canoni-

E O O id

S E

cal homomorphism S and consider the class , i.e. the extension

de®ned by the push-out diagram

p O q F F q F

E q F q G q F

id F S

where the extension in the bottom row is given by E . Note that is -¯at for the obvious

q F q F S reason that and are -¯at. 38 2 Families of Sheaves

2.2 Grothendieck's Quot-Scheme

The Quot-scheme is an important technical tool in many branches of algebraic geometry.

Grass V r r

In the same way as the Grassmann variety k parametrizes -dimensional quo-

V Quot F P

tient spaces of the k -vector space , the Quot-scheme parametrizes quo-

O F P

tient sheaves of the X -module with Hilbert polynomial . Recall the notion of a rep-

resentable functor:

o C

Let C be a category, the opposite category, i.e. the category with the same objects and

reversed arrows, and let C be the functor category whose objects are the functors

S ets

and whose morphisms are the natural transformations between functors. The Yoneda

C x ObC x y

Lemma states that the functor C which associates to the functor

C C C x Mor y x

C embeds as a full subcategory into . A functor in of the form is said to

be represented by the object x.

ObC F ObC

De®nition 2.2.1 Ð A functor F is corepresented by if there is a

F F F F

C ±morphism such that any morphism factors through a unique

F F F F F

morphism ; is universally corepresented by , if for any morphism

T F T T F T F

, the ®bre product F is corepresented by . And is represented by

F F C

F if is a ±isomorphism.

F F F F If F represents then it also universally corepresents ; and if corepresents then

it is unique up to a unique isomorphism. This follows directly from the de®nition. We can

y F F Mor y F Mor F C

rephrase these de®nitions by saying that represents if C for

F y y ObC F F Mor F y Mor y ObC C all , and corepresents if C for all .

Example 2.2.2 Ð We sketch the construction of the Grassmann variety. Let k be a ®eld,

r r dim V

let V be a ®nite dimensional vector space and let be an integer, . Let

V r S chk S ets k S

Grass be the functor which associates to any -scheme of

K O V F O

k S k

®nite type the set of all subsheaves S with locally free quotient r

V K of constant rank .

r W V G

For each -dimensional linear subspace we may consider the subfunctor W

GrassV r k S O V

which for a -scheme consists of those locally free quotients S

F O W O V F S

such that the composition S is an isomorphism. In this case, the

g O V O W S

inverse of this isomorphism leads to a homomorphism S which splits

W V G

the inclusion of in . From this one concludes that W is represented by the af®ne sub-

HomV W G H om V W Sp ec S

space W correspondingto homomorphismsthat

W V O V F GrassV r S

split the inclusion map . Now for any element S

S S j G S

S W W

there is a maximal open subset W such that lies in the subset

GrassV r S W r

W . Moreover, if runs through the set of all -dimensional subspaces of

V S S

, then the corresponding W form an open cover of . Apply this to the universal fami-

G G W W V W

lies parametrized by W and for two subspaces : because of the universal

G G G

WW WW W property of W there is a canonical morphism . One checks that

2.2 Grothendieck's Quot-Scheme 39

W W

WW is an isomorphism onto the open subset and that for three subspaces the co-

WW WW W W

cycle condition W is satis®ed. Hence we can glue the spaces

V r G G GrassV r to produce a scheme Grass . Then represents the functor . Using

the valuative criterion, one checks that G is proper. The PlÈucker embedding

r r

GrassV r P V O V F O V det F

S S G exhibits G as a projective scheme. The local description shows that is a smooth irredu-

cible variety. 2

Example 2.2.3 Ð The previous example can be generalized to the case where V is replaced

k S

by a coherent sheaf V on a -scheme of ®nite type. By de®nition, a quotient module of

V q V F O

is an equivalence class of epimorphisms of coherent S -sheaves, where two

q V F i kerq kerq

epimorphisms i , , are equivalent, if , or, equivalently,

F F q q

if there is an isomorphism with . Here and in the following, quotient modules are used rather than submodules because the tensor product is a right exact functor,

so that surjectivity of a homomorphism of coherent sheaves is preserved under base change,

Grass V r S chS S ets

whereas injectivity is not. Let be the functor which

T S ObS chS q V

associates to the set of all locally free quotient modules T

V F O r Grass S V r O

T of rank . Then is represented by a projective -scheme

S S

Grass V r S

S . We reduce the proof of this assertion to the case of the ordinary

Grassmann variety of the previous example. First observe, that because of the uniqueness

S S Sp ecA

of Grass, if it exists, the problem is local in , so that one can assume that

b a

n n

V M A M A M A

and for some ®nitely generated -module . Now let be a

V F b

®nite presentation. Any quotient module T by composition with gives a quotient

F b

O . Thus induces an injection

n n

V r Grass O r S Grass k r b Grass

S S S k

S Grassk r

Clearly, the functor on the right hand side is represented by k . We must

Grass Grass O r V r

show that is represented by a closed subscheme of S . This fol-

O F

lows from the more general statement: if q is a locally free quotient module of

r T T g T T

rank , then there is closed subscheme such that any factors through

T g q a T T T

if and only if . Again this claim is local in , and by shrinking we may

F O q a r n B O

assume that . Then T is given by an -matrix with values in , and

g q a g

T vanishes if and only if factors through the closed subscheme corresponding to 2

the ideal which is generated by the entries of B .

S C S chS

We now turn to the Quot-scheme itself: let k , and be as in the second

f X S O f

example. Let be a projective morphism and X an -ample line bundle on

X H O P Q z

. Let be a coherent X -module and a polynomial. We de®ne a functor

40 2 Families of Sheaves

Q Quot S chS S ets

S C QT T

as follows: if T is an object in , let be the set of all -¯at coherent quotient

H O H F P g T T S T

sheaves T with Hilbert polynomial . And if is an -

g F Qg QT QT H F H T

morphism, let be the map that sends T to .

X V

Thus H here plays the rÃole of for the Grassmann scheme in the second example above.

Quot S H P

Theorem 2.2.4 Ð The functor is represented by a projective -scheme

Quot H P S

Sp ec k X P

Proof. Step 1. Assume that S and that . It follows from 1.7.6 that

H F QT

there is an integer m such that the following holds: If T is any quotient

K ker t T K H t

and if is the corresponding kernel, then for all the sheaves t , and

F m f O m

t are -regular. Applying the functor one gets a short exact sequence

f K m O H H m f F m

T T T

of locally free sheaves, and all the higher direct image sheaves vanish. Moreover, for any

m there is an exact sequence

f K m H O m m O H H m f F m

T T T

f K m

where the ®rst map is given by multiplication of global sections. Thus T completely

f F m F

determines the graded module T which in turn determines . This argu-

m m

H F O H H m H F m

T k ment shows that sending T to gives an injective

morphism of functors

Quot H H m P m H P Grass

T G Grass H H m P m

Thus we must identify those morphisms k which are

T GT

contained in the subset Q . Let

A O H H m B

G k

be the tautological exact sequence on G. Consider the graded algebra

S H P O

H P H S H A

and the graded -module . The subbundle generates a sub-

A S O H F O N O

k G

module G . Let be the -module corresponding to the graded -

O H A S Q

module G . Now it is straightforward to check that is represented by the

G G

locally closed subscheme P which is the component of the ¯attening strati®cation P for F corresponding to the Hilbert polynomial (see Theorem 2.1.5).

2.2 Grothendieck's Quot-Scheme 41 Q

It remains to show that Q is projective. Since we already know that is quasi-projective, R it suf®ces to show that Q is proper. The valuation criterion requires that if is a discrete

valuation ring with quotient ®eld L and if a commutative diagram

Sp ecL Q

Sp ec R Sp eck

q Sp ecR Q

is given, then there should exist a morphism R such that the whole di-

agram commutes. The diagram encodes the following data: there is a coherent sheaf F on

X P F P

L with and a short exact sequence

K H L F

H R K K

Certainly, there are coherent subsheaves k which restrict to over the generic

Sp ecR K F H

R k

point of . Let R be maximal among all these subsheaves, and put

K R K R

R . The maximality of implies that multiplication with the uniformizing parameter

F F F R

R R

induces an injective map R , which means that is -¯at. The classifying map

F q R

for R is the required .

X i X P

Step 2. Let S and be arbitrary. Choosing a closed immersion and replacing

H i H X P

by we may reduce to the case . By Serre's theorem there exist presentations

n n

O N m O N m H

P P

S S

O N m

As in Example 2.2.3 any quotient of H can be considered as a quotient of .

O N m H

Conversely, a quotient F of factors through , if and only if the composite

n n

N m O N m F

homomorphism O vanishes. The latter is equivalent

P P

T S

O H O N m f F

to the vanishing of the homomorphism T for some

P k

suf®ciently large integer . Hence by the same argument as in Example 2.2.3 the functor

Quot Quot N O m P H P

N is represented by a closed subscheme in

P S

S Quot N O m P 2

k .

P k

Q Quot H P Q Quot H P

Since represents the functor , we have

Mor Y Q QY S Y Q Y

S chS for any -scheme . Inserting for we see that the identity

map on Q corresponds to a universal or tautological quotient

e H F QQ

H F QT e

Any quotient T is equivalent to the pull-back of under a uniquely

S T Q

determined -morphism , the classifying map associated to .

S P Quot H P

In the case X the polynomial reduces to a number and simply

Grass H P S Sp eck H O H X is S . If and , then quotients of correspond to closed

subschemes of X . In this context the Quot-scheme is usually called the Hilbert scheme of

P Hilb X

closed subschemes of X of given Hilbert polynomial and is denoted by

Quot O P P Hilb X

X . In particular, if is a number, the Hilbert scheme parame-

X X trizes zero-dimensional subschemes of length in .

42 2 Families of Sheaves

H O O H F Quot

Proposition 2.2.5 Ð Let be a coherent X -module. Let be the

H P

universal quotient module parametrized by Quot . Then for suf®ciently large

L detf F O S

Quot X the line bundles are -very ample.

Proof. The arguments of the ®rst step in the proof of the theorem show that for suf®ciently

large there is a closed immersion

Quot H P Grass f H P

XS S

Recall the PlÈucker embedding of the Grassmannian: If V is a vector bundle on and if

V W GrassV r r

pr denotes the tautological quotient on , then the -th exterior power

r r

pr Grass V r P V S V det W

induces a closed immersion S of -

det W O P V

schemes, and is the pull-back of the tautological line bundle P on .

Combining the PlÈucker embedding with the Grothendieck embedding , we see that the line

F L detf S 2

bundles are very ample relative to .

In general, depends non-linearly on as we will see later (cf. 8.1.3).

We now turn to the study of some in®nitesimal properties of the Quot-scheme. Recall that

Y y

the Zariski tangent space of a k -scheme at a point is de®ned as

T Y Hom m m k y

y y

k y

m O Y y

where y is the maximal ideal of . Moreover, there is a natural bijection between tan-

Sp ec k y Y y Y gent vectors at y and morphisms with set-theoretic image . If

represents a functor, then one expects such morphisms to admit an interpretation in terms

of intrinsic properties of the object represented by y . We follow this idea in the case of the

Quot H P X S k

scheme Q , where is a projective morphism of -schemes,

O X S H S O X

X a line bundle on , ample relative to , and an -¯at coherent -module.

Ar tink k k

Let denote the category of Artinian local -algebras with residue ®eld . Let

A A Ar tink be a surjective morphism in and suppose that there is a commutative

diagram

Sp ecA Q

Sp ecA S

Sp ec A k q Q s S q

The images of the closed point of are -rational points and , and

K H F

corresponds to a short exact sequence A of coherent sheaves on

H X Sp ecA X H A

S A O

A with .

q Sp ec A Q

We ask whether the morphism q can be extended to a morphism such

q q

that q and , and if the answer is yes, how many different extensions are

I m I

there? The kernel of is annihilated by some power of A . We can ®lter by the ideals

m I

and in this way break up the extension problem in several smaller ones which satisfy

I m I A A the additional property that A . Assume that is an extension of this form.

2.2 Grothendieck's Quot-Scheme 43

Suppose that an extension q exists. It corresponds to an exact sequence

F K H

F X q q X X A

A A A

on A . That extends means that over the quotient is equivalent

F F Am F

A A

to . Let etc. Then there is a commutative diagram whose columns and

H F

rows are exact because of the ¯atness of A and :

I H I K I F

k k k

F K H

I F F I

In the ®rst row we have used the isomorphisms A etc. We can recover

i I K F K H i

as the cokernel of the homomorphism A induced by .

O i X

Conversely, any -homomorphism which gives when composed with de®nes an

F F F

A -¯at extension of (¯atness follows from 2.1.3). Thus the existence of is equiva-

lent to the existence of as above which in turn is equivalent to the splitting of the extension

I F B K k (2.2)

where B is the middle homology of the complex

j i

H F I K

B O I B X

Check that though a prioriis an -module it is in fact annihilated by , so that can

A O

be considered as an X -module. The extension class

K I F o q Ext

q q K A I F

de®ned by is the obstruction to extend to . Since is a -¯at and k is m

annihilated by A , there is a natural isomorphism

K F I Ext K I F Ext

k k

X X

s A

q o q Lemma 2.2.6 Ð An extension q of exists if and only if vanishes. If this is the

case, the possible extensions are given by an af®ne space with linear transformation group

K F I Hom

X . s 44 2 Families of Sheaves

Proof. The ®rst statement follows from the discussion above. For the second note that,

K I F

given one splitting , any other differs by a homomorphism k . As before the

K F I K Hom 2

¯atness of implies that these are elements in X .

X S k

Proposition 2.2.7 Ð Let f be a projective morphism of -schemes of ®nite type

O f X H S O P X

and X an -ample line bundle on . Let be an -¯at coherent -module, a

Q Quot H P S

polynomial and the associated relative Quot-scheme. Let

s S q s k H F s and be -rational points corresponding to a quotient with

kernel K . Then there is a short exact sequence

T o

K F Q T S Ext K F T Hom

s q X

k A k 2

Proof. This is just a specialization of the lemma to the case A , .

k H X

Proposition 2.2.8 Ð Let X be a projective scheme over and a coherent sheaf on .

q H F QuotH P k K kerq

Let be a -rational point and . Then

homK F dim QuotH P homK F ext K F

H P q

If equality holds at the second place, Quot is a local complete intersection near .

K F QuotH P q If ext , then is smooth at .

The proof will be given in the appendix to this chapter, see 2.A.13.

F C

Corollary 2.2.9 Ð Let F and be coherent sheaves on a smooth projective curve of

r F F F

positive ranks r and and slopes and , respectively. Let

Quot F P F

be an extension that represents the point s . Then

dim homF F ext F F F F r r g

k r dim V

Corollary 2.2.10 Ð Let V be a -vector space, , and let

A V O B

Grass

V r

be the tautological exact sequence on Grass . Thenthetangentbundleof ofthesmooth

V r

variety Grass is given by

H omA B T

Grass

GrassV r

Proof. Let G . Consider the composite homomorphism

B A V O p p

G G A O B p H omp

on the product and its adjoint homomorphism G . Since

clearly vanishes precisely along the diagonal, the image of is the ideal sheaf of the di-

omB A

agonal. Restricting this homomorphism to the diagonal, we get a surjection H

G of locally free sheaves of the same rank, which must therefore be an isomorphism. 2.3 The Relative Harder-Narasimhan Filtration 45

2.3 The Relative Harder-Narasimhan Filtration

In this section we give two applications to the existence of relative Quot-schemes: we prove the openness of (semi)stability in ¯at families and extend the Harder-Narasimhan ®ltration, which was constructed in Section 1.3 for a coherent sheaf, to ¯at families.

Proposition 2.3.1 Ð The following properties of coherent sheaves are open in ¯at families:

being simple, of pure dimension, semistable, or geometrically stable.

f X S O

Proof. Let be a projective morphism of Noetherian schemes and let X

X F d

be an f -very ample invertible sheaf on . Let be a ¯at family of -dimensional sheaves

P f s S F

with Hilbert polynomial on the ®bres of . For each , a sheaf s is simple if

hom F F

s s

s k . Thus openness here is an immediate consequence of the semiconti-

nuity properties for relative Ext-sheaves ([19], Satz 3(i)). The three remaining properties of

P P P

being of pure dimension , semistable , or geometrically stable have similar characteristics: they can be described by the absence of certain pure dimensional quotient sheaves.

Consider the following sets of polynomials:

fP j deg P d P P s S

A and there is a geometric point

F F P F P g

and a surjection s onto a pure sheaf with

A fP Aj deg P P d g

P P g A fP Ajp pg A fP Ajp p

and

where as usual, p is the reduced polynomial associated to etc. By the Grothendieck

P A

Lemma 1.7.9 the set A is ®nite. For each polynomial we consider the relative

QP Quot F P S

Hilbert scheme . Since is projective, the image

QP S P S F P i

is a closed subset of . We see that s has property if and only if

S P S s 2

is not contained in the ®nite ± and hence closed ± union .

P A

k f X S

Theorem 2.3.2 Ð Let S be an integral -scheme of ®nite type, let be a projec-

O f X F

tive morphism and let X be an -ample invertible sheaf on . Let be a ¯at family of f

d-dimensional coherent sheaves on the ®bres of . There is a projective birational morphism

T S k

g of integral -schemes and a ®ltration

HN F HN F HN F F

such that the following holds:

HN F HN F T i

1. The factors i are -¯at for all , and

U T HN F g HN F

2. there is a dense open subscheme such that g for

U all t .

46 2 Families of Sheaves

g HN F g T S

Moreover, is universal in the sense that if is any dominant mor-

F F

phism of integral schemes and if is a ®ltration of T satisfying these two properties,

HN F S h T T F h

then there is an -morphism with .

This ®ltration is called the relative Harder-Narasimhan ®ltration of F .

Proof. It suf®ces to construct an integral scheme T and a projective birational morphism

F g T S F g F

such that T admits a ¯at quotient which ®brewise gives the

F t T

minimal destabilizing quotient of t for all in a dense open subscheme of and such that

T is universal in the sense of the theorem. For in that case the kernel of the epimorphism

F F S S F T F

T is -¯at and we could iterate the argument with replaced by . This

would result in a ®nite sequence of morphisms

T T T T S

and the composition of theses morphisms would have the required properties.

A P A

As in the proof of the proposition consider the ®nite set of polynomials such

p S S P P A

that p . Then is the (set-theoretic) union of the closed subsets , .

A P P p p P P

De®ne a total ordering on as follows: if and only if and in

p p S P S P S P

case . Since is irreducible, there is a polynomial with . Let be

minimal among all polynomials with this property with respect to . Thus

S P

P A P P

is a proper closed subscheme of S . Let be its open complement. Consider the morphism

QP S P s S

. By de®nition of , is surjective. For any point the ®bre of

F P s V parametrizes possible quotients of s with Hilbert polynomial . If then any such

quotient is minimally destabilizing, by construction of V . By Theorem 1.3.4 the minimal

destabilizing quotient is unique and by Theorem 1.3.7 it is de®ned over the residue ®eld

s U V V t U s t

k . This implies that is bijective, and for each and

k s k t

one has . Moreover, according to Proposition 2.2.7 the Zariski tangent space to

F F t Hom F F F t

the ®bre of at is given by X , where is the

s s s t t

F HomF

short exact sequence corresponding to t. But by construction must vanish

t t

according to Lemma 1.3.3. This proves that the relative tangent sheaf U V is zero, i.e.

U V V U V

is unrami®ed and bijective. Since is integral, is an isomorphism.

T U QP U T

Now let be the closure of in with its reduced subscheme structure. Then

f j T S

is an open subscheme, and T is a projective birational morphism. To

see that T is universal, suppose that is an integral scheme with a dominant morphism

G g T S F P G P t T

and a quotient morphism T such that for general .

QP g h T QP

By the universal property of , factors through a morphism with

g g T QP

. The image of is a reduced irreducible subscheme of and contains an

U hj g j h T 2

g V g V open subscheme of , since . Thus factors through .

2.3 The Relative Harder-Narasimhan Filtration 47 S

Remark 2.3.3 Ð If under the hypotheses of the theorem the family F is not ¯at over or

F d s S

if s is -dimensional only for points in some open subset of , one can always ®nd an

S S F

open subset such that the conditions of the theorem are satis®ed for S . Making

S even smaller if necessary, we can assume that the relative Harder-Narasimhan ®ltration

HN F S F S S is de®ned over . This ®ltration can easily be extended to a ®ltration of over by coherent subsheaves (cf. Exc. II 5.15 in [98]), which, however, can no longer be expected

to be S -¯at or to induce the (absolute) Harder-Narasimhan ®ltration on all ®bres. This ®l- tration satis®es some (weaker) universal property. Nevertheless, we will occasionally use the relative Harder-Narasimhan ®ltration in this form in order to simplify the notations. 48 2 Families of Sheaves Appendix to Chapter 2

2.A Flag-Schemes and Deformation Theory

2.A.1 Flag-schemes Ð These are natural generalizations of the Quot-schemes. Let f

O f X S

be a projective morphism of Noetherian schemes, X an -ampleline bundleon

H S X P

X , and let be an -¯at coherent sheaf on with Hilbert polynomial . Fix polynomials

P P P P

with . Let

Drap H P S chS S ets

be the functor which associates to T the set of all ®ltrations

F H F H F H H O H

T T T T T

g r H T P i

such that the factors T are -¯at and have (®brewise) the Hilbert polynomial for

H P Quot i Drap H P

. Clearly, if then . In general,

XS XS

Drap S Drap H P H P

is represented by a projective -scheme which can be

S S X X H H i

constructed inductively as follows: let , and . Let , and

S X H ObCohX S

i i i i

suppose that i , and have already been constructed. Let

H P X Quot X S H

i i i i S i

, i and let be the kernel of the tautological

i X S

i i

S Drap H P S

surjection parametrized by i . Then . XS

2.A.2 Ext-groups revisited ÐIf H is a coherent sheaf together with a ¯ag of subsheaves,

Hom H H H

we can consider the subgroup of those endomorphisms of which preserve

the given ¯ag. In analogy to ordinary Ext-groups one is lead to the de®nition of correspond- Ext

ing higher -groups, which play a rÃole in the deformation theory of the ¯ag-schemes:

k X k K L O

let be a ®eld and a -scheme of ®nite type. Let and be complexes of X -

K L

modules which are bounded below. Let Hom be the complex with homogeneous

i iq q n i

HomK L K L d f

components Hom and boundary operator

ni i n i i

d f f d K F K

. A ®nite ®ltration of is a ®ltration by subcomplexes p

g r K F K F K

p p

such that only ®nitely many of the factor complexes p are nonzero.

L HomK L If K and are endowed with ®nite ®ltrations then inherits a ®ltration as

well: let

F HomK L ff jf F K F L j g

j j p

p for all

L F HomK L Hom K

Let and

Hom K L HomK L F HomK L

A ®ltered injective resolution of the ®ltered complex L consists of a ®nitely ®ltered com-

I O

plex of injective X -modules and a ®ltration preserving augmentation homomorphism

L I g r I

such that all factor complexes consist of injective modules and in-

F F

L g r I

duces quasi-isomorphisms g r . Such resolutions always exist.

p p

2.A Flag-Schemes and Deformation Theory 49

i i

K L Ext K L

De®nition and Theorem 2.A.3 Ð Let Ext and be the cohomol-

Hom K I Hom K I ogy groups of the complexes and , respectively. These are

up to isomorphism independent of the choice of the resolution. 2

From the short exact sequence of complexes

Hom K L HomK L Hom K L

one gets a long exact sequence of Ext-groups:

q q q

Ext K L Ext K L Ext K L Ext K L

Theorem 2.A.4 Ð There are spectral sequences

Ext g r K g r L p

i ip

pq pq

Ext K L E

pq pq

Ext K L E

Ext g r K g r L p

i ip

Hom K I 2

Proof. Use the natural induced ®ltrations on .

2.A.5 Deformation Theory Ð This is a very short sketch of some aspects of deformation theory which is by no means intended to provide a systematic treatment of the theory. Not

all assertions will be justi®ed by explicit computations.

m k Ar tin

Let be a complete Noetherian local ring with residue ®eld , and let k

be the category of local Artinian -algebras with residue ®eld . We want to study covariant

Ar tin S ets D k

functors D with the property that consists of a single element.

A A Ar tin

Suppose we are given a surjective homomorphism in . What is the

D A D A

image of the induced map D , and what can be said about the ®-

Aa a ker

bres? We can always factor through the rings , , and in this way reduce

a m

ourselves to the study of those maps which satisfy the additional hypothesis A ,

m A

A denoting the maximal ideal of . We will refer to such maps as small extensions, de-

viating slightly from the use of this notion by Schlessinger [229]. The functor D is said to U

have an obstruction theory with values in a (®nite dimensional) k -vector space , ifthe fol-

A a

lowing holds: (1) For each small extension A with kernel , there is a map (of sets)

D A U a D A D A U a A A

o such that the sequence is exact. (2) If

B a b A B

and B are small extensions with kernels and , respectively, and if is

a b

a morphism with , then the diagram

D A U a

D B U b 50 2 Families of Sheaves

commutes. There are essentially two types of examples that concern us here: The problem of deforming a sheaf, and that of deforming a subsheaf, or more generally, a ¯ag of subsheaves within

a given sheaf.

k X

2.A.6 Sheaves Ð Let be an algebraically closed ®eld, let be a smooth projective

k F O EndF k

variety over , and let be a coherent X -module which is simple, i.e. . If

A ObAr tink D A F A

, let F be the set of all equivalence classes of pairs where

F X Sp ecA F k

A A

A is a ¯at family of coherent sheaves on parametrized by and

O F F F A

is an isomorphism of X -modules. and are equivalent if and only if

F F

there is an isomorphism A with .

I d O F I

Let be a complex of injective X -modules and a quasi-isomorphism.

The following assertions can be checked easily with the usual diligence and patience nec-

I I

essary in homological algebra: the cohomology of the complex Hom computes

ExtF F A ObAr tink d

. Let and suppose we are given a collection of maps A

A I A I d A d

Hom which restrict to over the residue ®eld of . If then

A I d F H A I d

is in fact an exact (!) complex except in degree 0, and A is

A A

A F A D A A

an -¯at extension of over , i.e. an element in F (use induction on the length of

D A

and the local ¯atness criterion 2.1.3). Conversely, any element in F can be represented

d d

this way. Suppose that such a boundary map A with is given, de®ning an element

F D A A A a d

F A

A . Let be a small extension with kernel . Choose a lift of

d d d I I a

A . Since , the square factors through a homomorphism .

A A

d d

This homomorphism is a 2-cocycle, i.e. d , and its cohomology class

oF Ext F F a d

k A

A is independent of the choice of the extension . If

d oF oF d

then A , and conversely, if then is the boundary

I I a d d d

of some homomorphism , and A satis®es .

A A

d d d

Moreover, if A and are two boundary maps extending then they differ by a 1-cocyle A

, and they are equivalent, if this cocycle is a coboundary (it is at this place that we need the

F D D A F

assumption that be simple). We summarize: the ®bres of the map F

Ext F F a D D A

k F

F are af®ne spaces with structure group , and the image of is

o D A Ext F F a k the preimage of under the obstruction map F .

2.A.7 Flags of Subsheaves Ð We turn to the description of the deformation obstructions

for points in Drap. We will proceed in a similar way as sketched in the previous paragraph. This yields independent proofs of the results of Section 2.2 and a bit more. Again we leave

out the details.

X O k G

Let X be a polarized smooth projective -scheme, and let be a ¯at family of

k S s S

coherent sheaves on X parametrized by a Noetherian -scheme . Let be a closed

O k

point and the completion of the local ring Ss . A -algebra structure of an Artinian -

Sp ec A S A

algebra A corresponds to a morphism that maps the maximal ideal of to

s G G G s

. Let A be the corresponding deformation of the ®bre of at the point .

2.A Flag-Schemes and Deformation Theory 51

G G G G

Suppose we are given a ¯ag of submodules in the

O G D D Ar tin S ets

coherent X -module . De®ne a functor as follows:

D A G G G A

A A A

Let be the set of all ®ltrations with -¯at factors,

k Am G G G

whose restriction to A equals the given ®ltration .

I n

There is an injective resolution G of the following special form: in each degree

n n

I d

the module decomposes into a direct sum I , such that the boundary map

I d I d i j d

ij ij

ij , , has upper triangular form, i.e. for , and the sub-

I I G G

complexes are injective resolutions for the subsheaves p . (To get

g r G I

such a resolution ®rst choose injective resolutions p and then choose appropriate

d i j

homomorphism ij , .)

A ObAr tin G G

Let . The associated deformation A of can be described by an

d HomA I A I d G

element A with . A deformation of the ¯ag over

A b HomA I A I A

is given by an endomorphism of the form A ,

m I A I

A Aij

where A is a strictly lower triangular matrix with entries ; in

i j

b b A

particular, A is invertible. Moreover, is subject to the condition that the boundary map

d b d b

A A

A is ®ltration preserving, i.e. upper triangular. (To see this observe, that a A

deformation of the ¯ag is given by (1) deformations of the boundary maps of the complexes

I I

I and (2) deformations of the inclusion maps . Since we are free to change

p p p

these by deformations of the identity map of the complex I , we can in fact assume that the b

latter are given by a matrix A as above. Clearly, the boundary maps of the subcomplexes

I then are already determined by the requirement that they commute with the inclusion

maps.)

a A A Ar tin

Suppose now that is a small extension in . Let

A d G d d A

A A A A A

A be a homorphism that yields , and assume that de®nes an -

G G

¯at extension of the ®ltration . Choose an arbitrary (strictly lower triangular) ex-

A A A

tension A of and let . Let denote the strictly lower triangular part

e e

b d b d d

A A A

of A . Since the strictly lower triangular part of vanishes by the as-

Hom I I a

sumptions, de®nes an element in . As before, is in fact a 1-cocycle,

and its cohomology class is independent of the choice of A . Let this class be denoted by

oG Ext G G a G G

A A

A . An extension of the ®ltration exists if and only

if this obstruction class vanishes. Moreover, if the obstruction vanishes then any two ad-

Hom I I a

missable choices of A differ by a cocyle in and are isomorphic if and

D A D A G A

only if this cocycle is a coboundary. This means that the ®bre of over

F F

is an af®ne space with structure group Ext .

2.A.8 Comparison of the Obstructions Ð As before, let X be a smooth projective vari-

H F K F ety, and let be an epimorphism of coherent sheaves with kernel , such that

is simple. In the last two paragraphs we de®ned obstruction classes for the deformation of

F H K F as an `individual' sheaf and of as a quotient of , or what amounts to the same, of

as a submodule of H . We want show next that these obstructions are related as follows:

A A Ar tink a Let be a small extension in with kernel and let

52 2 Families of Sheaves

K A H F

A k A

F A Ext K F Ext F F

be an extension of the quotient H over . Let be the

K H F

boundary map associated to . Then

id oK A H oF Ext F F a

a A A

H H Ext K F

(Note that Ext .) Following the recipe above, we choose resolu-

d d I K I F I I F

tions K and , and a homomorphism such that

K F F K

the complex

I I d

K F

H d d d F

is an injective resolution of . Note that implies K , which means

Ext F K

that is a 1-cocyle. Its class is precisely the extension class in corresponding

to H . A deformation of the inclusion over an Artinian algebra is given by a matrix

subject to the condition that

d b d b

d d d

F K F

d d K

be upper triangular. Let F . Moreover, the induced deformation

F d d F

of is described by the lower right entry F . Let . Then

d d d

F F

d d d d d

F K K F

The ®rst two terms in the last line vanish ( is a cocycle!). Thus . Assuming

that the deformation exists over A means that and, therefore, vanish when restricted

HomI I a HomI I a

to A and thus induce the obstruction classes in and ,

K F F F

respectively. Note that multiplication by gives the boundary map . Hence indeed,

oK A H idoF

A A

Moreover, if the obstruction vanishes for a given , then any other choice is of the form

for a cocyle . Note that the boundary operator of A then changes by . Thus

D A D A K H A

H K H A

the natural map of the ®bre of K over into the

D A D A F

F A ®bre of F over is compatible with the boundary map between the structure groups of these af®ne spaces.

2.A Flag-Schemes and Deformation Theory 53

m k

2.A.9 Dimension Estimates Ð Let be a complete Noetherian local -algebra with

k R m k R

residue ®eld , and let R be a complete local -algebra with residue ®eld . Let

Hom R Ar tin S ets D Ar tin

alg

denote the functor , and let

S ets R

be a covariant functor as in the previous sections. Though is not in the category

i i

Ar tin R m lim D R m

, the quotients are. Any element i de®nes a natural

R R

R D R D transformation . The pro-couple is said to pro-represent , if is an

isomorphism of functors.

G S X p Y DrapG P S

For example, let be an -¯at family of sheaves on , let be

y Y G G

a relative ¯ag scheme, and let be a closed point that corresponds to a ¯ag

s py S D G

s , . Then the functor as de®ned in 2.A.7 is pro-represented by the pro-

b b b

O O G O

Y y Y y

couple Y y , where is the completion of the local ring at its maximal ideal,

b G

and is the limit of the projective system of ¯ags obtained from restricting the tautological

Y X Sp ecO m X fy g X

¯ag on to the in®nitesimal neighbourhoods Y y of . In

y D particular, Y and have the same tangent spaces. The results of Section 2.A.7 say:

Theorem 2.A.10 Ð There is an exact sequence

T o

G G G G T Y T S Ext Ext

y s

R D If D is pro-represented by then the deformation theory for provides information

about the number of generators and relations for R :

Proposition 2.A.11 Ð Suppose that D is pro-represented by a couple and has an

r U d dim m m

obstruction theory with values in an -dimensional vector space . Let R R

be the embedding dimension of R . Then

d dimR d r

R d r R r R Moreover, if dim , then is a local complete intersection, and if , then

is isomorphic to a ring of formal power series in d variables.

t t m k m m R

d R R

Proof. Choose representatives of a -basis of . Then

k t t J J J r d

for some ideal . It suf®ces to show that is generated by at most ele-

n t t d

ments: all statements of the proposition follow immediately from this. Let

k t t d

be the maximal ideal in . According to the Artin-Rees Lemma one has an in-

n J n N a A

clusion J for suf®ciently large . Consider the small extension

N N N

A A R m k t t J n A k t t nJ n

d d

with ,

N N

J n nJ n J nJ R A

and a . The natural surjection de®nes an element

D A D A

A A A , and the obstruction to extend to an element is given by an element

54 2 Families of Sheaves

o f U a

f g U f f J A

where is a basis of and are elements in . Consider the algebra

o A o A f f

A r

. The obstruction to extend to is the image of under the map

U a U af f

r A

and therefore vanishes. The existence of an extension

R A

corresponds to a lift of the natural ring homomorphism q to a ring homomor-

q R A t t d

phism . And picking pre-images for the generators we can also

k t t R A d

lift the composite homomorphism to a homomorphism

k t t k t t

d d

such that the following diagram commutes:

k t t R A

q k

k t t A A

In this diagram all horizontal arrows are natural quotient homomorphisms. is an isomor-

nn x k t t x d

phism, since it induces the identity on . For any one has

N N

J J n J mo dJ n

x , which implies . By construction of , one has

nJ f f n d

. Combining these two inclusions one gets

N N

J nJ f f n J n

n nJ

Recall the inclusion J we started with. From

N N N N N

J J n J nJ J n n nJ n f f n

J nJ f f J d

one deduces . Nakayama's Lemma therefore implies that is gen-

f f 2 r

erated by .

k O

Note that if R is the completion of a local -algebra of ®nite type, then the statements

d dim O d r dim O d

of the proposition will hold for O as well, i.e. . If

r r O

, then O is a local complete intersection, and if , then is a regular ring: clearly,

R dimO dimO O

dim ; is regular if and only if its completion is isomorphic to

O k x x I I R

a ring of power series. Finally, write for some ideal . Then

b b b b

I k x x I I mI I mI

, and . Hence by Nakayama's Lemma, if is generated by,

I s O say, s elements, then is also generated by elements. This shows that is a local complete

intersection, if this is true for R . We can apply the previous proposition and the observation above to ¯ag-schemes and

conclude:

X y

Proposition 2.A.12 Ð Let G be a coherent sheaf on a projective scheme . Let be a

Drap G P G

closed point in , de®ning a ®ltration of . Then

G G G G ext G G dim Drap G P ext ext

Drap G P

If equality holds at the second place, then is a local complete intersection

y Ext G G Drap G P y

near . If , then is smooth at .

Xk 2.B A Result of Langton 55

Proof. This follows at once from Subsection 2.A.7, Proposition 2.A.11 and the remark

following it. 2

Note that these estimates can be sharpened if one can show in special cases that all de-

G G

formation obstructions are contained in a proper linear subspace of Ext .

Remark 2.A.13 Ð Note that Proposition 2.A.12 contains 2.2.8 as the special case :

i i

K Ext G G Ext K GK

if the ®ltration of G is given by a single subsheaf , then .

2.B A Result of Langton

k R m

Let X be a smooth projective variety over an algebraically closed ®eld . Let

k K X X

be a discrete valuation ring with residue ®eld and quotient ®eld . We write K

Sp ec etc.

R d X

Theorem 2.B.1 Ð Let F be an -¯at family of -dimensional coherent sheaves on such

X F F K Coh d d

K R dd

that K is a semistable sheaf in for some . Then there is

X E F E F E Coh

K k dd

a subsheaf such that K and such that is semistable in .

Proof. It suf®ces to show the following claim: If d and if in addition to the

F Coh E F

assumptions of the theorem k is semistable in then there is a sheaf such

E F E Coh

K k d

that K and such that is semistable in . Clearly, the theorem follows from

this by descending induction on , beginning with the empty case .

Suppose the claim were false. Then we can de®ne a descending sequence of sheaves F

n n

F F F F F F Coh X d

with K and not semistable in as follows: Sup-

K k

n n n

B F

pose F has already been de®ned. Let be the saturated subsheaf in which represents

n n n n n

F Coh X G F B F

the maximal destabilizer of in d . Let and let be the ker-

k k

n n n

F G R

nel of the composite homomorphism F . As a submoduleof an -¯at sheaf,

n R

F is -¯at again. There are two exact sequences

n n n n n

B F G F B G

and (2.3)

n n n n n n

F F C G B (To get the second one use the inclusions F ). If is

non-zero, then

n n n n

pC p G pF pB mo d Q T

max

n n n n

B C B p B

Thus, in any case isisomorphictoa nonzerosubmoduleof and d

n n n n

C F p B C p B

k d

d with equality if and only if . Since is semistable

n n

Coh p B p F p G n

d d k d

in d it follows that for all . In partic-

p B p F T mo d Q T

d k n n n

ular, d for a rational number . As is a

descending sequence of strictly positive numbers in a lattice Z , it must become sta-

tionary. We may assume without loss of generality that n is constant for all . In this case

56 2 Families of Sheaves

n n

B n

we must have G for all . In particular, there are injective homomorphisms

n n n n

B B G G n n n

and . Hence there is an integer such that for all we have

n n n n

P B P B mo d Q T P G P G mo d Q T

and .

n G G

(Again we may and do assume that ). Now is an increasing sequence

of purely d-dimensional sheaves which are isomorphic in dimensions . In particu-

n DD n

G G lar, their re¯exive hulls are all isomorphic. Therefore, we may consider the as

a sequence of subsheaves in some ®xed coherent sheaf. As an immediate consequence all

n n

G G

injections must eventually become isomorphisms. Again we may assume that

B B

for all n . This implies: the short exact sequences (2.3) split, and we have ,

n n n n n

G G F B G n Q F F n Q G

and for all . De®ne , . Then and there

k k

n n

G Q Q n

are short exact sequences for all . It follows from the local

n n n

R F F n

¯atness criterion 2.1.3 that Q is an -¯at quotient of for each . Hence the

Quot F P G Sp ecR

image of the proper morphism contains the closed

X R

Sp ecR n F

subscheme for all . But this only possible if is surjective, so that K must

also admit a (destabilizing!) quotient with Hilbert polynomial P for some ®eld exten-

K K F 2

sion . This contradicts the assumption on K .

R d

Excercise 2.B.2 Ð Use the same technique to show: if F is an -¯at family of -dimensional sheaves

F E F E F E

K K K

such that is pure, then there is a subsheaf such that and k is pure. Moreover,

F E k there is a homomorphism k which is generically isomorphic.

Comments: Ð For a discussion of ¯atness see the text books of Matsumura [172], Atiyah and Macdonald [8] or Grothendieck's EGA [94]. The existence of a ¯attening strati®cation in the strong form 2.1.5 is due to Mumford [191]. The weaker form 2.1.6 is due to Grothendieck, cf. [172] 22.A Lemma 1. A detailed discussion of determinant bundles can be found in the paper of Knudson and Mumford [126]. Ð The notion of a scheme corepresenting a functor is due to Simpson [238]. Quot-schemes were introduced by Grothendieck in his paper [93]. There he also discusses deformations of quotients. Other presentations of the material are in Altman and Kleiman [3], KollÂar [128] or Viehweg [258]. Ð Openness of semistability and torsion freeness is shown in Maruyama's paper [161]. Relative Harder-Narasimhan ®ltrations are constructed by Maruyama in [164]. Ð Proposition 2.A.11 is based on Prop. 3 in Mori's article [181] with an additional argument from Li [149] Sect. 1. Flag-schemes and their in®nitesimal structure are discussed by Drezet and Le Potier [51]. Ð The presentation in Appendix 2.A is modelled on a similar discussion of the deformation of modules over an algebra by Laudal [137]. Deformations of sheaves are treated in Artamkin's papers [5] and [7] and by Mukai [186]. For an intensive study of deformations see the forthcoming book of Friedman [69]. For more recent results on deformations and obstructions see the articles of Ran [225] and Kawamata [120, 121]. Ð The theorem of Appendix 2.B is the main result of Langton's paper [135]. In fact, the original

version deals with -semistable sheaves. The analogous assertion for semistable sheaves was formulated by Maruyama ( Thm. 5.7 in [163]). We have stated and proved it in a more general form which covers both cases. 57

3 The Grauert-MulichÈ Theorem

One of the key problems one has to face in the construction of a moduli space for semistable sheaves is the boundedness of the family of semistable sheaves with given Hilbert polynomial. In fact, this boundedness is easily obtained for semistable sheaves on a curve, as we have seen before (1.7.7). On the other hand, the Kleiman Criterion for boundedness (Theorem 1.7.8) suggests to restrict semistable sheaves to appropriate hyperplane sections

and to proceed by induction on the dimension. In order to follow this idea we would like the

F j F restriction H of a semistable sheaf to be semistable again. There are three obstacles:

The right stability notion that is well-behaved under restriction to hyperplane sections

is -semistability. There is no general restriction theorem for semistability.

F j

In general, the restriction H will have good properties only for a general element

H in a given ample linear system. Even for a general hyperplane section the restriction might well fail to be -semistable. But this failure can be numerically controlled.

The Grauert-MÈulich Theorem gives a ®rst positive answer to the problem. In its original

form, it can be stated as follows:

Theorem 3.0.1 Ð Let E bea -semistable locally free sheaf of rank on the complex pro-

n n

O b O b P L P E j

L L r

jective space . If is a general line in and L with

b b b

integers , then

b b

i i

r for all i .

In Section 3.1 we will prove a more general version of this theorem that suf®ces to establish the boundedness of the family of semistable sheaves in any dimension. This will be

done in Section 3.3. As a further application of the Grauert-MÈulich theorem we will show in Section 3.2 that the tensor product of two -semistable sheaves is again -semistable. The chapter ends with a proof of the famous Bogomolov inequality. For all these results it is essential that the characteristic of the base ®eld be zero. It is not known, whether the family of semistable sheaves is bounded, if the characteristic of the base ®eld is positive. The discussion of restriction theorems for semistable sheaves will be resumed in greater detail in Chapter 7. 58 3 The Grauert-MÈulich Theorem

3.1 Statement and Proof X

Let k be an algebraically closed ®eld of characteristic zero, and let be a normal projective

k n O

variety over of dimension endowed with a very ample invertible sheaf X . For

a V H X O a PV jO aj

X a a X

let a , and let denote the linear system

a Z fD x X jx D g a of hypersurfaces of degree . Let a be the incidence

variety with its natural projections

X Z

Z K

Scheme-theoretically a can be described as follows: let be the kernel of the evaluation

V O O a Z PK

X X a

map a . Then there is a natural closed immersion

q PV X

a . In particular, is the projection morphism of a projective bundle, and the rel-

ative tangent bundle is given by the Euler sequence

q K p O T O

q Z

Z X

a a

We slightly generalize this setting: let be a ®xed ®nite sequence of positive

n pr

a i a

integers, . Let a with projections , and let

Z Z Z

X X a

a with natural morphisms

Z X

pr Z Z q a

as above and projections i . Then is a locally trivial bundle map with ®bres i

isomorphic to products of projective spaces. The relative tangent bundle is given by

pr T T pr T

X X Z ZX Z

a a

s D D

If is a closed point in parametrizing an -tuple of divisors , then the ®bre

Z p s q D D X

s is identi®ed by with the scheme-theoretic intersection .

X F q E

Next, let E be a torsion free coherent sheaf on and let .

S p

Lemma 3.1.1 Ð i) There is a nonempty open subset such that the morphism S

S Z s S Z S

is ¯at and such that for all the ®bre s is a normal irreducible complete X

intersection of codimension in .

S S F q E j Z

ii) There is a nonempty open subset such that the family S is ¯at over

E j S s S F

and such that for all the ®bre s is torsion free. s 3.1 Statement and Proof 59

Proof. Part i) follows from Bertini's Theorem and 1.1.15. For ii) take the dense open sub-

s s E E xt E

set of points which form regular sequences for and for all ,

i F E j 2 Z

, which implies that S is ¯at and that is torsion free by 1.1.13.

s F

Now we apply Theorem 2.3.2 to the family S and conclude that there exists a relative

Harder-Narasimhan ®ltration

F F F F

j S

F F S S

such that all factors i are ¯at over some nonempty open subset and such

E j s S F F

Z s s

that for all the ®bres form the Harder-Narasimhan ®ltration of .

S S S

Without loss of generality we can assume that . Since is connected, there are

F F s S

j i i i s

rational numbers with for all . We de®ne

maxf ji j g

i i

j E j s

if and else. Then Z for a general point , and vanishes if

E j s

and only if Z is -semistable for general . Using these notations we can state the general s

form of the Grauert-MÈulich Theorem:

Theorem 3.1.2 Ð Let E be a -semistable torsion free sheaf. Then there is a nonempty

s S

open subset S such that for all the following inequality holds:

maxfa g a deg X E j

i i Z

Proof. If , there is nothing to prove. Assume that is positive, and let be an

F F F F F s S

i i

index such that i . Let and , so that for all the

F F F F Z

i max i

sheaves and are torsion free, and min , . Let be the

s s s s

Z F j F j r

Z Z

maximal open subset of S such that and are locally free, say of rank and

F j r F j X Z Grass E r

Z X

. The surjection Z de®nes an -morphism . We

T D T j

Grass E r X

are interested in the relative differential ZX of the map .

Recall that the relative tangent sheaf of a Grassmann variety can be expressed in terms of B

the tautological subsheaf A and the tautological quotient sheaf (cf. 2.2.10):

T H omA B H om A B H omF F j

Grass E r X

Thus we can consider D as the adjoint of a homomorphism

F j F T j

Z Z

s S

Suppose s were zero for a general point . This would lead to a contradiction:

S X

®rst, making smaller if necessary, this supposition would imply that is zero. Let

Z X q Z X X

be the image of in . Since is a bundle, is open. In fact, since for

s S Z Z Z

s s s

any point the complement of in has codimension in , we also have

co dimX X X E j X

. Moreover, is locally free. Thus we are in the following situation:

60 3 The Grauert-MÈulich Theorem

r Z GrassE j

where is a smooth map with connected ®bres. If , then is constant on the ®bres of

r q X GrassE j

and hence factors through a morphism (Here we make use

of the assumption that the characteristic of the base ®eld is zero). But such a map corre-

E E j r E j

Z X

sponds to a locally free quotient X of rank with the property that is

F j s F

isomorphic to Z for general . Since by assumption is a destabilizing quotient of

s s s

F E E

s , any extension of as a quotient of is destabilizing. This contradicts the assumption

that E is -semistable.

s S C

We can rephrase the fact that s is nonzero for general as follows: let be the

Coh Z

n s s

quotient category n as de®ned in Section 1.6. Then is a nontrivial el-

F Hom F T j

C Z

ement in ZX . The appropriately modi®ed version of 1.3.3 says that

s s s

in this case the following inequality must necessarily hold:

F F T j

max min Z

ZX (3.1)

s s

e V O O a

X X

The Koszul complex associated to the evaluation map a provides us

V O a kere K X

with a surjection a and hence a surjection

V q O a p O q K p O T

a X

Z X

T pr T

ZX Z

Using X this yields a surjection

T j O a j V

Z k X i Z a

s s i i

From this we get the estimate

O a F j V F T j

k X i Z a min min Z

s i s s

a F g min f O

i min Z

s s

F maxfa g deg Z

min i s s

Combining this with inequality (3.1) one gets

F F

i i min max

s s

maxfa g degZ maxfa g a deg X

i s i i

a a deg X Note that if , then the bound for is just . 3.1 Statement and Proof 61

Remark 3.1.3 Ð In the proof above we used an argumentinvolving the relative Grassmann

HomT F F E

scheme to show the following: if ZX , then there is subsheaf ,

E E j q E F j Z

namely the kernel of X , such that . This fact can also be interpreted

k r q E q E rs e ds e

as follows: Consider the -linear map ZX given by ,

q E q E s e O q E

Z O

where q . This is an integrable relative connection in ,

s e s re ds e e q E s

i.e. r where is a local section in and a local section

O HomF F HomT F F r

Z ZX

in . Since ZX , the connection

F r r F F

preserves , i.e. induces a relative integrable connection ZX . Then

F q E E q E F E de®nes a coherent subsheaf of . That we indeed have is a local problem, which can be solved by either going to the completion or using the analytic category ([41],[32]), where Deligne has proved an equivalence between coherent sheaves with relative integrable connections and relative local systems.

The last step of the proof of the theorem indicates that there is space for improvement.

F j T j

Z min Z

Indeed, if the inequality for ZX can be sharpened then we automati-

s s

E j T j

Z min Z

cally get a better bound for . In order to relate and ZX we need the

s s following important theorem:

Theorem 3.1.4 Ð Let X be a normal projective variety over an algebraically closed ®eld

F F F F

of characteristic zero. If and are -semistable sheaves then is -semistable,

too.

F F

Remember that even if and are torsion free, their tensor product might have torsion,

F F F F

though only in codimension 2. Thus under the assumption of the theorem , and are locally free in codimension 1. The proof of Theorem 3.1.4 will be given in the next section (3.2.8). It uses the coarse

version of the Grauert-MÈulich Theorem proved above. If the -semistability of the tensor

product is granted one can prove a re®ned version of 3.1.2.

Theorem 3.1.5 Ð Let E be -semistable. Then there is a nonempty open subset

such that for all s the following inequality holds:

T j E j

min Z Z

s s

F F

Proof. Indeed, it is an immediate consequence of Theorem 3.1.4 that min

F F

min

min . In particular,

F j T j F j T j

min Z min Z Z min Z

ZX ZX

s s s s

E j T j 2

Z min Z

Hence, ZX follows from 3.1.

s s

Therefore any further analysis of the minimal slope of the relative tangent bundle is likely to improve the crude bound of the Grauert-MÈulich Theorem. This analysis was carried out

by Flenner and led to an effective restriction theorem: if the degrees of the hyperplane sec-

E j Z s

tions are large enough then Z is semistable for a general complete intersection . This s result will be discussed in Section 7.1.

62 3 The Grauert-MÈulich Theorem

X n O

Corollary 3.1.6 Ð Let be a normal projective variety of dimension and let X be

F O r Y

a very ample line bundle. Let be a -semistable coherent X -module of rank . Let be

s n jO j

the intersection of general hyperplanes in the linear system X . Then

r r

F j F deg X F j F degX

Y max Y

min and

F F j

Proof. We may assume that is torsion free. If Y is -semistable there is nothing to

r r

j j

prove. Let and be the slopes and ranks of the factors of the -Harder-

F j deg X

i i

Narasimhan ®ltration of Y . By Theorem 3.1.2 one has , and

i degX

summing up terms from 1 to i: i . This gives

j j

X X

r r

i i

F i degX

r r

i i

deg X r

i F j degX

max Y

F j 2 Y and similarly for min .

3.2 Finite Coverings and Tensor Products

In this section we will use the Grauert-MÈulich Theorem to prove that the tensor product of -semistable sheaves is again -semistable. This in turn allows one to improve the Grauert- MÈulich Theorem, as has been shown in the previous section, and paves the way to Flen-

ner's Restriction Theorem. The question how -semistable sheaves behave under pull-back for ®nite covering maps enters naturally into the arguments. Conversely, some boundedness problems for pure sheaves can be treated by converting pure sheaves into torsion free ones via an appropriate push-forward. At various steps in the discussion one needs that the characteristic of the base ®eld is 0, though some of the arguments work in greater generality. We therefore continue to assume

that k is an algebraically closed ®eld of characteristic 0.

Y X d k

Let f be a ®nite morphism of degree of normal projective varieties over of

n O O f O

Y X

dimension . Let X be an ample invertible sheaf. Then is ample as f

well. The functor on coherent sheaves is exact and the higher direct image sheaves van-

i i i

H Y F m H X f F m H X f F m

ish. Therefore and in particular

P F P f F A f O O

Y X

. The sheaf of algebras is a torsion free coherent -module

d f Y

of rank , and gives an equivalence between the category of coherent sheaves on and

X A f

the category of coherent sheaves on with an -module structure. Moreover, preserves A

the dimension of sheaves and purity. On the other hand, since X is normal and is torsion f free, A is locally free in codimension 1, which means that is ¯at in codimension 1. Thus

3.2 Finite Coverings and Tensor Products 63

n F ObCohX

f is exact modulo sheaves of dimension . Moreover, if has no

f F

torsion in dimension n ,thenthe sameis truefor . It is therefore appropriate to work

Coh

in the categories nn as we will do throughout this section.

f F

We need to relate rank and slope of F and :

F O n

Lemma 3.2.1 Ð Let be a coherent X -module of dimension with no torsion in codi-

rkf F rkF f F d F G O

mension 1. Then and . Let be a coherent Y -

rkf G d rkG G d

module with no torsion in codimension 1. Then and

f G A

G f G

Proof. For the last assertion note the following identities: , in particular

O A A deg X A O X

Y . Moreover, and are related by ,

G degY G O f G deg X f G O

Y and (See the

Y d deg X remark after De®nition 1.6.8). Finally, deg . The assertion follows from

this. 2

F n O F

Lemma 3.2.2 Ð Let be an -dimensional coherent X -module. Then is -semistable

F if and only if f is -semistable.

Proof. Certainly, F has no torsion in codimension 1 if and only if the same is true for

F F F F F f F f F f . If is a submodule with then by the

previous lemma. This shows the `if'-direction. For the converse, let K be a splitting ®eld of

Y K X Z Y K

the function ®eld K over and let be the normalization of in . This gives

Y X

®nite morphisms Z and, because of the ®rst part of the proof, it is enough

X Y X

to consider the morphism Z instead of . In other words we may assume that

Y K X G F

K is a Galois extension with Galois group . Suppose now that is a torsion

f F F f F

free sheaf on X such that is not -semistable, and let be the maximal

destabilizing submodule. Because of its uniqueness, F is invariant under the action of .

F f F F

By descent theory, there is a submodule F such that is isomorphic to in

F Coh Y F F 2 f F

codimension 1, i.e. in nn . Thus destabilizes . Y

This lemma can be extended to cover the case of polystable sheaves:

n X F

Lemma 3.2.3 Ð Let F be an -dimensional coherent sheaf on . Then is -polystable

if and only if f is -polystable.

tr A O

Proof. Again, we prove the `if'-direction ®rst. There is a natural trace map X .

E ndA

This map is obtained as the composition of the homomorphism A given by the

A E ndA O

algebra structure of and the trace map X . The latter is ®rst de®ned in the

U X Aj

usual way over the maximal open set where U is locally free: it extends uniquely

X X

over all of , since is normal. The homomorphism tr splits the inclusion morphism

i O A

X .

64 3 The Grauert-MÈulich Theorem

We may assume that F is torsion free. Suppose that is a nontrivial proper submodule

F f F

with the property that the homomorphism f splits. Such a splitting is given by

O F A F

an X -homomorphism which makes the diagram

F A F

x x

O F F

A F F tr

commutative. Thus composing with F de®nes a splitting

of the inclusion F . If is de®ned outside a set of codimension 2, then the same is

f F F true for . Now if is -polystable, then is -semistable by the previous lemma, and

the arguments above show, that any destabilizing submodule in F is a direct summand (in

codimension 1).

Y X

For the converse direction we may again assume that f is a Galois covering

E f F

because of the ®rst part of the proof. Let F be -stable and let be a destabilizing

g E f F

stable subsheaf. Then E is a -polystable subsheaf which

g Gal Y X

is invariant under the Galois action and is therefore the pull-back of a submodule F .

f F F F F E 2 Since is stable, we must have . Thus .

The next step is to relate semistability to ampleness. A vector bundle E on a projective

X O PE k -scheme is ample if the canonical line bundle on is ample. On curves a line bundle is ample if and only if its degree is positive. For arbitrary vector bundles of higher rank the degree condition is of course much too weak to imply any positivity properties. However, this is true if the vector bundle is semistable. Before we prove this result

of Gieseker, recall some notions related to ampleness: k

Let X be a projective -scheme. X

De®nition 3.2.4 Ð A Cartier divisor D on is pseudo-ample, if for all integral closed

dimY

X Y D D Y D

subschemes Y one has . And is called nef, if for all closed

X integral curves Y .

This notion of pseudo-ampleness is justi®ed in view of the following ampleness criterion: X

Theorem 3.2.5 (Nakai Criterion) Ð A Cartier divisor D on is ample, if and only if for

dimY

X Y D all integral closed subschemes Y one has .

Proof. See [100]. 2

The following theorem of Kleiman says that it is enough to test the nonnegativity of a

divisor on curves in order to infer its pseudo-ampleness.

X 2 Theorem 3.2.6 Ð A Cartier divisor D on is pseudo-ample if and only if it is nef. 3.2 Finite Coverings and Tensor Products 65

The analogous statement for ampleness is wrong. For counterexamples and a proof of the

theorem see [100]. However, if D is contained in the interior of the cone dual to the cone

generated by the integral curves, then D is indeed ample. This result due to Kleiman and references to the original papers can also be found in [100].

Theorem 3.2.7 Ð Let X be a smooth projective curve over an algebraically closed ®eld of

r X PE

characteristic zero and E a semistable vector bundle of rank on . Denote by

X O PE

the canonical projection and by the tautological line bundle on .

degE O

i) is pseudo-ample.

degE O

ii) is ample.

Proof. One direction is easy: assume that is pseudo-ample or ample. Then the self-

intersection number is or , respectively. But this number is the leading

O m

coef®cient of the polynomial . By the projection formula and the Riemann-Roch

formula we get:

m m m

O m O m S E deg S E rkS E g

Now

m r

m m

S E detS E det E

rk and

Thus the leading term is indeed deg . Now to the converse:

O C PE

i) By Theorem 3.2.6, it suf®ces to show that is nef. Let be any integral

C C f C X C

closed curve, its normalization and . If is mapped to a

point by then it is contained in a ®bre. But the restriction of to any ®bre is ample,

C O f C X

hence . Thus we may assume that is a ®nite map of smooth

C O deg O f E O

curves. We have and a surjection . Accord-

f E deg O deg f E r

ing to Lemma 3.2.2, is semistable. This implies

E deg f r

deg .

Y X deg f r

ii) Choose a ®nite morphism f of smooth curves of degree , and

P Y E f E O P

let be a closed point. The vector bundle Y still has positive

degree:

deg E deg f deg E rkE

E L O

E Moreover, is semistable by Lemma 3.2.2, so that by i) the line bundle P

is pseudo-ample. Under the isomorphism

PE Pf E Y PE

L LF L O F P

f E

can be identi®ed with , where P and is the ®bre over . Now

Pf E s V

let V be any closed integral subscheme of of dimension . If is contained in a

L V

®bre, then V L , since is very ample on the ®bres. If is not contained in a ®bre, it

Z F maps surjectively onto Y and has a proper intersection with . Now

66 3 The Grauert-MÈulich Theorem

s s s s

V L V L F V L s V F L

F F Lj L j L V L F

since and F . We knowthat is pseudo-ample. Therefore .

s s

Lj F V F L Z Lj F

Moreover, F is very ample on the ®bre . Therefore .

L L O

Using the Nakai Criterion we conclude that is ample. But is the pull-back of

Pf E PE O 2

under the ®nite map . Therefore is ample, too.

We are now ready to prove the theorem on the -semistability of tensor products as stated

in the previous section.

O F

3.2.8 Proof of Theorem 3.1.4: Ð We may assume that X is very ample. Let

rkQ F F F F Q

be a torsion free destabilizing quotient, i.e. and

F Q

F F Q deg X rkF rkF

Step 1. Assume that . A

X C

general complete intersection of dim hyperplane sections is a smooth curve , and

F F Q C

the restrictions of the sheaves , , and to are locally free. By the Corollary 3.1.6

g r F j C

to the Grauert-MÈulich Theorem their Harder-Narasimhan factors satisfy i

F deg X rkF i

i . De®ne

F rkF

i i

deg X

Then

g r F n j F deg X n rkF

i i C i i i

g r F n j

i C

Thus i is a semistable vector bundle of positive degree. By Theorem 3.2.7 j

it is ample. As Hartshorne shows [97], the tensor product of two ample vector bundles is

HN HN

F n g r F n g r

again ample (in characteristic 0). Thus is ample. Hence

F F j n n

is an iterated extension of ample vector bundles and therefore itself

Qj n n

ample, and ®nally, being a quotient of an ample vector bundle, C is ample as

Qn n well. To get a contradiction it suf®ces to show that the slope of is negative.

But

Qn n Q n n deg X

F F degX n n rkF rkF

rkF

n deg X F

rkF

F n deg X

Step 2. To reduce the general case to the situation of Step 1, we apply the following theorem which allows us to take `roots' of line bundles. 3.2 Finite Coverings and Tensor Products 67

Theorem 3.2.9 Ð Let X be a projective normal variety over an algebraically closed ®eld

k O

of characteristic zero and let X be a very ample invertible sheaf. For any positive

d X O

integer there exist a normal variety with a very ample invertible sheaf X and a

O d f X X f O X X

X ®nite morphism such that X . Moreover, if is smooth, can be chosen to be smooth as well.

Proof. See [166, 258] 2

X X

Using this theorem the proof proceeds as follows: choose a ®nite map f as

in the theorem with suf®ciently large d. Observe that if slope and degree on are de®ned

O F X

with respect to X , then for any coherent sheaf on one has

F f F

deg X deg X

f F f F O

And according to Lemma 3.2.2, and are -semistable with respect to ,

f Q f F f F f F F d

and destabilizes . Choosing the degree large enough it

is easy to satisfy the condition

f F f F f Q F F Q rkF rkF

deg X deg X

of Step 1. This ®nishes the proof. 2

E ndF F

Corollary 3.2.10 Ð If F is a -semistable sheaf, then , all exterior powers

and all symmetric powers S are again -semistable.

Proof. F is -semistable by Theorem 3.1.4. Since the characteristic of the base ®eld

F S F F

is 0, and are direct summands of and therefore -semistable. Up to sheaves

F F E ndF

of codimension 2 one has , so again the assertion follows from the

theorem. 2

X O

Theorem 3.2.11 Ð Let be a smooth projective variety and X an ample line bundle.

The tensor product of any two -polystable locally free sheaves is again -polystable. In particular, the exterior and symmetric powers of a -polystable locally free sheaf are - polystable.

We do not know a purely algebraic proof of this theorem. Using transcendental meth-

C ods, one can argue as follows: ®rst reduce to the case k . Then a complex algebraic

vector bundle on X is polystable if and only if it admits a Hermite-Einstein metric with

X O

respect to the KÈahler metric of induced by X . (This deep theorem, known as the Kobayashi-Hitchin Correspondence, was proved in increasing generality by Narasimhan- Seshadri [201], Donaldson [44, 45] and Uhlenbeck-Yau [253]. For details see the books

[127],[46] and [157].) Now, if E has a Hermite-Einstein metric, it is not dif®cult to see that

the induced metric on any tensor power E is again Hermite-Einstein. The assertion of the theorem follows. 68 3 The Grauert-MÈulich Theorem

3.3 Boundedness II

The Grauert-MÈulich Theorem allows one to give uniform bounds for the number of global

sections of a -semistable sheaf in terms of its slope. This is made precise in a very elegant

x maxf xg x

manner by the following theorem. Let for any real number .

Theorem 3.3.1 (Le Potier- Simpson) Ð Let X be a projective scheme with a very ample

O d F r F

line bundle X . For any purely -dimensional coherent sheaf of multiplicity

F H H d

there is an -regular sequence of hyperplane sections , such that

h X F j r F r F d

max

r F

d X H H

d for all and .

We prove this theorem in several steps.

d F

Lemma 3.3.2 Ð Supposethat X is a normal projective variety of dimension and that is

F F

a torsion free sheaf of rank rk . Then for any -regular sequence of hyperplane sections

H H X H H d

d d

and the following estimate holds for all .

h X F j F j

X max X

rkF deg X deg X

F F j X

Proof. Let for brevity. The lemma is proved by induction on .

h X g r F h X F

Let . Since and since the right hand side

i i

of the estimate in the lemma is monotonously increasing with , we may assume without

F F F

loss of generality that max , i.e. that is -semistable. For one gets

estimates

h X F h X F r degX

(3.2)

F h X F homO F degX

But if by Proposition 1.2.7.

bF degX c

Put . Then (3.2) is the required bound in the case .

Suppose the inequality has been proved for , . From the standard exact se-

quences

F k F k F k k

one inductively derives estimates

X X

h X F i h X F i h X F h X F

i i Of course, the sum on the right hand side is in fact ®nite. Using the induction hypothesis and replacing the sum by an integral, we can write

3.3 Boundedness II 69

h X F F

max

t dt

rkF degX deg X

where C is the maximum of and the smallest zero of the integrand. Evaluating the inte-

gral yields the bound of the lemma. 2

Corollary 3.3.3 Ð Under the hypotheses of the lemma there is an F -regular sequence of

H H d

hyperplane sections such that

F rkF h X F j

max X

rkF deg X degX

d for all .

Proof. Combine the lemma with Corollary 3.1.6. 2

Corollary 3.3.3 gives the assertion of the theorem in the case that F is torsion free on a normal projective variety. In order to reduce the general case to this situation we use the

same trick that was already employed in the proof of Grothendieck's Lemma 1.7.9.

X P

Proof of the theorem. Let i be the closed embedding induced by the complete

O Z d F i F

linear system of X . Let be the ( -dimensional) support of , and choose

P N d Z

a linear subspace L of dimension which does not intersect . Linear

L Z Y O j P

projection with centre then induces a ®nite map such that X

O F F r F rk F

. Since is pure, is also pure, i.e. torsion free. Moreover,

F F F O F d F

and . A -regular sequence

H Y F H X

of hyperplanes in induces an -regular sequence of hyperplane sections i on . If

F j h F j Y H H F j h F j

Y Y X X

, then and hence .

F F

max

We need to relate max and . For that purpose consider the sheaf of algebras

A O Z

A A rkA rk F

Lemma 3.3.4 Ð is a torsion free sheaf with min

r F

F A

Proof. carries an -module structure. The corresponding algebra homomorphism

F A E nd Z F

O is injective, since by de®nition is the support of . This implies that

A rk F r F Z

is torsion free and has rank less or equal to . By construction, is a

closed subscheme of the geometric vector bundle

P n L Sp ec S W Y

N d

W A W O S

where Y . Hence, there is a surjection . Consider the

A F A O W S W A

ascending ®ltration of given by the submodules p . Since

F A

is coherent, only ®nitely many factors g r are nonzero. Moreover, since the multiplication

F F F

g r A g r A g r A

map W is surjective, it follows that, once is torsion, the same

p p p

F F

A i g r A p rkA

is true for all g r , . In particular, if is not torsion then . In

rkA

O S W A

other words, the cokernel of Y is torsion. This implies that

rkA

A S W rkA 2 min min .

70 3 The Grauert-MÈulich Theorem

F F r F d

max

Lemma 3.3.5 Ð max .

G F G

Proof. Let be the maximal destabilizing submodule of , and let be the image of

A G A F F G A F

the multiplication map , i.e. is the -submodule of

G G G O G F

generated by . Then for some X -submodule . It follows that

F G G G O

max Y

A G O

min Y

G A O Y

min because of 3.1.4

F r F d

max

G F G

where for the last inequality we have used that max by the choice of ,

O d A r F 2 min Y , and by the previous lemma.

As a consequence of Lemma 3.3.5 we have

rk F r F d

F F r F

max max

d F

for any . Applying Corollary 3.3.3 to and using this estimate we get the

inequality of the theorem. 2

A slight modi®cation of the proof of 3.3.2 gives the following proposition:

X O

Proposition 3.3.6 Ð Let be a smooth projective surface and X a globally generated

ample line bundle. Let E and be torsion free -semistable sheaves. Then

rkE rkF rkE rkF

homE F F E deg X

degX

F To see this, apply Corollary 3.1.6 to both sheaves E and , and use the same induction

process as in Lemma 3.3.2. The bound of the proposition is slightly sharper than the one

F E 2

obtained by applying the theorem to E , say in case is locally free.

As a major application of the Le Potier-Simpson Estimate we get the boundedness of the

family of semistable sheaves:

X S k

Theorem 3.3.7 Ð Let f be a projective morphism of schemes of ®nite type over

O f P d

and let X be an -ample line bundle. Let be a polynomial of degree , and let be f

a rational number. Then the family of purely d-dimensional sheaves on the ®bres of with

P Hilbert polynomial and maximal slope max is bounded. In particular, the family

of semistable sheaves with Hilbert polynomial P is bounded.

3.4 The Bogomolov Inequality 71

S O

Proof. Covering by ®nitely many open subschemes and replacing X by an appro-

priate high tensor power, if necessary, we may assume that f factors through an embedding

N N

S P S Sp ec k X P

X . Thus we may reduce to the case , . According to The- F

orem 3.3.1 we can ®nd for each purely d-dimensional coherent sheaf a regular sequence

C H H h F j i d

d H H

of hyperplanes such that for all where

i X

C is a constant depending only on the dimension and degree of and the multiplicity and

F P

maximal slope of . Since these are given or bounded by and , respectively, the bound is uniform for the family in question. The assertion of the theorem follows from this and the

Kleiman Criterion 1.7.8. 2

For later reference we note the following variant of Theorem 3.3.1. Let X be a projective

O F i i

scheme with a very ample line bundle X . Let , , be the factors of

d F r r

the Harder-Narasimhan ®ltration of a purely -dimensional sheaf , and let i and denote

F F h F h F i

the multiplicities of i and . Then , and applying the Le Potier-

Simpson Estimate to each factor individually and summing up, we get

X X

h F r r r r d h F

i i i i i

r r r r d

i i

F F i F F F m F m

max Using i for , and ,

one ®nally gets:

r r d

Corollary 3.3.8 Ð Let C . Then

h F m r

d d

F C m F C m

max

r r d r d 2

3.4 The Bogomolov Inequality

Another application of Theorem 3.1.4 on the semistability of tensor products is the Bogo- molov Inequality. This important result has manifold applications to the theory of vector bundles and to the geometry of surfaces. We begin with the de®nition of the discriminant of

a sheaf.

F X c

Let be a coherent sheaf on a smooth projective variety with Chern classes i and F

rank r . The discriminant of by de®nition is the characteristic class

F r c r c

72 3 The Grauert-MÈulich Theorem

If X is a complex surface, we will denote the characteristic number obtained by evaluating

F X on the fundamental class of by the same symbol. (Warning: This de®nition of the discriminant differs from many other conventions in the literature, partly by the sign, partly

by a multiple or a power of r , each of which has its own virtues.) Clearly, the discriminant

F F

of a line bundle vanishes. If F is locally free, then . The Chern character

of F is given by the series

chF r c c c

F r y y

Hence ch for some nilpotent element . Following Drezet we write

F c

log chF log r

r r

X H X Q The Chern character is a ring homomorphism from K to , and the logarithm

converts multiplicative relations into additive ones. From this it is clear that for locally free

sheaves F and one has

F F F F

(3.3)

r r r r

rkF r rkF

where r and . In particular, the discriminant of a coherent sheaf n is invariant under twisting with a line bundle, and if F is locally free and is a positive

integer, then

n n

F nr F E ndF r F

and (3.4)

F c E ndF

The latter equation also implies the relation . H

Theorem 3.4.1 (Bogomolov Inequality) Ð Let X be a smooth projective surface and

an ample divisor on X . If is a -semistable torsion free sheaf, then

F F F

Proof. Let r be the rankof . The double dual of is still -semistable, and the dis-

F F F r F F F

criminants of F and are related by . Hence

F F E ndF

replacing F by , if necessary, we may assume that is locally free. Moreover,

E ndF r F F E ndF is also -semistable and , so that by replacing by we

may further reduce to the case that F has trivial determinant and is isomorphic to its dual

F k k H H K

. Let be a suf®ciently large integer so that X and that there is a smooth

jk H j

curve C . Recall that -semistable sheaves of negative slope have no global sec-

n n n

S F O C S F S F j C tions. The standard exact sequence X and

Serre duality lead to the estimates:

n n n n

h S F h S F C h S F j h S F j

C C

n n n

h S F h S F K h S F K

X X

n n n

h S F K C h S F j K j h S F j K j

X C X C C X C

3.4 The Bogomolov Inequality 73

n F

Thus we can bound the Euler characteristic of S by

n n n n n

S F h S F h S F h S F j h S F j K j

C C X C

Y PF X F Y Y C X

Now let be the projective bundle associated to , C ,

O Y n

and consider the tautological line bundle on . Then for all we have

n i

O n S F R O n i

and for all

S F O n

In particular, , and by the projection formula we get

M h C S F j M h Y O nj

C C Y

M PicC dim Y r M

for any line bundle . Since C , there are constants such that

M n h Y O nj n

M Y

C for all C

This shows that

r n

n S F n

O j

K for all (3.5)

X C

S F

On the other hand, we can compute by the Hirzebruch-Riemann-Roch formula ap-

O n

plied to the line bundle :

S F O n

(3.6)

c O n

where we have set and suppressed terms of lower order in . The cohomol-

r r r

c F c F c F

ogy class satis®es the relation . Since ,

r r r

c F O

we get . Finally, has degree 1 on the ®bres

of , so that (3.6) yields:

F n

S F n

terms of lower order in

r r

F 2 If were negative, this would contradict (3.5). 74 3 The Grauert-MÈulich Theorem Appendix to Chapter 3

3.A e-Stability and Some Estimates K

Throughout this appendix let X be a smooth projective surface, its canonical divisor,

O H and X a very ample line bundle. The following de®nition generalizes the notion of

-stability.

F r De®nition 3.A.1 Ð Let e be a nonnegative real number. A coherent sheaf of rank is

e-stable, if it is torsion free in codimension 1 and if

ejH j

F F

F r r r

for all subsheaves F of rank , .

H j H H

The factor j is thrown in to make the inequality invariant under rescaling

H e

H . Obviously -stability is the same as -stability, and -stability is stronger than

e e e -stability if . The same arguments as in the proof of Proposition 2.3.1 show that

e-stability is an open property. The following proposition due to O'Grady [208] is rather technical. It will be needed

in Section 9 to give dimension bounds for the locus of -unstable sheaves in the moduli space of semistable sheaves. The main ingredients in the proof are the Hirzebruch-Riemann- Roch formula, the Le Potier-Simpson Estimate 3.3.1 for the number of global sections of

-semistable sheaves and the Bogomolov Inequality 3.4.1.

Let F be a torsion free -semistable sheaf of rank and slope which, however, is not

e-stable for some . Let

F F F F

F F F F r

i i i

i i be a ®ltration of with factors of rank and slope such that the

following holds: all factors are torsion free and -semistable and satisfy the conditions

ejH j

and

F e F F F F

i.e. is -destabilizing, and is a -Harder-Narasimhan ®ltration of .

Ext F F

For a ®ltered sheaf F we de®ned groups in the appendix to Chapter 2. We use

i i

dim Ext A B ext A B

ext for and for the alternating sum of the dimensions (cf.

6.1.1).

X H r

Proposition 3.A.2 Ð There is a constant B depending on , and such that the fol-

r e

lowing holds: if F is a -semistable torsion free sheaf of rank which is not -stable, and

F F

if is a ®ltration of as above, then

3.A e-Stability and Some Estimates 75

r K H

ext F F F r e e B

r jH j Ext

Proof. First, the spectral sequence 2.A.4 for and Serre duality allow us to write

ext F F ext F F

j i

ext F F ext F F F F

j i j i j i

X X

F F F F homF F homF F K

j i j i i j

By Le Potier-Simpson 3.3.1 we have

r r

i j

homF F r H

j i i j

r r

i j

K H r H homF F K

j i i j

so that

X X

r r

i j

homF F homF F K

j i i j i j

ij ij

r r

i j

K H

j i

r H r H K H

r r

i j

The Hirzebruch-Riemann-Roch formula yields

F F r O

i j ij

F F r r r r O

j i j i j i X

r r

i j

where we have used the abbreviations

c F c F

i j

F F

i ij

i and

r r

i j

c c r ch r c

Using the additivity of the Chern character and , the

following identities are easily veri®ed:

X X

c F c F

i i

r r r r

i i

H i

i j i and, clearly, ij . The Bogomolov Inequality implies for all . Hence

76 3 The Grauert-MÈulich Theorem

X X X

i i i j

r r r r

j i i

r r r r

i j i i

ij i i

r r

i j

r r ij

This shows:

X X X

r r r

i j

F F F F r r O K

j i i j X ij

r r

ij ij

The ®rst term on the right hand side has already the required shape, the second one is clearly

r O

bounded by X . For the third term we use quadratic completion and the Hodge

H H Index Theorem, which says that for any class .

This leads to

r r r

i j

X X

r r K r K r r

i j

r r

ij i j

r r r

ij ij

X X

r r r K H r K r

i j

r r

i j i j

H r r r

ij ij

X X

r r r r r K H r

i j i j

H r H r

ij ij

K H r

r r K

i j

r H ij

Note that the term in brackets in the last summand is nonnegative by the Hodge Index The-

r r j

orem, so that the sum i has to be bounded from above (its maximum value being

r r ).

Putting things together and using the abbreviation

K H r

r O K B r X H

r H r H K H

r r

we have proved so far that

r r K H

F F ext a b B r X H

r H H

P P

r r r r

i j i j

a b a b

j i j

where and stand for i and , and we

ij ij

r r

a e H b ejH j are left with the assertions and . 3.A e-Stability and Some Estimates 77

We have

X X

r r

i j

r a

i i i j

i ij

r F i

since i . Because of the -semistability of and the assumptions on the

i , we have , and for all , and, ®xing for a moment,

i i

the problem is to maximize subject to the conditions i and

r r

i . A moment's thought yields:

a r r e H

r F i F

i i i Now let and denote rank and slope of , the -th step in the ®ltration of . Note

that the following relations hold:

From this we get

b r

i i

i j j

Moreover, i is negative when . Hence

r r r

b r r r r r r ejH j

i i

r r r

This ®nishes the proof of the proposition. 2

Comments: Ð In [20] Barth proves Theorem 3.0.1 for vector bundles of rank 2 and attributes it to Grauert and MÈulich. This result was extended to vector bundles of arbitrary rank by Spindler [240]. As Schnei- der observed, Spindler's theorem together with results of Maruyama [162] implied the boundedness of the family of semistable vector bundles of ®xed rank and Chern classes. For this result see also [54]. Shortly afterwards, Spindler's theorem was further extended to arbitrary projective manifolds

by Maruyama [166] and Forster, Hirschowitz and Schneider [66]. The bound for in terms of the minimal slope of the relative tangent bundle was given by Hirschowitz [102], based on Maruyama's results on tensor products of semistable sheaves.

Ð Lemma 3.2.2 and Theorem 3.2.7 are contained in [78]. In this paper Gieseker also gives an algebraic proof that symmetric powers of -semistable sheaves are again -semistable if the characteristic is zero. This had been proved in the curve case by Hartshorne [99] using the relation between stable

bundles on a curve and representations of the fundamental groups established by Narasimhan and Se- shadri [201]. The fact that tensor products of -semistable sheaves are again -semistable is due to Maruyama [166]. His proof uses Hartshorne's corresponding result on ampleness [97] and the tech-

niques developed by Gieseker. More results on -stability in connection with unrami®ed coverings can be found in Takemoto's article [243]. 78 3 The Grauert-MÈulich Theorem

Ð The boundedness theorem for surfaces was proved by Takemoto [242] for sheaves of rank 2, and by Maruyama [160] and Gieseker [77] for semistable sheaves of arbitrary rank. Our approach via Theorem 3.3.1 follows the papers of Simpson [238] and Le Potier [145]. Ð Theorem 3.4.1 ®rst appears in a special case in Reid's report [226]. A detailed account then was given by Bogomolov in [28]. In fact, he proves a stronger statement, that we will discuss in Section 7.3. In this stronger form the Bogomolov Inequality has interesting applications known as Reider's method. See Reider's paper [227] and the presentation in [139]. Gieseker gave a different proof of the Bogomolov Inequality in [78]. Ð The proof of the estimate in the appendix follows O'Grady [208]. Our coef®cients differ slightly from his paper. 79

4 Moduli Spaces

The goal of this chapter is to give a geometric construction for moduli spaces of semistable sheaves, the central object of study in these notes, and some of the properties that follow from the construction. As the chapter has grown a bit out of size, here is a short introduction: Intuitively, a moduli space of semistable sheaves is a scheme whose points are in some

`natural bijection' to equivalence classes of semistable sheaves on some ®xed polarized pro-

X H jective scheme . The phrase `natural bijection' can be given a rigorous meaning in terms of corepresentable functors. The correct notion of `equivalence' turns out to be S- equivalence. This is done in Section 4.1.

The moduli space can be constructed as a quotient of a certain Quot-scheme by a natural F

group action: instead of sheaves F one ®rst considers pairs consisting of a sheaf and a ba-

X F m m m

sis for the vector space H for some ®xed large integer . If is large enough,

h F m

H O m F

such a basis de®nes a surjective homomorphism X and hence

H P F H F

a point in the Quot-scheme Quot . An arbitrary point in this Quot-

scheme is of this particular form if and only if F is semistable and induces an isomorphism

P F m

H F m R QuotH P F

k . The subset of all points satisfying both con- M

ditions is open. The passage from R to the moduli space consists in dividing out the

F m ambiguity in the choice of the basis of H . We collect the necessary terminology and results from Geometric Invariant Theory in Section 4.2. The construction itself is carried out in Section 4.3 following a method due to Simpson. In fact, the proofs of the more technical theorems are con®ned to a separate section. The in®nitesimal structure of the moduli space is described in Section 4.5. It also contains upper and lower bounds for the dimension of the moduli space. Once the existence of the moduli space is established, the question arises as to what can be said about universal families of semistable sheaves parametrized by the moduli space. Section 4.6 gives partial answers to this problem. This chapter has three appendices. In the ®rst we sketch an alternative and historically earlier construction of the moduli space due to Gieseker and Maruyama, which has the virtue of showing that a certain line bundle on the moduli space is ample relative to the Picard

scheme of X . The second contains a short report about `decorated sheaves', and in the third we state some results about the dependence of the moduli space on the polarization of the base scheme. 80 4 Moduli Spaces

4.1 The Moduli Functor

X O k

Let X be a polarized projective scheme over an algebraically closed ®eld . For

Q z

a ®xed polynomial P de®ne a functor

M S chk S ets

ObS chk M S S

as follows. If S , let be the set of isomorphism classes of -¯at fam-

P f S S

ilies of semistable sheaves on X with Hilbert polynomial . And if is a

S chk M f f

morphism in , let be the map obtained by pulling-back sheaves via X

f id

X :

M f M S M S F f F

M S S L

If F is an -¯at family of semistable sheaves, and if is an arbitrary line bundle

S F p L S F F p L F

s s s

on , then is also an -¯at family, and the ®bres and k

s s S

L are isomorphic for each point . It is therefore reasonable to consider the quotient

functor M , where is the equivalence relation:

F F F F M S F F p L L PicS

for if and only if for some

If we take families of geometrically stable sheaves only, we get open subfunctors

M M and M . In 2.2.1 we explained the notion of a scheme corepresenting a functor.

De®nition 4.1.1 Ð A scheme M is called a moduli space of semistable sheaves if it core-

presents the functor M.

M M P

Recall that this characterizes up to unique isomorphism. We will write O

M P M M

and O instead of and , if the dependence on the polarization and the Hilbert X

polynomial is to be emphasized.

k A

If A is a local -algebra of ®nite type, then any invertible sheaf on is trivial. Hence the

Sp ec A MSp ec A

map M is a bijection. This implies that any scheme corepre-

senting M would also corepresent and conversely. We will see that there always is a M projective moduli space for M. In general, however, there is no hope that can be repre-

sented. M

Lemma 4.1.2 Ð Suppose M corepresents . Then S-equivalent sheaves correspond to F identical closed points in M . In particular, if there is a properly semistable sheaf , (i.e.

semistable but not stable), then M cannot be represented.

F F F Proof. Let be a short exact sequence of semistable sheaves

with the same reduced Hilbert polynomial. Then it is easy to construct a ¯at family F of

semistable sheaves parametrized by the af®ne line A , such that

F F F F F t

t and for all 4.2 Group Actions 81

Either take F to be the tautological extension which is parametrized by the af®ne line in

F F Ext through the point given by the extension above, or, what amounts to the same,

let F be the kernel of the surjection

q F i F

q A X X i X fg X A X

where is the projection and is the

n fg F

inclusion. Since over the punctured line A the modi®ed family and the constant

O A M F F k

family A are isomorphic, the morphism induced by must be constant

n fg F F F

on A , hence everywhere. This means that and , or more generally all

M M

sheaves which are S-equivalent to F , correspond to the same closed point in . Hence 2

does not represent M. s

Such phenomena cannot occur for the subfunctor M of stable families. The question s whether M is representable will be considered in Section 4.6.

4.2 Group Actions

In this section we brie¯y recall the notions of an algebraic group and a group action, various notions of quotients for group actions and linearizations of sheaves. We then list without proof results from Geometric Invariant Theory, which will be needed in the construction of moduli spaces. For text books on Geometric Invariant Theory we refer to the books of Mumford et al. [194], Newstead [202] and, in particular, Kraft et al. [131].

Let k be an algebraically closed ®eld of characteristic zero.

Group Actions and Linearizations

k G

An algebraic group over k is a -scheme of ®nite type together with morphisms

G G G Sp ec k G G G and

de®ning the group multiplication, the unit element and taking the inverse, and satisfying

S chk

the usual axioms for groups. This is equivalent to saying that the functor G

S ets factors through the category of (abstract) groups. Since the characteristic of the base

®eld is assumed to be zero, any such group is smooth by a theorem of Cartier. An algebraic

group is af®ne if and only if it is isomorphic to a closed subgroup of some GL .

k X X G X A (right) action of an algebraic group G on a -scheme is a morphism

which satis®es the usual associativity rules. Again this is equivalent to saying that for each

T GT X T

k -scheme there is a an action of the group on the set and that this action

T X Y k G Y

is functorial in . A morphism of -schemes with -actions X and ,

G id G

G X respectively, is -equivariant, if Y . In the special case that

82 4 Moduli Spaces

Y Y G Y

acts trivially on , i.e. if Y is the projection onto the second factor, an

X Y

equivariant morphism f is called invariant .

X G X x X

Let be a group action as above, and let be a closed point. Then

fxg G X G X x G

the orbit of is the image of the composite x . It is a G

locally closed smooth subscheme of X , since acts transitively on its closed points. The

x G G G

®bre x is a subgroup of and is called the isotropy subgroup or the

G V G V stabilizer of x in . If is a -representation space, let denote the linear subspace of

invariant elements.

X G X

De®nition 4.2.1 Ð Let be a group action. A categorical quotient for is Y

a k -scheme that corepresents the functor

X G S chk S ets T X T GT

X G

If Y universally corepresents , it is said to be a universal categorical quotient.

Y X G id X G X Y X

Suppose that corepresents the functor .Theimageof X in

X Y

corresponds to a morphism . This morphism has the following universal prop-

X Y G k

erty: is invariant, and if is any other -invariant morphism of -schemes

Y Y f then there is a unique morphism f such that . Indeed, it is straightfor-

ward to check that this characterizes Y as a categorical quotient.

Even if a categorical quotient exists, it can be far from being an `orbit space': let the mul-

Sp eck T T G A

tiplicative group m act on by homotheties. Then the projection

Sp ec k A is a categorical quotient. However, clearly, it is not an orbit space. We will

need notions which are closer to the intuitive idea of a quotient:

k k X

De®nition 4.2.2 Ð Let G an af®ne algebraic group over acting on a -scheme . A mor-

X Y

phism is a good quotient, if

is af®ne and invariant.

U Y U X

is surjective, and is open if and only if is open.

O O

The natural homomorphism Y is an isomorphism.

W X W Y W

If is an invariant closed subset of , then is a closed subset of . If

W X W W

and are disjoint invariant closed subsets of , then .

is said to be a geometric quotient if the geometric ®bres of are the orbits of geometric

X Y X Y

points of . Finally, is a universal good (geometric) quotient if Y is a good

Y k

(geometric) quotient for any morphism Y of -schemes.

Any (universal) good quotient is in particular a (universal) categorical quotient. If

X Y

is a good quotient and if X is irreducible, reduced, integral, or normal, then the

X XG same holds for Y . We will denote a good quotient of , if it exists, by .

4.2 Group Actions 83

X Y G

Let G be an algebraic group and let be an invariant morphism of -schemes.

G Y Y

is said to be a principal -bundle, if there exists a surjective Âetale morphism and

G Y G Y X X

a -equivariant isomorphism Y , i.e. is locally (in the Âetale toplogy)

Y G

isomorphic as a G-scheme to the product . Principal bundles are universal geomet-

X Y

ric quotients. Conversely, if is a ¯at geometric quotient and if the morphism

p X G X X G Y

is an isomorphism, then is a principal -bundle.

X Y G Z k

Let be a principal -bundle, and let be a -scheme of ®nite type with a

G X Z

G-action. Then there is a geometric quotient for the diagonal action of on . It is a

Z X Z

bundle (in the Âetale topology) over Y with typical ®bre , and is denoted by .

Y k F O

Example 4.2.3 Ð Let be a -scheme of ®nite type, let be a locally free Y -module

r r

H om O F Sp ec S H omO F Y

of rank r and let be the geometric vec-

Y Y

r r

F X IsomO F

tor bundle that parametrizes homomorphisms from O to . Let

Y Y

om O F X Y

H be the open subscheme corresponding to isomorphisms, and let

F GLr

be the natural projection. X is called the frame bundle associated to . The group

y Y k g GLr k f k y F y

acts naturally on X by composition: if , , and if

f g f g X Y GLr is an isomorphism, then . Then is a principal -bundle,

which is locally trivial even in the Zariski topology. (In fact, as Serre shows in [232], any

principal GL -bundle is locally trivial in the Zariski topology.) Similarly, we can construct

r X X Pro jS H omO F

a principal PGL -bundle by taking the image of in . We

F 2

will refer to X as the projective frame bundle associated to .

H G

Example 4.2.4 Ð Let G be an algebraic group and a closed subgroup. Then there

G H nG H G

is a geometric quotient for the natural (left) action of on , which in fact 2 is a (left) principal H -bundle.

The following gives the precise de®nition for a group action on a sheaf that is compatible

with a given group action on the supporting scheme.

k G k X

De®nition 4.2.5 Ð Let X a -scheme of ®nite type, an algebraic -group and

G X G O F

a group action. A -linearization of a quasi-coherent X -sheaf is an isomor-

O F p F p X G X

phism of X -sheaves , where is the projection, such

that the following cocycle condition is satis®ed:

id p id

X G

p X G G X G

where is the projection onto the ®rst two factors.

x k G X

Intuitively this means the following: if g and are -rational points in and , respec-

x g F

tively, and if we write xg for , then provides an isomorphism of ®bres of

F xg F x xg 84 4 Moduli Spaces

And the cocycle condition translates into

F xg h F x

xg xg h xg h

O G

Note that a given X -sheaf might be endowed with different -linearizations. A homo-

F F G O

morphism of -linearized quasi-coherent X -sheaves is a homomorphism of

O G F F

X -sheaves which commutes with the -linearizations and of and , respectively,

p G

in the sense that . Kernels, images, cokernels of homomorphisms of -

linearized sheaves as well as tensor products, exterior or symmetric powers of G-linearized

f X Y

sheaves inherit G-linearizations in a natural way. Similarly, if is an equivariant

G f F R f F

morphism of -schemes, then the pull-back and the derived direct images ,

G F F Y X i of any -linearized sheaves and on and , respectively, inherit natural lin-

earizations. In the case of the derived direct image functor this follows from the fact that

X G X a group action is ¯at and that taking direct images commutes with ¯at base change. In particular, the space of global sections of a linearized sheaf on a projective

scheme naturally has the structure of a G-representation.

A G-linearization on a sheaf induces an `ordinary' action on all schemes which are func- k

torially constructed from the sheaf: Let X be a -scheme with an action by an algebraic

G A O G

group and let be a quasi-coherent sheaf of commutative X -algebras with a -linear-

O A Sp ec A X

ization that respects the X -algebra structure. Let be the

associated X -scheme. Then induces a morphism

X G A X G A A G A

X A k X p

such that the diagram

A A G

X G X

G A

commutes. The cocycle condition for implies that A is group action of on , and

A X A

the commutativity of the diagram says that is equivariant. Similarly, if is

Z Pro jA G

a G-linearized -graded algebra, then inherits a natural -action that makes the

A X A S F projection Pro j equivariant. Typically, will be the symmetric algebra of

a linearized coherent sheaf F .

Apply this to the following special situation: suppose that X is a projective scheme with

L G G

a G-action and that is a -linearized very ample line bundle. Then acts naturally on

H X L H X L O L X

the vector space , the natural homomorphism k is equiv-

G X G PL PH X L

ariant and induces a -equivariant embedding . Thus the - X

linearization of L linearizes the action on in the sense that this action is induced by the

H X L projective embedding given by L and a linear representation on .

4.2 Group Actions 85

Y k F O r

Example 4.2.6 Ð Let be a -scheme of ®nite type, a locally free Y -moduleofrank ,

X Y

and let be the associated frame bundle (cf. 4.2.3). It follows from the de®nition

F O F

of X that there is an isomorphism , the universal trivialization of . If we

r O

give F the trivial linearization and the linearization which is induced by the standard

r k GL r

representation of GL on , then is equivariant. Similarly, there is a -equivariant

F O O X

isomorphism of sheaves on the projective frame bundle

X F Y associated to .

Geometric Invariant Theory (GIT)

In general, good quotients for group actions do not exist. The situation improves if we restrict to a particular class of groups, which fortunately contains those groups we are most interested in.

De®nition 4.2.7 Ð An algebraic group G is called reductive, if its unipotent radical, i.e.

its maximal connected unipotent subgroup, is trivial.

GLn

For the purposes of these notes it suf®ces to notice that all tori G and the groups ,

n PGLn SL , are reductive.

The main reason for considering reductive groups is the following theorem:

k X

Theorem 4.2.8 Ð Let G be a reductive group acting on an af®ne -scheme of ®nite type.

G G

X X Y Sp ecAX AX

Let A be the af®ne coordinate ring of and let . Then is

Y k X Y ®nitely generated over k , so that is of ®nite type over , and the natural map

is a universal good quotient for the action of G.

Proof. See Thm. 1.1 in [194] or Thm. 3.4 and Thm. 3.5 in [202] 2

G L

Assume that X is a projective scheme with an action of a reductive group and that

H X L X R

is a G-linearized ample line bundle on . Let be the associated

Z k

homogeneous coordinate ring. Then R is a ®nitely generated -graded -algebra as well.

G G

Pro jR R R X Y

Let Y . The inclusion induces a rational map which is

R R Pro jR X

de®ned on the complement of the closed subset V , i.e. on all

n G s H X L

points x for which there is an integer and a -invariant section with

s . This property is turned into a de®nition:

X G

De®nition 4.2.9 Ð A point x is semistable with respect to a -linearized ample line

n s H X L

bundle L if there is an integer and an invariant global section with

sx x G G x

. The point is stable if in addition the stabilizer x is ®nite and the -orbit of

is closed in the open set of all semistable points in X .

A point is called properly semistable if it is semistable but not stable. The sets X

L G and X of stable and semistable points, respectively, are open -invariant subsets of

X , but possibly empty.

86 4 Moduli Spaces X

Theorem 4.2.10 Ð Let G be a reductive group acting on a projective scheme with a

L Y

G-linearized ample line bundle . Then there is a projective scheme and a morphism

X L Y G

such that is a universal good quotient for the -action. Moreover, there

s s s s s

Y X L Y X L Y is an open subset Y such that and such that is

a universal geometric quotient. Finally, there is a positive integer m and a very ample line

M Y L j M

bundle on such that X .

Pro j H X L

Proof. Indeed, Y . For details see Thm. 1.10 and the

remarks following 1.11 in [194] or Thm. 3.21 in [202]. 2

Suppose we are in the set-up of the theorem. The problem arises how to decide whether a

given point x is semistable or stable. A powerful method is provided by the Hilbert-Mum-

G G G

ford criterion. Let m be a non-trivial one-parameter subgroup of . Then the

G X G X X G m

action of on induces an action of m on . Since is projective, the orbit map

x t f A X X t extends in a unique way to a morphism such that the

diagram

G g G

X x g A

G A n fg A

commutes, where m is the inclusion. We write symbolically

lim x t f

f G X G m

Now is a ®xed point of the action of m on via . In particular, acts on the ®bre

f r L f t

of L with a certain weight , i.e. if is the linearization of , then

r L

t id x r

L . De®ne the number .

Theorem 4.2.11 (Hilbert-Mumford Criterion) ÐApoint x is semistable if and only

G G

if for all non-trivial one-parameter subgroups m , one has

x And x is stable if and only if strict inequality holds for all non-trivial .

Proof. See Thm. 2.1 in [194] or Thm. 4.9 in [202]. 2

Once a good quotient is constructed, one wants to know about its local structure.

Theorem 4.2.12 (Luna's Etale Slice Theorem) Ð Let G be a reductive group acting on a

X X XG x X

k -scheme of ®nite type, and let be a good quotient. Let be a point

G G G x

with a closed -orbit and therefore reductive stabilizer x . Then there is a -invariant

X x S G X

locally closed subscheme S through such that the multiplication

G X S

induces a G-equivariant Âetale morphism . Moreover, induces an Âetale

SG XG

morphism x , and the diagram

4.2 Group Actions 87

G X S

SG XG

x S is cartesian. Moreover,if X is normal or smooth, then can be taken to be normal or smooth as well.

Proof. See the Appendix to Chapter 1 in [194] or [131]. 2

X Y G

Corollary 4.2.13 Ð If the stabilizer of x is trivial, then is a principal -bundle

x 2 in a neighbourhood of .

Some Descent Results

k k

Let G be a reductive algebraic group over a ®eld that acts on a -scheme of ®nite type.

X Y F G

Assume that there is a good quotient . Let be a -linearized coherent sheaf

F Y E Y

on X . We say that descends to , if there is a coherent sheaf on such that there is

F G E

an isomorphism of -linearized sheaves.

X Y G F G Theorem 4.2.14 Ð Let be a principal -bundle, and let be a -linearized

coherent sheaf. Then F descends.

X G X X

Proof. If is a principal bundle then there is an isomorphism Y .

G F p F p F

Under this isomorphism the -linearization of induces an isomorphism ,

p p X X X

Y where are the two projections. Moreover, the cocyle condition for the linearization translates precisely into the cocycle condition for usual descent theory for

faithfully ¯at quasi-compact morphisms (cf. Thm. 2.23 in [178]). 2

In general, we only have the following

X Y F G

Theorem 4.2.15 Ð Let be a goodquotient,andlet bea -linearized locally F

free sheaf on X . A necessary and suf®cient condition for to descend is that for any point

x X G G x F x

in a closed -orbit the stabilizer x of acts trivially on the ®bre . 2

Proof. See `The Picard Group of a G-Variety' in [131].

X G Let Pic denote the group of all isomorphism classes of -linearized line bundles

on X , the group structure being given by the tensor product of two line bundles.

X XG

Theorem 4.2.16 Ð Let be a good quotient. Then the natural homomor-

PicXG Pic X phism is injective.

Proof. Loc. cit. 2 88 4 Moduli Spaces

4.3 The Construction Ð Results k

Let X be a connected projective scheme over an algebraically closed ®eld of characteristic

O X P Q z

zero and let X be an ample line bundle on . If we ®x a polynomial , then

according to Theorem 3.3.7 the family of semistable sheaves on X with Hilbert polynomial

m F m

equal to P is bounded. In particular, there is an integer such that any such sheaf is -

m h F m P m V

regular. Hence, F is globally generated and . Thus if we let

P m

k H V O m X

and k , then there is a surjection

H F

H F m O m F

obtained by composing the canonical evaluation map X with

H F m

an isomorphism V . This de®nes a closed point

H F QuotH P

QuotH P

In fact, this point is contained in the open subset R of all those quotients

H E E

, where is semistable and the induced map

V H H m H E m

is an isomorphism. The ®rst condition is open according to 2.3.1 and the second because of

R the semicontinuity theorem for cohomology. Moreover, let R denote the open sub-

scheme of those points which parametrize geometrically stable sheaves F . P

Thus R parametrizes all semistable sheaves with Hilbert polynomial but with an ambi-

F m GLV

guity arising from the choice of a basis of the vector space H . Thegroup

Aut H H P

acts on Quot from the right by composition:

g g

g QuotH P GLV R for any two S -valued points and in and , respectively. Clearly,

is invariant under this action, and isomorphism classes of semistable sheaves are given by

k GLV k M M P the set R . Let be the functor de®ned in Section 4.1. The next lemma relates the moduli problem with the problem of ®nding a quotient for the group ac-

tion.

M GLV M

Lemma 4.3.1 Ð If R is a categorical quotient for the -action then core-

M M R M

presents the functor M . Conversely, if corepresents then the morphism ,

induced by the universal quotient module on R , is a categorical quotient. Similarly,

s s s s

M M M

R is a categorical quotient if and only if corepresents .

S k F m O

Proof. Suppose that is a Noetherian -scheme and a ¯at family of -regular X -

P S V p F

sheaves with Hilbert polynomial which is parametrized by . Then F

q O m O P m X is a locally free S -sheaf of rank , and there is a canonical surjection

4.3 The Construction Ð Results 89

p V q O m F

F F X

R F IsomV O V V

F F

Let S be the frame bundle associated to (cf. 4.2.3) with the

R F S V F

natural projection . Composing F with the universal trivialization of

on R we obtain a canonically de®ned quotient

q O H F

F k

R F X q

on . This quotient F gives rise to a classifying morphism

R F QuotH P

V R F

As discussed earlier, the group GL acts on from the right by composition, so

R F S GLV

that becomes a principal -bundle. The morphism F is clearly

R S

equivariant. It follows directly from the construction that , where

S fs S jF g S

s is semistable . In particular, if parametrizes semistable sheaves only,

e e

R F R R F R F

then F . In this case, the morphism induces a transforma-

F GLV RGLV R F S

tion of functors R and, since is a principal bundle

GLV S

and therefore a categorical quotient as well, de®nes an element in R . In this

RGLV R

way, we have constructed a transformation M . The universal family on

yields an inverse transformation. Hence, indeed it amounts to the same to corepresent M

GLV 2

and to corepresent R .

The construction used in the proof of the is functorial in the following sense: if f

F f F

S is a morphism of ®nite type of Noetherian schemes and if we set then there is

f R F R F V f

a canonical GL -equivariant morphism commuting with and the

e e

f S S S F

projections to and , respectively, such that F . As a consequence, if and

G G R F

F carry in addition compatible -actions for some algebraic group , then inherits

GLV

a natural G-structure commuting with the action of such that is equivariant and

F is invariant.

H F QuotH P F m

Lemma 4.3.2 Ð Let be a closed point such that is

m H H m H F m

globally generated and such that the induced map H

F GLV

is an isomorphism. Then there is a natural injective homomorphism Aut

GLV

whose image is precisely the stabilizer subgroup of the point .

F GLV

Proof. Consider the map Aut de®ned by

H m H m H m

Since F is globally generated, this map is injective. By the de®nition of equivalence for

GLV

two surjective homomorphisms representing the same quotient, an element g is in

GLV F g

the stabilizer if and only if there is an automorphism of such that . 2

90 4 Moduli Spaces

GLV

The lemma implies that the centre Z is contained in the stabilizer of any point

H P GLV

in Quot . Instead of the action of we will therefore consider the actions of

V SLV QuotH P

PGL and . There is no difference in the action of these groups on ,

V PGLV

since the natural map SL is a ®nite surjective homomorphism. But it is

a little easier to ®nd linearized line bundles for the action of SL than for the action of

V L PGLV

PGL , though not much: if carries a -linearization, then a fortiori it is also

V L SLV

SL -linearized. Conversely, if carries an -linearization, then all that could pre-

V Z SLV

vent it from being PGL -linearized is the action of the group of units of order

dimV

V L dim . But this action becomes trivial if we pass to the tensor power . The next step, before we can apply the methods of Geometric Invariant Theory as de-

scribed in the previous section, is to ®nd a linearized ample line bundle on R :

q H F QuotH P X V

Let be the universal quotient on , and let

V GLV O V O

V GL V

GL be the `universal automorphism' of parametrized by . Let

p p QuotH P GLV

and denote the projection from to the ®rst and the second factor,

H q H p F

respectively. The composition q is a family of quotients

H P GLV

parametrized by Quot , whose classifying morphism

QuotH P GLV QuotH P

V QuotH P

is, of course, just the GL -action on , which we de®ned earlier in terms of

point functors. By the de®nition of the classifying morphism, the epimorphisms and

p F p

yield equivalent quotients. This means that there is an isomorphism

X X

e F

p such that the diagram

q H p F

x x

q H F

commutes. It is not dif®cult to check that satis®es the cocyle condition 4.2.5. Thus is

GLV a natural -linearization for the universal quotient sheaf F . We saw in Chapter 2, (cf.

Proposition 2.2.5), that the line bundle

L det p F q O

QuotH P L

on is very ample if is suf®ciently large. Since the de®nition of commutes

GLV L

with base change (if is suf®ciently large), induces a natural -linearization on .

R QuotH P

Thus we can speak of semistable and stable points in the closure R of in

L SLV

with respect to and the -action (!). Remember that the de®nition of the whole set-

up depended on the integer m.

4.3 The Construction Ð Results 91

Theorem 4.3.3 Ð Suppose that m, and for ®xed also , are suf®ciently large integers.

ss s

R R L R R L

Then and . Moreover, the closures of the orbits of two points

JH JH

H F i R g r F g r F

i i

, , in intersect if and only if . The orbit of

H F R F a point is closed in if and only if is polystable.

The proof of this theorem will take up Section 4.4. Together with Lemma 4.3.1 and The-

orem 4.2.10 it yields:

M P

Theorem 4.3.4 Ð There is a projective scheme O that universally corepresents

M P M P

the functor O . Closed points in are in bijection with S-equivalence

X X

classes of semistable sheaves with Hilbert polynomial P . Moreover, there is an open subset

s s

P M P 2

M that universally corepresents the functor .

O O

X X

R M M P

More precisely, Theorem 4.3.3 tells us that O is a good quo-

s s s

R M M P

tient, and that is a geometric quotient, since the orbits of

stable sheaves are closed. According to Lemma 4.3.2 the stabilizer in PGL of a closed s

point in R is trivial. Thus:

s s

R M PGLV Corollary 4.3.5 Ð The morphism is a principal -bundle.

Proof. This follows from Theorem 4.3.3 and Luna's Etale Slice Theorem 4.2.12. 2

k S X n

Example 4.3.6 Ð Let X be a projective scheme over , and let be its -th sym-

X n X

metric product, i.e. the quotient of the product X of copies of by the per-

S M n

mutation action of the symmetric group n . And let denote the moduli space of zero- X

dimensional coherent sheaves of length n on . It is easy to see that any zero-dimensional

F n n lengthF x X F x

sheaf of length is semistable. Moreover, if x for each , then is

S-equivalent to the direct sum k of skyscraper sheaves. This shows that the

f S X M O

following morphism n is bijective. Consider the structure sheaf of the

X X X X

diagonal as a family of sheaves of length one on parametrized by .Thuson

X X X p X X X p O i

we can form the family , where

X i

X is the projection onto the product of the -th and the last factor. This family induces a

f X X M S n

morphism which is obviously n -invariant and therefore descends

f S X M f

to a morphism n . In fact, is an isomorphism. In order to see this, we shall

g M S X

construct an inverse morphism n . In general, there is no universal family on

X M g M S n

n which we could use. Instead, we construct a natural transformation

M F

for the moduli functor corepresented by n . Let be a ¯at family of zero-dimensional

X S s S

sheaves of length n on parametrized by a scheme . Let be a closed point repre-

F X F X s

senting a sheaf s on . Since the support of is ®nite, and since is projective, there

U Sp ec B X F

is an open af®ne subset containing the support of s . Then there is

V Sp ec A S s Supp F U

an open af®ne neighbourhood of such that t for all

t V V H p F j V . Moreover, making smaller if necessary, we may assume that is free

92 4 Moduli Spaces

n O F V

of rank . Choose a basis of sections. Then the U -module structure of is determined

k B E E End H

by a -algebra homomorphism , where A is isomorphic to the ring

n A of n -matrices with values in .

Recall the notion of the linear determinant: there is a natural equivariant identi®cation

n n n

End H E S H

with respect to the actions of the symmetric group n on and

n n n S n

E E H

E . Hence is the subalgebra of those endomorphisms of which

n S

S E

commute with the action of n . In particular, commutes with the anti-symmetriza-

tion operator

n n

a H H h h sgn h h

and therefore acts naturally on the image of a, which is and, hence, free of rank 1.

n S

A E

This gives a ring homomorphism l d . An equivalent description is the

E E A b

following: let b be the polar form of the determinant. Then restricted

l d bn

to symmetric tensors is formally divisible by n , and .

n n

F V S U S X Using the linear determinant we can ®nish our argument: let g

be the morphism induced by the ring homomorphism

n S n S

n n

A E B

Check that the morphisms thus obtained for an open cover of S glueto giveamorphism

X g S , that this construction is functorial, and that the natural transformation constructed

in this way provides an inverse of f .

X X

Consider now the Hilbert scheme Hilb of zero-dimensional subschemes of of

n O Z Hilb X X

length . The structure sheaf Z of the universal subscheme induces

Hilb X M

a morphism n . Using the above identi®cation, we obtain the Hilbert-to-

X S X X Chow morphism Hilb , which associates to any cycle in its support

counted with the correct multiplicity.

X M O n X Assume now that is a smooth projective surface, and let X denote the

moduli space of rank one sheaves with trivial determinant and second Chern number n.

Hilb X M O n

Then there is a canonical isomorphism X obtained by sending a

Z X I M O n

X X

subscheme to its ideal sheaf Z . In this context the morphism

X S appears as a particular case of the `Gieseker-to-Donaldson'morphism which we will

discuss later (cf. 8.2.8 and 8.2.17). 2

Occasionally, one also needs to consider relative moduli spaces, i.e. moduli spaces of se-

S mistable sheaves on the ®bres of a projective morphism X . It is easy to generalize the

previous construction to this case.

X S k

Theorem 4.3.7 Ð Let f be a projective morphism of -schemes of ®nite type with

O X

geometrically connected ®bres, and let X be a line bundle on very ample relative

S P M P S

to . Then for a given polynomial there is a projective morphism XS which universally corepresents the functor

4.4 The Construction Ð Proofs 93

M S chS S ets

XS T

which by de®nition associates to an S -scheme of ®nite type the set of isomorphism classes

T X T X T S

of -¯at families of semistable sheaves on the ®bres of the morphism T

P s S M P

with Hilbert polynomial . In particular, for any closed point one has XS

P M M P M P X

. Moreover, there is an open subscheme XS that universally

M M

corepresents the subfunctor XS of families of geometrically stable sheaves.

M M XS

Proof. Because of the assertion that XS universally corepresents , the statement S of the theorem is local in S . We may therefore assume that is quasi-projective. The family

of semistable sheaves on the ®bres of f with given Hilbert polynomial is ®nite and hence

P m

m m H O m R

-regular for some integer . As in the absolute case, let X and let

H F F Quot H P

denote the open subset of all points s where is a semistable

X s S H X H m H X F m

s s s

sheaf on s , , and induces an isomorphism . If

e e

H F O L detp F q O

O X

R denotes the universal quotient family, is

well-de®ned and very ample relative to S for suf®ciently large . For any such there is a

B S L g B R g R S

very ample line bundle on such that is very ample on (where is

H F

the structure morphism). Then the following statements about a closed point s

R s S

in the ®bre s over are equivalent:

R L q B

1. is a (semi)stable point in with respect to the linearization of .

R L 2. is a (semi)stable point in s with respect to the linearization of . This follows either directly from the de®nition of semistable points (4.2.9), or can be de-

duced by means of the Hilbert-Mumford Criterion 4.2.11. The theorem then is a conse-

M P R SLP m

quence of this easy fact, Theorem 4.3.3 and the fact that XS is

a universal good quotient (Theorem 4.2.10). We omit the details. 2

4.4 The Construction Ð Proofs

The proof of Theorem 4.3.3 has two parts: in order to determine whether a given point

V O m F R

X in is semistable or stable by means of the Hilbert-Mumford G

Criterion we must compute the weight of a certain action of m . In this way we shall ob-

tain a condition for the semistability of (in the sense of Geometric Invariant Theory) in

F terms of numbers of global sections of subsheaves F of , which then must be related to

the semistability of F . We begin with the second problem and prove a theorem due to Le

Potier that makes this relation precise.

d r

Theorem 4.4.1 Ð Let p be a polynomial of degree , and let be a positive integer. Then d

for all suf®ciently large integers m the following properties are equivalent for a purely -

r p dimensional sheaf F of multiplicity and reduced Hilbert polynomial . 94 4 Moduli Spaces

(1) F is (semi)stable

pm h F m h F m r pm F F

(2) r , and for all subsheaves of

r r

multiplicity r , .

pm h F m F F r r

(3) r for all quotient sheaves of multiplicity ,

r .

F F Moreover, for suf®ciently large m, equality holds in (2) and (3) if and only if or ,

respectively, are destabilizing.

r p

Proof. (1) (2): The family of semistable sheaves with Hilbert polynomial equal to

F m

is bounded by 3.3.7. Therefore, if m is suf®ciently large, any such sheaf is -regular, and

pm h F m F F r r r

r . Let be a subsheaf of multiplicity , . In order to

show (2) we may assume that F is saturated in . We distinguish two cases:

F F r C

r r d

where C is the constant that appears in Corollary 3.3.8. The family of (sat-

urated!) subsheaves F of type B is bounded according to Grothendieck's Lemma 1.7.9.

F m h F m P F m

Thus for large m, any such sheaf is -regular, implying , and,

P F g

moreover, since the set of Hilbert polynomials f is ®nite, we can assume that

P F m r pm P F r p

For subsheaves of type A we use estimate 3.3.8 to bound the number of global sections

F F F F F r C

directly. Note that max by the semistability of , and ,

since F is of type A. Thus

r h F m

d d

F C m F C m

max

r r d r d

d d

F C m F r C m

r d r d

Hence for large m we get

d d

m m h F m

(4.1)

r d d

m d

where stands for monomials in of degree smaller than with coef®cients that

d d

m m

F F F pm

depend only on r d C and , but not on . Since ,

d d

m m

the right hand side of (4.1) is strictly smaller than p for suf®ciently large .

F F F r r

(2) (3): Let be the kernel of a surjection and let and be the multi-

plicities of F and , respectively. Then (2) implies:

h F m h F m h F m pm r pm r pm r

4.4 The Construction Ð Proofs 95

F F (3) (1): Apply (3) to the minimal destabilizing quotient sheaf of . Then,byCorol-

lary 3.3.8

h F m

pm F C m

r d

F F F max This shows that min is bounded from below and consequently

is bounded uniformly from above. Hence by 3.3.7 the family of sheaves F satisfying (3)

d F F

is bounded. Now let F be any purely -dimensional quotient of . Then either

F F F F F

and is far from destabilizing , or indeed, . But according to

Grothendieck's Lemma 1.7.9, the family of such quotients F is bounded. As before, this

h F m P F m

implies that for large m one has and

P F m r pm P F r p 2 Hence indeed, (3) (2)

This theorem works for pure sheaves only. The following proposition allows us to make

the passage to a more general class of sheaves: d

Proposition 4.4.2 Ð If F is a coherent module of dimension which can be deformed to

P E P F

a pure sheaf, then there exists a pure sheaf E with and a homomorphism

F E ker T F

with d .

F T F

Proof. If itself is pure there is nothing to show. Hence, assume that d is non-

X F

trivial and let Y be its support. The condition on means that there is a smooth con-

F C C F d X F

nected curve and a -¯at family of -dimensional sheaves on such that for

C F s C n fg

some closed point and such that s is purefor all . (Note that this implies

d that F is pureof dimension : any torsion subsheaf supported on a ®bre would contradict

¯atness, and any other torsion subsheaf could be detected in the restriction of F to the ®bre

C n fg t O

over a point in ). Let be a uniformizing parameter in the local ring C . Consider

N F F F the action of t on the cokernel of the natural homomorphism from to its

re¯exive hull (cf. 1.1.9). Since F is pure, this homomorphism is injective (cf. 1.1.10). The

N t N N

kernels n of the multiplication maps form an increasing sequence of sub-

N N t N N

modules and hence stabilize. Let be the union of all n . Then is injective on ,

C E F N N which is equivalent to saying that N N is -¯at. Let be the kernel of .

Thus we get the following commutative diagram with exact columns and rows:

N N N N

F F N

F E N

96 4 Moduli Spaces

d E

F is re¯exive and therefore pure of dimension , and the same holds for . In partic-

ular, both sheaves as well as N N are -¯at. Restricting the middle column to the special

fg X E F N N

®bre we get an exact sequence . By

F E E

Corollary 1.1.14 the sheaf is pure, and being a subsheaf of a pure sheaf,

fg X F E C n fg

is pure as well. Since N has support in , and are isomorphic over ,

P F P E

and since both are C -¯at, they have the same Hilbert polynomial: . Note

N dimF d F E

that dim (cf. 1.1.8). This implies that has at

d ker

most -dimensional cokernel and kernel. In particular, is precisely the torsion 2 submodule of F .

After these preparations we can concentrate on the geometric invariant theoretic part of

V O m F R

the proof. Let X be a closed point in . In order to apply the Hilbert-

lim t

Mumford Criterion we need to determine the limit point t for the action of

G SLV

any one-parameter subgroup m on . Now is completely determined

V V V V n Z n n

by the decomposition of into weight spaces n , , of weight , i.e.

v t t v v V V n n

for all n . Of course, for almost all . De®ne ascending F

®ltrations of V and by

V F V O m V

n n n X

and

V O m F F F

n X n n n Then induces surjections n . Summing up over

all weights we get a closed point

F F V O m

n n n X

H P

in Quot .

lim t

Lemma 4.4.3 Ð t

V O m k T F

Proof. We will explicitly construct a quotient X parametrized

A Sp eck T by such that and for all . The assertion

follows from this. Let

F F T F k T T

n k

n F

Only ®nitely many summands with negative exponent n are non-zero, so that can be con-

A X N V

sidered as a coherent sheaf on . Indeed, let be a positive integer such that n

F n N F F T k T k

and n for all . Then . Similarly, de®ne a module

n N

V V O m T V O m T k T

n X k X k n

4.4 The Construction Ð Proofs 97

V F A A X

Clearly, induces a surjection of -¯at coherent sheaves on . Finally,

V T T id V k T j

n V V

de®ne an isomorphism (!) k by for all ;

and let be the surjection that makes the following diagram commutative:

n N

F T F F T k T

x x x

V O m k T V V O m T k T

X X

fg X

n n

First, restrict to the special ®bre : it is easy to see that ; for we have

F F T F F A n fg

n n etc. Restricting to the open complement corresponds to

inverting the variable T : all horizontal arrows in the diagram above become isomorphisms.

Thus we get:

F k T T F k T T

k T

x x

V O m k T T V O m k T T

k X k k X k

Note that describes precisely the action of ! Hence has the required properties, and we

are done. 2

G L Lemma 4.4.4 Ð The weight of the action of m via on the ®bre of at the point is

given by

n P F

F F G n

Proof. decomposes into a direct sum of subsheaves on which m acts via a

n n G n

character of weight . Hence for each integer the group m acts with weight on the

H F i

complex which de®nes the cohomology groups n , , (cf. Section 2.1). This

P F G m

complex has (virtual) total dimension n , so that acts on its determinant with

detH F n P F L

weight n . Since , the weight of the action on

L 2 n P F

is indeed n .

n dim V

We can rewrite this weight in the following form: use the fact that n n

since the determinant of is 1, and that in both sums only ®nitely many summands are non-

zero:

X X

n P F n dimV P F dimV P F

n n n

dimV

nZ nZ

dimV P F dim V P F

n n

dimV

98 4 Moduli Spaces

H F R

Lemma 4.4.5 Ð A closed point is (semi)stable if and only if for all

V F V

non-trivial proper linear subspaces V and the induced subsheaf

O m F

X , the following inequality holds:

V P F dimV P F

dim (4.2) V

Proof. De®ne a function on the set of subspaces of by

V dim V P F dimV P F

Then, with the notations of Lemma 4.4.4, we have

X X

x n P F V

n n

dim V

nZ nZ

Hence, according to the Hilbert-Mumford Criterion 4.2.11, a point is (semi)stable, if for

V V V

n n

any non-trivial weight decomposition , the condition is satis-

V V V

®ed. Hence if for any non-trivial proper subspace , then is (semi)sta-

V V V V

ble. Conversely, if V is a subspace with and is any complement

of V , de®ne a weight decomposition of by

V V V V V

dimV dimV

and else.

2 V dimV V n

Then . This proves the converse.

H F R

Lemma 4.4.6 Ð If is suf®ciently large, a closed point is (semi)stable

F V V H F m if and only if for all coherent subsheaves F and the following

inequality holds:

V P F dimV P F

dim (4.3)

H F m

Here and in the following we use the more suggestive notation V instead

m H F m

of H .

V V

Proof. If V runs through the linear subsets of then the family of subsheaves

F V fP F g

F generated by is bounded. Hence, the set of polynomials is ®nite, and

if is large, the conditions (4.2) and (4.3) are equivalent (with still denoting the subsheaf

F V V V H F m

generated by V ). Moreover, if is generated by , then , and

F V V H F m

conversely, if F is an arbitrary subsheaf of and , then the subsheaf

V F of F generated by is contained in . This shows that the condition of Lemma 4.4.6 is

equivalent to the condition of Lemma 4.4.5. 2

Corollary 4.4.7 Ð For to be semistable, a necessary condition is that the induced ho-

H F m F F

momorphism V is injective and that no submodule of dimension

d has a global section.

4.4 The Construction Ð Proofs 99

Proof. Indeed, if this were false, let V be a non-trivial linear subspace such that the

F V P F d subsheaf F generated by is trivial or torsion. Then has degree less than

and we get a contradiction to (4.3). 2

4.4.8 Proof of Theorem 4.3.3 Ð Let m be large enough in the sense of Theorem 4.4.1

r p

and such that any semistable sheaf with multiplicity and Hilbert polynomial is

m-regular. Moreover, let be large enough in the sense of Lemma 4.4.6.

H F R R V

First assume that is a closed point in . By de®nition of , the map

F m F F r r

H is an isomorphism. Let be a subsheaf of multiplicity and

V H F m

let V . According to Theorem 4.4.1 one has either

pF pF P F r P F r

, i.e. , or

h F m r pm

m dimV h F m r pm

In the ®rst case F is -regular, and we get and therefore

dimV P F r pm r p r pm r p dim V P F

In the second case

dimV r r r pm h F m r dimV r

These are the leading coef®cients of the two polynomials appearing in (4.3), so that indeed

V P F dim V P F

dim and hence Criterion (4.3) is satis®ed. This proves:

s ss s

s s

R R R n R R n R

and .

V O m F R

Conversely, suppose that X is semistable in the GIT sense.

R Because of the ®rst part of the proof it suf®ces to show that .

By Lemma 4.4.6 we have an inequality

dim V P F dim V P F

F V V H F m for any F and . Passing to the leading coef®cients of the

polynomials we get

m r r dimV r dim V r

p (4.4)

R F As is in the closure of by assumption, the sheaf can be deformed into a semistable

sheaf, hence a fortiori into a pure sheaf. Thus we can apply Theorem 4.4.2 and conclude

F E E

that there exists a generically injective homomorphism to a pure sheaf

E P F F

with P and whose kernel is the torsion of . According to Corollary 4.4.7

H F m H E m

the composite map V is injective, since any element in the

T F E E

kernel would give a section of d . Let be any quotient module of of multi-

r r F F E E

plicity r , . Let be the kernel of the composite map and

V H F m V . Using inequality (4.4) we get:

100 4 Moduli Spaces

h E m h F m h F m

dim V dimV

r pm r pm r pm

h E m dim V V

Thus E is semistable by Theorem 4.4.1. In particular, . Since

V E V E m

H is injective, it is in fact an isomorphism, and generates . But the map

O m E F F E

X factors through , forcing the homomorphism to be surjective.

F F

Since E and have the same Hilbert polynomial, must be an isomorphism. Hence, is

H F m R semistable and V is bijective. This means that is a point in .

Remark 4.4.9 Ð The last paragraph is the only place where have used the fact that the

R QuotH P

given semistable point lies in rather than just in . Sometimes this restric- F tion is not necessary: Suppose that X is a smooth curve. Then any torsion submodule of

is zero-dimensional and can therefore be detected by its global sections. Hence Corollary

H F 2 4.4.7 implies that F is torsion free if is semistable.

We are almost ®nished with the proof of 4.3.3. What is left to prove is the identi®cation

of closed orbits. Observe ®rst that we can read the proof of Lemma 4.4.3 backwards: let

H F R JH F F V

be a point in and a Jordan-HÈolder ®ltration of . Let

H JH F m V n V V

n n

n for all , and choose linear subspaces which split the

V O m g r F X

®ltration. Summing up the induced surjections n one gets a point

H g r F lim t

, and a one-parameter subgroup such that t . Thus, loosely speaking, any semistable sheaf contains its associated polystable sheaf in the

closure of its orbit. Now is a good quotient and separates closed invariant subschemes.

H F R F

It therefore suf®ces to show that the orbit of a point is closed in if is

H F R

polystable. Suppose is in the closure of the orbit of . It suf®ces

F F C

to show that in this case . The assumption implies that there is a smooth curve

F E X E

parametrizing a ¯at family of sheaves on such that for some closed point

F C E F O F

C nfg nfg

and C . Let be the (unique) decomposition of

F F

into isotypical components. Formally, we can think of i as running through a complete

set of representatives of isomorphism classes of stable sheaves with reduced Hilbert poly-

p n homF F E i

nomial , where the i are given by . Since the family is ¯at, the function

C N t homF E

i t

i n t n homF F n

i i

is semicontinuous for each and equals i for all . Thus .

F HomF F F

i k i

The image of the homomorphism i is polystable with all

F F F i

summands isomorphicto i . Moreover, must be injective. Finally, the sum

n n F 2 F F

must be direct. This is possible only if i and . We are done.

i i 4.5 Local Properties and Dimension Estimates 101

4.5 Local Properties and Dimension Estimates

In this section we want to derive some easy bounds for the dimension of the moduli spaces

of stable sheaves on a projective scheme X . This is done by showing that at stable points the moduli space pro-represents the local deformation functor. From this we get a smooth-

ness criterion and dimension bounds by applying Mori's result 2.A.11. If the scheme X is a smooth variety these results can be re®ned by exploiting the determinant map from the

moduli space to the Picard scheme of X .

X F M

Theorem 4.5.1 Ð Let F be a stable sheaf on represented by a point . Then the

O D

completion of the local ring M pro-represents the deformation functor . (cf. 2.A.5).

D O

Proof. Clearly, there is a natural map of functors F by the openness of stabil-

M F

ity 2.3.1 and the universal property of M . To get an inverse consider the geometric quotient

s s s

R M q H F R

constructed in Section 4.3. Let be a point in the ®bre

F S R

over . By Luna's Etale Slice Theorem there is a subscheme through the closed

b b

O O q S M q

point such that the projection is Âetale near . Then as func-

Sq M F

Ar tink R X S X

tors on , and the universal family on , restricted to , induces a map

O 2 D

F which yields the required inverse.

As a consequence of this theorem and Proposition 2.A.11 we get

M F

Corollary 4.5.2 Ð Let F be a stable point. Then the Zariski tangent space of at is

T M Ext F F M F Ext F F

F canonically given by . If , then is smooth at .

In general, there are bounds

ext F F dim M ext F F ext F F

F 2

If X is smooth, these estimates can be improved. Recall that to any ¯at family of sheaves S

on X parametrized by a scheme we can associate the family of determinant line bundles

PicX which in turn induces a morphism S . By the universal property of the moduli

space we also obtain a morphism

det M PicX

which coincides with the morphism induced by a universal family on M in case such

F D

a family exists. Similarly, if is a stable sheaf, there is a natural map of functors F

F D

F det from deformations of to deformations of its determinant. We want to relate the obstruction spaces for these functors and their tangent spaces.

102 4 Moduli Spaces

E tr E ndE O tr

If is a locally free sheaf, then the trace map X induces maps

i i

H E ndE H O Ext E E

X . We shall see later (cf. Section 10.1) how to con-

tr Ext F F H O F

struct natural maps X for sheaves which are not necessarily

locally free. These homomorphisms are surjective if the rank of F is non-zero as an ele-

i i

k Ext F F tr ext F F ment of the base ®eld . Let denote the kernel of , and let be

its dimension.

det M PicX F Theorem 4.5.3 Ð Let F be a stable sheaf. The tangent map of at

is given by

T PicX Ext F F H O tr T M

detF F

ker A A Ar tink m

Moreover, if is an extension in with A , and if

F D A F

A , then the homomorphism

Ext det F detF tr Ext F F H O

oF F A odetF

A A maps the obstruction A to extend to onto the obstruction to extend the determinant.

Proof. The proof of this theorem requires a description of the deformation obstruction which differs from the one we gave, and a cocycle computation. We refer to Artamkin's

paper [5] and in particular to Friedman's book [69]. 2

X Now Pic naturally has the structure of an algebraic group scheme: the multiplication being given by tensorizing two line bundles. A theorem of Cartier asserts that (in characteristic zero) such a group scheme must be smooth (cf. II.6, no 1, 1.1, in [43]). In particular,

all obstructions for extending the determinant of a sheaf F vanish.

X F O

Theorem 4.5.4 Ð Let be a smooth projective variety and let be a stable X -module

Q M Q det

of rank r and determinant bundle . Let be the ®bre of the morphism

M PicX Q T M Q Ext F F Ext F F

over the point . Then . If ,

M Q F

then M and are smooth at . Moreover,

ext F F dim M Q ext F F ext F F

r B

Proof. Tensorizing a sheaf E or rank by a line bundle twists the determinant bundle

E B B E B

det by . Moreover, if is numerically trivial, is semistable or stable if and

det M PicX

only if E is semistable or stable, respectively. It follows from this that

Q M Q

is surjective in a neighbourhood of and is, in fact, a ®bre bundle with ®bre in

Q M Q F

an Âetale neighbourhood of . Then 4.5.3 implies that the tangent space of at is

tr Ext F F H O F

the kernel of the trace homomorphism X , and moreover, that

Ext F F Ext F F

has an obstruction theory with values in . Thus the vanishing of

M Q implies smoothness for M and hence for . Finally, the estimates stated in the theorem

follow as above from Proposition 2.A.11. 2

4.5 Local Properties and Dimension Estimates 103

Corollary 4.5.5 Ð Let C be a smooth projective curve of genus . Then the moduli

O r

space of stable C -sheaves of rank with ®xed determinant bundle is smooth of dimension

r g

C ext F F F

Proof. As is one-dimensional, for any coherent sheaf . Thus the

moduli space is smooth according to the theorem, and, using the Riemann-Roch formula,

ext F F F F O r g 2 C its dimension is given by .

As a matter of fact, for a smooth projective curve the moduli space of stable sheaves is irreducible and dense in the moduli space of semistable sheaves [234].

If X is a smooth surface we can make the dimension bound more explicit: Note that for

a stable sheaf F

i i

ext F F ext F F ext F F O

which by the Hirzebruch-Riemann-Roch formula is equal to

F r O

F r c F r c F (Recall that .)

De®nition 4.5.6 Ð The number

exp dimM Q F r O

is called the expected dimension of M . r

Lemma 4.5.7 Ð Let X be a smooth polarized projective surface and let be a positive

X r

integer. There is a constant depending only on and such that for any semistable

r X

sheaf F of rank on one has

ext F F

ext F F homF F h

X X

Proof. By Serre Duality, . Applying Proposi-

degX r homF F X

tion 3.3.6 we get X Obviously,

degX

r 2 the right hand side depends only on X and .

Thus we can state F

Theorem 4.5.8 Ð Let X be a smooth polarized projective surface and let be a stable

sheaf of rank r and determinant . Then

exp dimM Q dim M Q exp dimM Q

exp dimM Q dim M Q M Q F

F If then is a local complete intersection at (cf.

2.A.12). 2

104 4 Moduli Spaces

n M What can be said about points in M ? In this case the picture is blurred because of the existence of non-scalar automorphisms. At this stage we only prove a lower bound

for the dimension of R , that will be needed later in Chapter 9. The setting is that of Section V

4.3 for the case of a smooth projective surface: m is a suf®ciently large integer, a vector

P m H V O m R QuotH P X

space of dimension and k . Let be the open

H F F V H F m

subscheme of those quotients where is semistable and

H F R F m

is an isomorphism. Let be ®xed. As is -regular, it follows from the

EndH Ext H F i K HomH F properties of , that and for all . Let be

the kernel of . Then there is an exact sequence

EndF HomH F HomK F Ext F F

i i

Ext F F Ext K F i

and isomorphisms for . Recall that the boundary map

K F Ext F F

Ext maps the obstruction to extend onto the obstruction to ex-

F Ext F F

tend (cf. 2.A.8), and that the latter is contained in the subspace . This leads

to the dimension bound

dim R homK F ext F F

homH F ext F F ext F F ext F F

endH h O exp dim M Q

det F det R PicX

where Q as before. Consider the map induced by the

X Q

universal quotient on R . This map is surjective onto a neighbourhood of , and

dim PicX h O

since X , we ®nally get the following dimension bound for the ®-

Q det Q

bre R :

dim R Q exp dimM Q endH 2

Proposition 4.5.9 Ð

Example 4.5.10 Ð Let X be a smooth projective surface, and consider the Hilbert scheme

Hilb X QuotO X

X of zero-dimensional subschemes in of length . It is easy

X X Hilb X X X

to see that Hilb and that is the quotient of the blow-up of

Hilb X

along the diagonal by the action of Z that ¯ips the two components. In fact, is

a smooth projective variety of dimension for all . We give two arguments: ®rst, let

X X Z x Hilb X X

denote the ideal sheaf of the universal family in Hilb . Let

I x k x I

be an arbitrary point. For any surjection Z we can consider the kernel of

I I x k x Z Hilb X Z

Z and the associated point . This construction yields

I Hilb X PI Hilb X X

a surjective morphism P . Note that the ®bres of

X are projective spaces and hence connected. In particular, by induction we see that Hilb is connected.

4.6 Universal Families 105

Now Hilb always contains the following -dimensional smooth variety as an open

U fx x jx x i j g X

i j subset: let be the quotient of the open subset in by the

permutation action of the symmetric group. Thus, if we can show that the dimension of the

Zariski tangent space at every point in Hilb is we are done: for then the closure of

Hilb X

U is smooth and cannot meet any other irreducible component of , hence is all of

X Z X Hilb X

Hilb as the latter is connected. Let be a closed point in . Recall that

T Hilb X HomI O

Z Z

Z . Moreover,

homI O ext O O homO O homO O

Z Z Z Z X Z Z Z

homO O O O ext O O

X Z Z Z Z Z

homO O homO O

X Z Z Z

lengthO

Hilb X

Using Theorem 4.5.4 we can give a shorter proof: observe that we can identify

Z X I M O Z

X by sending a subscheme to its ideal sheaf . In order to conclude

Ext I I

smoothness it suf®ces to check that Z . But

ext I I homI I K homO K

Z Z Z Z X X X

See also 6.A.1 for a generalization of this example. 2

4.6 Universal Families s

We now turn to the question under which hypotheses the functor M is represented by the

s s M moduli space M . If this is the case is sometimes called a ®ne moduli space.

Let X be a polarized projective scheme. Recall our convention that whenever we speak

S p q

about a family of sheaves on X parametrized by a scheme , and denote the projections

X S S X X

S and , respectively.

X M

De®nition 4.6.1 Ð A ¯at family E of stable sheaves on parametrized by is called

S X

universal, if the following holds: if F is an -¯at family of stable sheaves on with Hilbert

P S M L

polynomial and if F is the induced morphism, then there is a line bundle

S F p L M E E

on such that . An -¯at family is called quasi-universal, if there is

O W F p W E

a locally free S -module such that .

s s M Clearly, M represents the moduli functor if and only if a universal family exists.

Though this will in general not be the case, quasi-universal families always exist. Recall

GLV R R

that the centre Z of acts trivially on . Therefore the ®bre over any point

x R X GLV R R X

or of any -linearized sheaf on or , respectively, such as

F X Z

the universal quotient on R , has the structure of a -representation and decomposes

into weight spaces. We say that a sheaf or a particular ®bre of a sheaf has Z -weight , if

G t Z t

m acts via multiplication by .

106 4 Moduli Spaces

V R Z

Proposition 4.6.2 Ð There exist GL -linearized vector bundles on with -weight 1.

H omp A F E

If A is any such vector bundle then descends to a quasi-universal family , E and any quasi-universal family arises in this way. If A is a line bundle then is universal.

In the proof of the proposition we will need the following observation, which is the rel-

ative version of 1.2.8. X

Lemma 4.6.3 Ð Let F be a ¯at family of stable sheaves on a projective scheme , para-

S O p E ndF metrized by a scheme . Then the natural homomorphism S is an isomor-

phism.

O p E ndF O

Proof. The homomorphism S is given by scalar multiplication of on

F S

F . The assumption that is -¯at implies that the homomorphism is injective. Now, for

k s S F EndF k s

each -rational point the ®bre s is stable and therefore simple, i.e. ,

k s p E ndF s EndF s

so that the composite homomorphism is surjective.

p E ndF s EndF s

Hence is surjective and therefore even isomorphic by the semi-

Ext O p E ndF

continuity theorems for the functors . This means that S is surjective p

as well. 2

n A p F q O n

Proof of the proposition.If is suf®ciently large then n is a locally

P n GLV Z

free sheaf on R of rank and carries a natural -linearization of -weight 1. Let

GLV R Z Z

A be any -linearized vector bundle on with -weight 1. Then acts trivially on

omp A F PGLV

the bundle H , which therefore carries a -linearization and descends to

M X E F

a family E on by 4.2.14. We claim that is quasi-universal. Suppose that is a

S P

family of stable sheaves on X parametrized by a scheme with Hilbert polynomial . Let

R be the associated frame bundle and consider the commutative diagram

R R F

y y

M S

e e

F F

Then and hence

X F X

e e e

E E H omp A F

X F X F X X F X

e e e

H om p A F

F X F X

H omp A F

F X

R F S GLV A

Now is linearized in a natural way and is a -principal bundle

B S A B

so that for some vector bundle on . It follows that

H omp B F F p B E H om

X X X F X X and therefore

4.6 Universal Families 107

E H omp B F p B F

F X

Thus E is indeed a quasi-universal family. Conversely, let be a quasi-universal family on

s s

M X E R X

. Then applying the universal property of to the family F on we ®nd

e e

E E F A R p H om H omp A F

a vector bundle on such that . Then

X X

e e e

p H omp A F F A p E ndF A

by the previous lemma. This description

GLV Z

shows that A carries a -linearization of -weight 1 which is compatible with the

E E A H omp A F

isomorphism . It is clear that is universal if and only if is a line X

bundle. 2

Excercise 4.6.4 Ð Show by the same method: if E and are two quasi-universal families, then

W W M E p W E p W

there are locally free sheaves and on such that . c

For the remaining part of this section let X be a smooth projective variety. Let be a ®xed

s s

K X P M c M

class in num , let be the associated Hilbert polynomial, and let and

s s

c R R be the open and closed parts that parametrize stable sheaves of numerical class

c (see also Section 8.1). X

Suppose B is a locally free sheaf on . Then the line bundle

B det p F q B PicR c

c B B

as de®ned in Section 2.1 carries a natural linearization of weight . If is not lo-

B B B

cally free, we can still choose a ®nite locally free resolution and de®ne

P N

c B c B B B i

i . Then has weight .

i i

c B B Theorem 4.6.5 Ð If the greatest common divisor of all numbers , where runs

through some family of coherent sheaves on X , equals 1, then there is a universal family on

c X

M .

B B w w

Proof. Suppose there are sheaves and integers such that

P N

w c B B A Z

i i

i , then is a line bundle of -weight 1. Hence the theo-

i i

rem follows from the proposition. 2

Recall that the Hilbert polynomial P can be written in the form

n i

a P n

a a d dimF d

with integral coef®cients , where .

g cda a M X d

Corollary 4.6.6 Ð If then there is a universal family on .

O O d X

Proof. Apply the previous theorem to the sheaves X . It suf®ces to check

g cda a g cdP P d d that the . But this follows from the observation that the matrix

108 4 Moduli Spaces

i d

is invertible over the integers. 2

X r c c

Corollary 4.6.7 Ð Let be a smooth surface. Let , , be the rank and the Chern

c c K c c g cdr c H

classes corresponding to . If , then there is a

c X

universal family on M .

O O X

Proof. Apply the previous theorem to the sheaves X , , and the structure sheaf

O P X P P

P of a point . The assertion then follows by expressing and in terms of

c O r 2

Chern classes and using that P .

Remark 4.6.8 Ð The condition of Corollary 4.6.7 is also suf®cient to ensure that there are

c M c

no properly semistable sheaves, in other words that M . Namely, suppose that

c F r r

F is a semistable sheaf of class admitting a destabilizing subsheaf of rank and

Chern classes c and . Then we have the relations:

r c H r c H

and

r c c K c r c c K c

X X

r c H c c K c

If and are integers with , then

r K c c H c r r c

X , obviously a contradiction.

4.A Gieseker's Construction 109 Appendix to Chapter 4

4.A Gieseker's Construction

The ®rst construction of a moduli space for semistable torsion free sheaves on a smooth projective surface was given by Gieseker [77]. We brie¯y sketch his approach, at least where it differs from the construction discussed before, for the single reason that it gives a bit more: namely, the ampleness of a certain line bundle on the moduli space relative to the Picard

variety of X .

X O

Let X be a polarized smooth projective variety over an algebraically closed X

®eld of characteristic zero. Let P be a polynomial of degree equal to the dimension of P

(i.e. we consider torsion free sheaves only) and let r be the rank determined by . Recall V

the notations of Section 4.3: m is a suf®ciently large integer, a vector space of dimension

P m H V O m R QuotH P

, and X . Let be the subscheme of all quotients

H F F V H F m

such that is semistable torsion free and is an isomorphism.

R R QuotH P H O F

Let be the closure of in . The universal quotient induces R

an invariant morphism

det R PicX

det F det P PicX P p A

such that , where denotes the PoincarÂeline bundle on

X R m

and A is some line bundleon . We may assume that was chosen large enough so that

R PicX m

any line bundle represented by a point in the image det is -regular. From

e e

H O F V O detF q O m

we get homomorphisms X and

R R X

V O p det F m det p P r m A

which is adjoint to

det H om V p P r m A

Note that is everywhere surjective and therefore de®nes a morphism

R Z PH om V p P r m

A PicX O

of schemes over , such that Z . Moreover, is clearly equivariant for the

SLV Z F V O m F

obvious action of on . Observe, that if is torsion free then X

V H det F m

is, as a quotient, completely determined by the homomorphism .

j Aj PicX R

This means that R is injective, hence is ample relative to .

R Z Z

Theorem 4.A.1 Ð maps to the subscheme of semistable points in with respect

s s

SLV O R Z

to the -action and the linearization of Z , and . Moreover, as

ss s

R Z R R A

is ®nite, good quotients of and exist, and some tensor power of

PicX descends to a line bundle on the moduli space M which is very ample relative to . 110 4 Moduli Spaces

We sketch a proof of the ®rst statements. The last assertion then follows from these and

general principles of GIT quotients.

H F R G SLV

Let be a point in and let m be a one-parameter group

V V V V

n n n

given by a weight decomposition n . As in the proof of 4.3.3,let

V F V O m

n n X

be the induced ®ltration on , but de®ne as the saturation of in

F r F F F det F detF

n n n n n

, and let be the rank of . Then n and

O O

r r

detF m t V V H lim

n n

Hence the weight of the action at the limit point is, up to some constant, given by

X X X

r dimV r dimV n r dimV r dimV nr dimV

n n n

n n

n n n

The same reduction as in the proof of 4.3.3 shows that is (semi)stable, if and only if the

V r

following holds: If V is any non-trivial proper subspace of and if is the rank of the

subsheaf in F generated by , then

dim V r dimV r

At this point we can re-enter the ®rst half of the proof of 4.3.3 and conclude literally in the

same way. 2

4.B Decorated Sheaves

So far we have encountered two different types of moduli spaces: the Grothendieck Quot- scheme and the moduli space of semistable sheaves. The Grothendieck Quot-scheme parametrizes all quotients of a sheaf, i.e. sheaves together with a surjection from a ®xed one. In this spirit, one could, more generally, consider sheaves endowed with an additional structure such as a homomorphism to or from a ®xed sheaf, a ®ltration or simply a global section. For many types of such `decorated' sheaves one can set up a natural stability condition and then formulate the appropriate moduli problem. (Recall, there is no stability condition quotients parametrized by the Quot-scheme have to satisfy.) We are going to describe a moduli problem that is general enough to comprise various interesting examples. To a large extent the theory is modelled on things we have been explaining in the last sections. In particular, the boundedness and the actual construction of the moduli space, though involving some extra technical dif®culties, are dealt with quite similarly. However, two things in the theory of decorated sheaves are different. First, the stability condition is usually slightly more complicated and depends on extra parameters, which can be varied. Second, by adding the additional structure we make the automorphism group of the objects in question smaller. This can be used to construct ®ne moduli spaces in many instances.

4.B Decorated Sheaves 111 k

Let X be a smooth projective variety over an algebraically closed ®eld of characteristic

O E

zero. Fix an ample invertible sheaf X and a non-trivial coherent sheaf . Furthermore,

Q t F

let be a positive polynomial. A framed module is a pair consisting of a

F E

coherent sheaf F and a homomorphism . Its Hilbert polynomial is by de®nition

F P F

P , where if and otherwise. For

simplicity we give the stability condition only for framed modules of dimension dim :

F r

De®nition 4.B.1 Ð A framed module of rank is (semi)stable if for all framed sub-

F F F F j r P F rkF

modules , i.e. and F , one has

P .

Remark 4.B.2 Ð i) If , then this stability condition coincides with the stability con-

dition for sheaves. If , then the stability condition splits into the following two con-

ker r P F rkF P F rkF

ditions: for subsheaves F one requires and

F r P F rkF P F r rkF

for arbitrary F only the weaker inequality .

F F E

ii) If is a semistable framed module then embeds the torsion of into .

F deg dimX deg

iii) If is semistable and , then . Moreover, if

X F deg dimX dim and is semistable then is injective. Thus, for all framed

modules are just subsheaves of E . Since this case is covered by the Grothendieck Quot-

dimX

scheme, we will henceforth assume deg .

iv) By de®nition the stability of a framed module with depends on the poly-

nomial , but for `generic' the stability condition is invariant under small changes of .

Only when crosses certain critical values the stability condition actually changes. More-

over, for generic semistability and stability coincide.

v) For small and generic, e.g. is a positive constant close to zero, the underlying sheaf

F F

F of a semistable framed module is semistable. Conversely, if is a stable sheaf and

F E F

is non-trivial, then is stable with respect to small .

D X F

Example 4.B.3 Ð Let E be a sheaf supported on a divisor . Then a sheaf on

E X F j

together with an isomorphism D (a `framing') gives rise to a framed module

E F F F j E X

in our sense with D . Here, is considered as a sheaf on

X D fxg

with support on D . In the case of a curve and a point these objects are also

E O

called bundles with a level structure. Next, let be the trivial invertible sheaf X . In this

case, the underlying sheaf of a semistable framed module must be torsion free (this is true

X F F

whenever E is torsion free). Thus, on a curve semistable framed modules

X are locally free and, therefore, there is no harm in dualizing, i.e. instead of considering

F F H F

we could consider . This gives an equivalence between

semistable framed modules and semistable pairs, i.e. bundles with a global section. This correspondence holds also true in higher dimensions if we restrict to locally free sheaves. There are of course interesting types of decorations that are not covered by framed modules. Most important, parabolic sheaves and Higgs sheaves. 112 4 Moduli Spaces

Let us now introduce the corresponding moduli functor. As before, we ®x a positive poly-

dimX E O

nomial of degree less than , a coherent sheaf , an ample invertible sheaf X

Q z dim X

and a polynomial P of degree . Then the moduli functor

M S chk S ets

S ObS chk S F F E

maps to the set of isomorphism classes of -¯at families S

P F P P F P

t t

of semistable framed modules with t and for any closed point

S M

t . By we denote the open subfunctor of geometrically stable framed modules.

M P E

Theorem 4.B.4 Ð There exists a projective scheme O that universally co-

M M M P E

represents the functor . Moreover, there is an open subscheme of O

s 2

that universally represents M .

M P E

Analogously to the case of semistable sheaves, the closed points of O X parametrize S-equivalence classes of framed modules. We leave it as an exercise to ®nd the right de®nition of S-equivalence in this context. Note that the second statement is stronger

than the corresponding one in Theorem 4.3.4. It says that on the moduli space of stable

F framed modules with there exists a universal family. This will be essentially

used in the proof of the following proposition.

M M P

Proposition 4.B.5 Ð Let O be the moduli space of semistable sheaves with

s M

Hilbert polynomial P and let be the open subscheme of stable sheaves. Then there exists

M M M M E

a projective scheme , a morphism and an -¯at family such that:

s M

i) is birational over ,

ii) on the open set over M where is an isomorphism the family is quasi-universal,

t t M

iii) if F is the sheaf corresponding to for a closed point ,

E F b rkE rkF

then t is S-equivalent to , where .

M Of course, if M this is just the existence of a quasi-universal family (cf. 4.6.2). In

general, a quasi-universal family can not be extended to a family on the projective scheme

M E

M . The projective variety together with is a replacement for this. As it turns out, for

many purposes this is enough. Note, if is reduced, by desingularizing M and pulling-

back one can assume that M is in fact smooth.

M X b P

Proof. Let E be a quasi-universal family on with Hilbert polynomial (For the

existence of E see Proposition 4.6.2.). Let denote the moduli functor of semistable

sheaves with Hilbert polynomial P . We de®ne a natural functor transformation

b s

F F b MbP n M bP bP

M by . If , the image is contained in . The

M P M bP

induced morphism M is a closed immersion (see Lemma 4.B.6 be-

bP O n F F O n

low). Next, consider the moduli space M of framed modules .

F F q O n

For generic there exists a universal framed module on the product

bP O n X F F O n F M and for small generic the map de®nes a

4.B Decorated Sheaves 113

M bP O n M bP N M M bP O n

morphism . Let M . It suf®ces

M M M

to construct a section of N over a dense open subset of . Indeed, the closure

M N M bP O n F

of this section in N together with the pull-back of under s

satis®es i), ii), and iii). The construction of the section over a dense open subset of M goes

F M

as follows. For n and any sheaf there exists a non-trivial homomorphism

O n

F . Moreover, the generic homomorphism gives rise to a stable framed module,

M bP O n N p H omE q O n

i.e. a point in and hence in . The sheaf is free over

a dense open subscheme U . A generic non-vanishing section of this free sheaf in-

M U U duces a section of N over (We might have to shrink slightly in order to make

all framed modules semistable). 2

j M P M bP

Lemma 4.B.6 Ð The canonical morphism b is a closed immersion.

M P F F W

i i

Proof. As points in are in bijection with polystable sheaves i ,

F W HomF F

i i

where i are pairwise non-isomorphic stable sheaves and , and since the

j F F W k m

i i i

morphism b is given by , it is clearly injective. Let be a suf®ciently

P m bP m

QuotO m P R QuotO m bP

large integer, and let R and j

be the open subsets as de®ned in Section 4.3. Then b is covered by a natural morphism

b P m

R R F O m F

, . Let be a polystable sheaf as above and

R M P F GLP m

a point in the ®bre of over . The stabilizer subgroups of and

bP m

GL at the points and are given by

Y Y

GLW GLW k G G i

i and

i i

R R respectively. The normal directions to the orbits of in and in at these points

are

Ext F F HomW W E

i j i j ij

and

Ext F F HomW W Endk E

i j i j

on which G and act by conjugation. By Luna's Etale Slice Theorem, an Âetale neighbour-

F M P E G E G

hood of in embeds into . Therefore it suf®ces to show that embeds

E G id E E

into , or equivalently, that the diagonal embedding E induces

G G

O O

a surjective homomorphism . In fact, one can check that the partial trace map

E E

E Endk E 2 E induces a splitting.

The proposition above is one application of moduli spaces of framed modules. They also provide a framework for the comparison of different moduli spaces, e.g. the moduli space of rank two sheaves on a surface and the Hilbert scheme. For simplicity we have avoided the extensive use of framed modules in these notes, but some of the results in Chapter 5, 6, 11 could be conveniently and sometimes more conceptually formulated in this language. 114 4 Moduli Spaces

4.C Change of Polarization

The de®nition of semistability depends on the choice of a polarization. The changes of the moduli space that occur when the polarization varies have been studied by several people in greater detail. We only touch upon this problem and formulate some general results that will be needed later on.

Let X be a smooth projective surface over an algebraically closed ®eld of characteristic

PicX NumX PicX 0. Let denote numerical equivalence on , and let . This

is a free Z-module equipped with an intersection pairing

NumX NumX Z

The Hodge Index Theorem says, that, over R, the positive de®nite part is 1-dimensional.

Num u Num juj R

In other words, R carries the Minkowski metric. For any class let

u j

j . This is not a norm! Recall that the positive cone is de®ned as

K fx Num X jx xH H g

R and for some ample divisor X

It contains as an open subcone the cone A spanned by ample divisors. A polarization of

R H H A H K

is a ray , where . Let denote the set of rays in . This set can be identi®ed

fH K j jH j g

with the hyperbolic manifold . The hyperbolic metric is de®ned as

H H H

follows: for points let

H H

H H arcosh

jH jjH j

Recall that arcosh is the inverse function of the hyperbolic cosine.

Num X

De®nition 4.C.1 Ð Let r and be integers. A class is of type

if . The wall de®ned by is the real 1-codimensional submanifold

W fH H j H g H

r Lemma 4.C.2 Ð Fix r and as in the de®nition above. Then the set of walls of type

is locally ®nite in H .

H H

Proof. The lemma states that every point in has an open neighbourhood intersect-

r H H

ing only ®nitely many walls of type . Let be the class of length . Then

Num RH H u u aH u a R

R , and any class decomposes as with and

u H Num kuk a ju j bH

.De®neanormon by . Let be a class of

r B H H H

type and let be a positive number. is the open ball in with center

H W B H H H H H H

and radius . Suppose that . Write with .

H H jH j tanh H b H

Let . Check that . Then

j j j j b b j H j j j jH j tanh j j

and . Moreover,

tanh j j k k j j b tanh j j

and . Hence

4.C Change of Polarization 115

tanh r r

k k cosh

tanh

Thus is contained in a bounded, discrete and therefore ®nite set. This proves that the set

f jW B H g 2

is ®nite.

H F r

Theorem 4.C.3 Ð Let be an ample divisor, a H -semistable coherent sheaf of rank

F F r r r F

and discriminant , and let be a subsheaf of rank , , with H

F rc F r c F

H . Then satis®es:

H and

and if and only if .

c NumX H r

In particular, if is indivisible, and if is not on a wall of type , then

r c H

a torsion free sheaf of rank , ®rst Chern class and discriminant is -semistable if

and only if it is H -stable.

F F F F

Proof. We may assume that is saturated. Then is torsion free and H -

r r H

semistable of rank r . Since it follows from the Hodge Index Theorem

that with equality if and only if . Moreover, the following identity holds:

r r

F F

r r r r

F F

By the Bogomolov Inequality (3.4.1) one has and therefore

r r

c F

If is not divisible, then , hence . Thus, if a subsheaf as above exists, then

r 2

H lies on a wall of type .

Remark 4.C.4 Ð The assumption of the theorem that H be ample is too strong: if

K r c F

, it makes still sense to speak of H -(semi)stable sheaves in the sense that

F F r r r r c F H

for all saturated subsheaves of rank , . The proof of

the theorem goes through except for the following point: in order to conclude that

F and we need the Bogomolov Inequality in a stronger form (7.3.3) than proved so far (3.4.1). Chapter 7 is independent of this appendix. In the following we will therefore

make use of the theorem in this form, since in the applications we have in mind H will be

the canonical divisor K of a smooth minimal surface of general type, which is big and nef but in general not ample. 116 4 Moduli Spaces

It is clear from the proof that the conditions on classes whose walls could possibly affect

r the stability notion are more restrictive than just being of type as de®ned in 4.C.1.

Since we have no need for a more detailed analysis here, we leave the de®nition as it stands.

H H Num Recall that A is the closure of the ample cone. If and are elements in , we

write

H H ftH tH j t g

H A K F

Lemma 4.C.5 Ð Let H be an ample divisor and . Let be a torsion free

H H H

sheaf which is H -stable but not -stable. Then there is a divisor and a

F F F F F F

H H H

subsheaf such that , and and are -semistable of the

same slope.

F H H

Proof. If is H -semistable we can choose and there is nothing to prove.

Hence we may assume that is not even H -semistable. Then there exists a saturated sub-

F F F F F

H H sheaf with . If is any saturated subsheaf with this property,

let

F F

H H

F F

H H

H H tF H F F

H tF H tF H

so that H . Note that is ample. If

F F tF tF

H H

, then . By Grothendieck's Lemma 1.7.9, the family

of saturated subsheaves F with this property is bounded. This implies that there are only

tF tF F

®nitely many numbers which are smaller than . In fact, we may assume that

tF F H

was chosen in such a way that is minimal. Then and have the properties stated

in the lemma. 2

For the de®nition of e-stability see 3.A.1.

H A K r

Proposition 4.C.6 Ð Let H be an ample divisor, . Let and be

e sinh H H F

integers and put . Suppose that is a coherent sheaf of rank

r F e H F

and discriminant . If is -stable with respect to then is H -stable.

F H

Proof. Suppose that is H -stable but not -stable. By the previous lemma there ex-

H H H F H rc F

ists a divisor and a subsheaf such that for

rkF c F H H H H aH

. Let . Note that . Write

jH j

H bH H a b R tanh H H

and with and . Then . Moreover,

bjH j

H abjH j H

and therefore

j j H jH j ajH j tanh

Furthermore, by 4.C.3 and 4.C.4 the inequality

r j j a jH j j j a jH j sinh

4.C Change of Polarization 117

ajH j r sinh

holds, implying that . Finally we get:

sinh H ajH j jH j

F F jH j e

H H

r rkF r rkF rkF rkF

e 2

This means that F is not -stable, contradicting the assumption of the proposition.

Theorem 4.C.7 Ð Let H and be ample divisors. If , the moduli spaces

r c M r c M

H H

are birational.

Proof. We may assume that H and are very ample. Recall that we have an estimate

e M

for the -unstable locus of H :

r K H

r e e B X H dim M e

r jH j

e sinh

Inserting for some positive number , the coef®cient of on the

sinh

right hand side is , and this coef®cient is strictly smaller than 1 if

sinh arsinh H H H

. Fix . Subdivide the line in connecting and into

r r

®nitely many sections such that the division points have mutual distances and have

H H H H H

very ample integral representatives . Now choose large

r c M

enough such that H is a normal scheme of expected dimension (cf. Theorem

e dim M dim M i N H

9.3.3) and such that H for each . This is possible

i i

e M M

H H

since by our choice of the dimension of grows faster than the dimension of

i i

e M

considered as functions of . By the proposition only the -unstable sheaves in H can

H H

be unstable with respect to i or . Therefore the dimension estimate just derived

M M 2 H

shows H .

Comments: Ð The notion of S-equivalence is due to C. S. Seshadri [233]. He constructs a projective moduli space for semistable vector bundles on a smooth curve, which compacti®es the moduli space of stable bundles constructed by Mumford [190]. There exists an intensive literature on moduli of vector bundles on curves. We refer to Seshadri's book [234] and the references given there. In the curve case, G. Faltings [61] gave a construction without using GIT, see also the expository paper by Seshadri [235]. Ð The main reference for Geometric Invariant Theory is Mumford's book [194]. The lecture notes of Newstead [202] explain the material on a more elementary level. We also recommend the seminar notes of Kraft, Slodowy and Springer [131]. Theorem 4.2.15 is due to G. Kempf. For a proof see Thm 2.3. in [52] or Prop 4.2 in part 4 of [131]. 118 4 Moduli Spaces

Ð General constructions of moduli spaces of sheaves on higher dimensional varieties have been given by D. Gieseker [77] for surfaces and M. Maruyama [162, 163]. This Approach has been sketched in appendix 4.A. Our presentation in 4.3 and 4.4 follows the method of C. Simpson [238] and the J. Le Potier's exposÂe[145]. Theorem 4.4.1 and the proof of 4.4.2 are taken from Le Potier's exposÂe[145]. The observation of the dichotomie of the cases A and B in the proof of 4.4.1 as well as the statement of 4.4.2 are due to Simpson. This is one of the main technical improvements of his approach. Observe how well suited the Le Potier-Simpson Estimate is to mediate between the Euler characteristic and the number of globals sections of a sheaf. Theorem 4.4.2 is Lemma 1.17 in [238], where it is proved in a slightly different way. In a sense, this theorem is responsible for the projectivity of the moduli space of semistable sheaves. Thus in Gieseker's construction its rÃole is played by Lemmas 4.2 and 4.5 in [77]. In a certain sense, the properness of the moduli space had been proved by Langton [135] before the moduli space itself was constructed. See Appendix 2.B. In the proof of 4.4.5 we used a pleasant technical device we learned from A. King [123]. Ð The smoothness of Hilbert schemes of points on surfaces (Example 4.5.10) is due to Fogarty [65]. His argument for smoothness is the ®rst one given in the example, whereas his proof of the con-

nectivity is quite different and very interesting. He shows that the punctual Hilbert scheme, i.e. the

X X closed subscheme in Hilb of those cycles which are supported in a single, ®xed point in can

be considered as the set of ®xed points for the action of a unipotent algebraic group on a Grassmann

Hilb X M

variety and therefore must be connected. The existence of natural morphisms n

X S as discussed in Example 4.3.6 is asserted by Grothendieck [93] though without proof and using a different terminology. Our presentation of the linear determinant follows Iversen [117]. Ð Deformations of coherent sheaves are discussed in the papers of Mukai [186], Elencwajg and Forster [55] and Artamkin [5, 7]. See also the book of Friedman [69]. Theorem 4.5.1 was proved by Wehler in [259]. Ð The existence of universal families was already discussed by Maruyama [163]. The notion of quasi-universal families is due to Mukai [187]. Ð Theorem 4.B.4 can be found in [115]. For other constructions of similar moduli spaces see [147], [234], [244], [156]. The probably most spectacular application of stable pairs is Thaddeus' proof of the Verlinde formula for rank two bundles [244]. Recently, moduli spaces of stable pairs on surfaces

have found applications in non-abelian Seiberg-Witten theory. X Ð The changes that moduli spaces undergo when the ample divisor H on crosses a wall have been studied by several authors, often with respect to their relation to gauge theory and the computation of Donaldson polynomials. We refer to the papers of Qin, GÈottsche [84], Ellingsrud and GÈottsche [57], Friedman and Qin [72], Matsuki and Wentworth [171]. Ð We also wish to draw the reader's attention to the papers of Altman, Kleiman [3] and Kosarew, Okonek [130]. In these papers, moduli spaces of simple coherent sheaves are considered. In [3] the moduli space of simple coherent sheaves on a projective variety is shown to be an algebraic space in the sense of Artin. In general, however it is neither of ®nite type nor separated. The phenomenon of non-separated points in the moduli space was investigated in [203] and [130]. Part II

Sheaves on Surfaces

119

121

5 Construction Methods

The two most prominent methods to construct vector bundles on surfaces are Serre's construction and elementary transformations. Both techniques will be used at several occasions in these notes. Section 5.3 contains examples of moduli spaces on K3 surfaces and ®bred surfaces. In the latter case we discuss the relation between stability on the surface and stability on the ®bres. In order to motivate the ®rst two sections let us recall some general facts about globally

generated vector bundles.

V H W

Let be a short exact sequence of vector spaces and denote the

W v r s s minfv r g

dimension of V and by and , respectively. Let be an integer, ,

M HomV W

and let s be the general determinantal variety of all homomorphisms of

s M v sr s

rank . Then s is a normal variety of codimension and the singular part

M M x M M

s s

of s is precisely (cf. [4] II 2). Let be the intersection of the pre-image of

H W HomV W U HomH W

under the surjection Hom and the open subset of

surjective homomorphisms. Then M is either empty or a normal subvariety of codimension

M v sr s Sing M M GLW

with . Clearly, is invariant under the natural

s s s

M M U GrassH r

action on U . Let be the image of under the bundle projection .

s s

Then M has the analogous properties of .

s s

E r

Let X be a smooth variety. Suppose is a locally free sheaf of rank which is generated

H H X E H O

by its space of global sections . The evaluation homomorphism X

X GrassH r V H

E induces a morphism . If is a linear subspace of dimension

v M GrassH r X M X

and if is de®ned as above, then s is by construc-

s s

V O E

tion precisely the closed subscheme where the homomorphism X has rank less

GLH GrassH r than or equal to s. Since acts transitively on , we may apply Kleiman's

Transversality Theorem (cf. [98] III 10.8) and ®nd that for generic choice of V the morphism

GrassH r M n M

X is transverse to any of the smooth subvarieties . It follows that

X v sr s Sing X X

s s

s is either zero or is a subvariety of codimension with .

r H H X E Examples 5.0.1 Ð Let E be a globally generated rank vector bundle and

the space of global sections as above.

r d dim X v r d s r d X

1) Suppose and let , , so that s is precisely

V O E V H X s

the locus where X is not ®brewise injective. If is general, is either

r d d empty or has codimension r , hence is indeed empty. This means that

there is a short exact sequence

r d

O E E X

122 5 Construction Methods

d for some locally free sheaf E of rank . In words, any globally generated vector bundle of

rank bigger than the dimension of X is an extension of a vector bundle of smaller rank by

a trivial bundle.

X V H v r X

2) Let be a surface and let be a general subspace with . Then r

is empty or has codimension , and r is empty or has codimension 6 (hence is empty).

Thus for a general choice of r global sections there is a short exact sequence

O E F

where F is of rank almost everywhere, but has rank precisely at a smooth scheme

detE I X F r

Z r of dimension 0, i.e. . For this is part of the Serre correspon-

dence between -cycles and rank two bundles which will be discussed in Section 5.1.

X V H v r X

3) Again, let be a surface and let be a general subspace with . Then r

is empty or has codimension and r is empty or has codimension 4 (hence is empty).

Thus for a general choice of r global sections there is a short exact sequence

E L O

L X E

where is zero or a locally free sheaf of rank on the smooth curve r . We say is ob-

tained by an elementary transformation of the trivial bundle along the smooth curve r . The details will be spelled out in Section 5.2.

Thus, the theory of globally generated bundles and determinantal varieties providesa uniform approach to the Serre correspondence and elementary transformations.

For the rest of this chapter we assume that X is a smooth projective surface over an alge-

braically closed ®eld of characteristic 0 which, sometimes, will even be the ®eld of complex

K X numbers. By X we denote the canonical line bundle of .

For the convenience of the reader we recall the following facts discussed in Chapter 1 and

specify them for our situation. If F is a re¯exive sheaf of dimension on then all sheaves

E xt F O q

X have codimension by Proposition 1.1.10 and must therefore vanish

F F F F

for q , which means that is locally free. If is only torsion free then is

E xt F O

a canonical embedding into a locally free sheaf. Again by Proposition 1.1.10 X

E xt F O F

has dimension 0 and X . Hence is locally free outside a ®nite set of points

E F

in X and has homological dimension 1, i.e. if is any surjection with locally free

ker

E then is also locally free, or, still rephrasing the same fact, any saturated subsheaf

D X dhO

of a locally free sheaf is again locally free. If is a divisor then clearly D .

D O dh G

Since locally any vector bundle G on is isomorphic to , one gets as well.

X dhk x F F F

If x is a point, then . Finally, if is a short

F maxfdhF dh F g F F F

exact sequence then dh . If is torsion free and is locally free, then F F cannot contain 0-dimensional submodules. 5.1 The Serre Correspondence 123

5.1 The Serre Correspondence

The Serre correspondence relates rank two vector bundles on a surface X to subschemes of

codimension 2. We begin with some easy observations.

F L I F L

If F is a torsion free sheaf of rank 1, then is a line bundle and

O Z F L I Z

X is the ideal sheaf of a subscheme of codimension at least 2, i.e. . Using the

c F c L c F c L I

Hirzebruch-Riemann-Roch Theorem one gets and

c O Z F Z . Any torsion free sheaf of arbitrary rank has a ®ltration with torsion

free factors of rank 1: simply take any complete ¯ag of linear subspaces of the stalk of F F at the generic point of X and extend them to saturated subsheaves of . For example, any

torsion free sheaf of rank 2 admits an extension

F L I L I

Z Z

(5.1)

F detF L L c F

and the invariants of are given by the product formula: ,

c L c L Z Z

, and

F c F c F Z Z c L c L

(5.2)

c L c L c L c F

(5.3)

F Z Z If is locally free then must be empty and if in addition is not empty then the exten-

sion cannot split.

Theorem 5.1.1 Ð Let Z be a local complete intersection of codimension two, and let

M X

L and be line bundles on . Then there exists an extension

L E M I

E L M K Z

such that is locally free if and only if the pair X has the Cayley-

Bacharach property:

Z Z Z Z s H X L M K

(CB) If is a subscheme with and X

sj sj Z with Z , then .

Proof. Let us ®rst show the `only if' part. Assume the Cayley-Bacharach property does

Z Z s H X L M K

not hold, i.e. there exist a subscheme and a section X such

Z Z sj sj Z

that and Z but . We have to show that given any extension

L E M I E

Z the sheaf is not locally free. Use the exact sequence

I I k x Z Z Z

Z induced by the inclusion and the assumption to show

H X L M K I H X L M K I

Z X Z

that X is injective. The dual

L Ext M I L Ext M I Z of this map is the natural homomorphism Z which

is, therefore, surjective. Hence any extension ®ts into a commutative diagram of the form

124 5 Construction Methods

L E M I

k x k x

L M I E

Since and Z are torsion free, is torsion free as well. Hence the sequence

E k x E E is non-split and cannot be locally free.

For the other direction we use the assumption that Z is a local complete intersection. This

Z Z Z

implies that there are only ®nitely many subschemes Z with . For Z

let x be a closed point in the support of . Then there is presentation

f f

O I O

Zx X x

X x

H omk x E xt k x I f f m k x

Applying the functor , we ®nd Z , since ,

k x I Z Z I

Z so that there is precisely one subscheme with Z .

Suppose now that

L E M I

is a non-locally free extension. Then there exists a non-split exact sequence

x E L E k x

E where is a singular point of . The saturation of in can differ from

x L E L only in the point . Since is locally free, it is saturated in as well. Thus we get a

commutative diagram of the above form. Hence, the extension class is contained in the

L Ext M I L Ext M I Z

image of the homomorphism Z . Since the Cayley-

L Ext M I L Ext M I Z Bacharach property ensures that the map Z is not

surjective, we can choose such that it is not contained in the image of this map for any of

E 2

the ®nitely many Z that could occur. The corresponding will be locally free.

Z H X L M K

The Cayley-Bacharach property clearly holds for all if X .

P x X

Examples 5.1.2 Ð i) Let X and . Using Serre duality and the exact sequence

I O k x Ext I O H X I

x X x X

, we ®nd that x

O H X k x k

. Hence, up to scalars there is a unique non-split extension X

X K E I H

X x

x . Since , the Cayley-Bacharach Condition is satis®ed and,

therefore, this extension is locally free. Moreover, x is -semistable. Thus every point

x X E

corresponds to a -semistable vector bundle x .

x X

ii) Let X be an arbitrary smooth surface and let be a base point of the linear system

jL M K j L M K x L M K x

X X

X , i.e. all global sections of vanish in . Then

L E

satis®es (CB). Hence there exists a locally free extension of the form

M I

x .

5.1 The Serre Correspondence 125

Analogously, if x y are two points which cannot be separated by the linear system

jL M K j L E

X , then there exists a locally free extension of the form

M I

xy g f . These two examples are important for the study of surfaces of general type (cf. [227]).

Though the Serre correspondence works for higher dimensional varieties as well, it is in

general not easy to produce vector bundles in this way. The reason being that codimension two subschemes of a variety of dimension are dif®cult to control.

On surfaces Serre's construction can be used to describe -stable rank two vector bundles c

with given determinant and large Chern number .

H Q PicX

Theorem 5.1.3 Ð Let X be a smooth surface, an ample divisor, and a line

c c c

bundle. Then there is a constant such that for all there exists a -stable rank two

E detE c E c Q

vector bundle with and .

Proof. First observe that it suf®ces to prove the theorem under the additional assumption

Q Q QnH n

that deg is suf®ciently positive. For if the theorem holds for , ,

E Q c c

and gives a -stable vector bundle with determinant and second Chern class

c E E O nH Q

for some constant , then X is also -stable, has determinant and

nH c Q nH c E c nH c Q nH c c

Chern class . Hence will do.

Q E Thus we may assume that deg . The idea is to construct as an extension of the

form

O E Q I Z

X (5.4)

detE Q c E Z h K Q

so that indeed and . Let . Then for a generic 0-

Z Z K Q I

X Z

dimensional subscheme of length the sheaf has no non-trivial

Z Z K Q Z X sections, so that for a generic subscheme of length the pair has the

Cayley-Bacharach property (CB). Hence, under this hypothesis there exists an extension as

M E

above with locally free E . Suppose were a destabilizing line bundle. It follows from

c QH O M E M X

the inequality that cannot be contained in

O M E Q I Z

X . Thus the composite homomorphism is nonzero. It vanishes along

c QH d D Z D degD Q M

a divisor with and . The family

of effective divisors of degree less than or equal to d is bounded. (This can be proved using the techniques developed in Chapter 3 or more easily using Chow points. For a proof see

Lecture 16 in [191].) Let Y denote the Hilbert scheme that parametrizes effective divisors

Y X d maxf g

on of degree , and let be its dimension. For any integer let

Z D D Y

be the relative Hilbert scheme of pairs where is an effective divisor and

Y Z D D D Y

isatupleof distinct closed points on . Then for each the ®bre of the

e e e

dimY Hilb X Y Y Y

projection has dimension , so that . The imageof in

Z X Z dimHilb X

under the projection has dimension .

D Z d

Hence, if Z is a generic -tuple of points, a divisor containing and having degree

2 does not exist, which implies that the corresponding E is indeed -stable.

126 5 Construction Methods E

Remark 5.1.4 Ð The same method allows to construct -stable bundles with vanishing

Ext E E

obstruction space . Such bundles correspond to smooth points in the moduli

Q c space M . (Note that the vanishing condition is twist invariant, hence we may again

assume that Q is as positive as we choose.)

E K Q K K

X X

Indeed, tensorizing the exact sequence (5.4) by X , and , respectively,

we get sequences

E K E ndE K E I K

X Z X

X (5.5)

Q K E K I K

X Z X

X (5.6)

K E K Q I K

X Z X X (5.7)

From these we can read off that

ext E E h O h E ndE h E ndE K

X X

h E K h E I K

X Z X

h E K h E K

X X

h Q K h I K

X Z X

h K h Q K I

X X Z

ext E E h Q K h I K h Q K I

X Z X X Z

thus . The®rst term on

degQ deg K

the right hand side vanishes if X . The second and the third term vanish 2 for generic Z of suf®ciently great length.

The theorem above asserts the existence of -stable vector bundles for large second Chern

numbers. It is not known if one can ®nd stable bundles with given second Chern class

c Q PicX CH X

and . More precisely, one should ask if for a given line bundle

CH X

and a class c of degree zero one can construct a -stable rank two vector bundle

E cE c c E x x X c E with , where is a ®xed base point and is considered as an integer.

For the rest of Section 5.1 we assume for simplicity that our surface X is de®ned over

the complex numbers. Since the Albanese variety Alb is a ®rst, though in general very

X rough, approximation of CH , the following result can be regarded as a partial answer.

Before stating the result, let us brie¯y recall the notion of the Albanese variety of a smooth

X AlbX H X H X Z

variety . By de®nition, is the abelian variety X and, after

having ®xed a base point x , the Albanese map is the morphism de®ned by

A X AlbX y x

5.1 The Serre Correspondence 127

AlbX

The image of X generates as an abelian variety. In particular, the induced mor-

X Alb X

phism A , given by addition, is surjective for suf®ciently large . Note

X AlbX

that A is invariant with respect to the action of the symmetric group

on X . It therefore factors through the symmetric product and thus induces a morphism

A CH X AlbX Hilb X AlbX

A . There is also a group homomorphism

A X CH X x x A which commutes with A and the map . Both and depend on

the choice of the base point x. As a general reference for the Albanese map we recommend

[252].

PicX x X a Alb X

Proposition 5.1.5 Ð For given Q , , , and a polarization

H c c c

one can ®nd an integer such that for there exists a -stable rank two vector

Ac E a E det E c E c Q

bundle with , and .

Q Z

Proof. As above, we may assume that deg .If is a codimension two subscheme

E O E Q

and if is a locally free sheaf ®tting into a short exact sequence X

Ac E AZ U I

Z , then . Hence it is enough to show that the open subset

Z X

Hilb of those subschemes , for which a -stable locally free extension exists, maps

X U U

surjectively to Alb . In the proof of 5.1.3 we have seen that contains the set of all

D d

reduced Z which are not contained in any effective divisor of degree (notations as

h Q K I Z Z Z Z Z

in 5.1.3) and satisfy X for all with . We

Hilb X D

have also seen that the set of Z that are contained in some divisor as above

has codimension . Choosing large enough we can make this codimension greater

q h O AlbX

than X which is the dimension of and hence an upper bound for the

Hilb X AlbX

codimension of any ®bre of the morphism A . Hence it suf®ces to

A Hilb X AlbX Hilb X Hilb X

show that is surjective, where is the

Z Hilb X h Q K I Z Z Z open subscheme of all reduced with X for all

of colength one.

jmH j x

Let C be a smooth ample curve containing the ®xed base point . Sincethegroup

H X O C H X O H C O

X C X vanishes, the restriction homomorphism

is injective. Using Hodge decomposition, this map is complex conjugate to the restriction

H X H C H C

C C

map X . It follows that the dual homomorphism

H X AlbC AlbX

X is surjective, and therefore the group homomorphism is

C Alb C Pic C

surjective as well. The Albanese map S can also be described

C Z O Z x Pic C

by C . Hence it suf®ces to ®nd for any given line bundle

Pic C Z C

M a reduced subscheme such that the following conditions are satis®ed:

X Q K I Z O Z x h Z M

X Z

(1) C and (2) for every of colength

m H X Q K O C X

one. If then X , so that property(2)follows from the

H C Q K O Z Z C C

fact that X for suf®ciently large and any scheme

M x M

of length . Finally, is very ample for independently of . Hence we

Z C O Z 2 M x

easily ®nd a reduced with C , i.e. satisfying condition (1).

It is only natural to ask if Serre's construction can also be used to produce higher rank 128 5 Construction Methods

bundles. This is in fact possible as will be explained shortly. As a generalization of 5.1.3,

5.1.4, and 5.1.5 for the higher rank case one can prove

PicX r a Alb X x X

Theorem 5.1.6 Ð For given Q , , , , and a polarization

H c c c

one can ®nd a constant such that for there exists a -stable vector bundle

E rkE r detE Q c E c H X E nd E

with , , and , such that and

Ac E a

Proof. We only indicate the main idea of the proof. The details, though computation- ally more involved, are quite similar to the ones encountered before. First, one generalizes

Serre's construction and considers extensions of the form

M I L E

i Z

Z Z Z i j

i j

Assuming that all i 's are reduced and ( ), one can prove that a locally free

L M K Z

X i

extension exists if and only if i satis®es the Cayley-Bacharach property

r for all i . In order to construct vector bundles as asserted by the theorem one

considers extensions of the form

nH I Q r nH E

n Q O r nH

for some suf®ciently large integer . Twisting with X yields the exact

sequence

Q r nH I O E

Z X

E Q r nH

where E . Then the Cayley-Bacharach property holds for generic

Z Z h Q r nH K F E

i X

i with . Suppose now, that is a destabilizing

r n Q r nH

locally free subsheaf of rank s .If was chosen large enough so that ,

F Q r nH I

then must be contained in Z , and passing to the exterior powers there

is a nonzero and therefore injective homomorphism

detF Q r snH I

Z Z

i i

i i r

I Z

(Note that the sheaf on the right hand side is the quotient of i by its torsion sub- i

module.) Thus there is an effective divisor D of degree

degD s Q r nH s F Q r nH

s r Z

which contains at least of the subschemes i . As in the proof of 5.1.3 this is impos- Z

sible if all i are general and have suf®ciently great length.

H X E ndE Ext E E The vanishing of is achieved as in 5.1.4. It is also not

dif®cult to see that the proof of 5.1.5 goes through. 2 5.2 Elementary Transformations 129

5.2 Elementary Transformations

Now, we come to the third example discussed in the introduction.

X F G

De®nition 5.2.1 Ð Let C be an effective divisor on the surface . If and are vector

C E X

bundles on X and , respectively, then a vector bundle on is obtained by an elementary G

transformation of F along if there exists an exact sequence

E F i G

C X

where i denotes the embedding .

G i G G O X If no confusion is likely, we just write instead of , meaning with its natural -

structure.

G X C

Proposition 5.2.2 Ð If F and are locally free on and , respectively, then the kernel

E F i G G

of any surjection is locally free. Moreover, if denotes the rank of , one

detF O C detE c E c F C c F C C

has and

K G

X .

O G O C O O

X C

Proof. Since locally and X is a locally free

X i G dh E

resolution on , the sheaf is of homological dimension . This implies that

E detE detF det i G detF C

, i.e. is locally free. The isomorphism

follows from the fact that G is trivial on the complement of ®nitely many points on . Thus

deti G deti O

and are isomorphic on the complement of ®nitely many points, hence

O C deti O deti G

. The formula for the second Chern class follows from

E F G E F F

and the Hirzebruch-Riemann-Roch formula for and :

E c F c F K c F rkF O c E c E K

X X X

and

c E rkF O c E c F C 2

. Inserting gives the desired result.

Note that for a smooth (or at least reduced) curve C the characteristic can be written

C K C G deg G g C degG c E c F

as . Hence

C deg G C c F

O C O X

Example 5.2.3 Ð A trivial example is X , which is the elementary transform of

O C log C X

along C . Another example is provided by the sheaf of differentials with

logarithmic poles along a smooth curve C . This is the locally free sheaf that is locally

dx x dx x x x

generated by and , where is a local chart and is the equation for

C O C C

. The restriction map X twisted by yields an exact sequence

log C C C

X X C

f dx x f dx x C f g x

Indeed, is mapped to zero in if and only if . Thus

log C C C

X C X is the elementary transform of along .

130 5 Construction Methods

r X

Let E be any vector bundle of rank on a smooth projective surface . For suf®ciently

E nH

large n the bundle is globally generated. The discussion in the introduction tells

us that there is a short exact sequence

E nH M O

C X

for some line bundle M on a smooth curve . Dualizing this sequence and twisting

O nH

with X yields

E O nH L

M O nH L E xt L C

with X . Note that is a line bundle on , as can easily be seen

O O O C M E xt L O M

C X C

from the fact that, locally, C and . In fact

O C nH

C . Thus we have proved: r

Proposition 5.2.4 Ð Every vector bundle E of rank can be obtained by an elementary

nH n C X 2

transformation of O , with ,alonga linebundleon a smooth curve . X

Similarly to Serre's construction, elementary transformations can be used to produce -

stable vector bundles on X .

Q PicX r H c Z

Theorem 5.2.5 Ð For given , , ample divisor and integer ,

E detE rkE r c E c Q

there exists a -stable vector bundle with , and .

Proof. Let C be a smooth curve. According to the Grothendieck Lemma 1.7.9, the torsion

F O F rkF r

free quotients of with C H and form a bounded family

C homO O nH homF O nH F C

. Now C grows much faster than for any in the

C n O O nH

family . Thus, if is suf®ciently large, a general homomorphism C is

C E E

surjective and does not factor through any F . Let be the kernel of . Then is locally

detE O C c E nH C E

free with X and . In order to see that is -stable,

E F E O

let E be a saturated proper subsheaf, let be the saturation of in and consider

F E O nH F E det E det F O C X the subsheaf C .If is nonzero, then ,

hence

C H C H

E F E

rkE rkE

E F O E

and we are done. If on the other hand F then is torsion free and

C H F C F

factors through F . By construction cannot be contained in , hence . It r

follows that

r rkE r rkE r C H

E F C H E

rkE rkE r r So E is indeed -stable.

5.3 Examples of Moduli Spaces 131

m Q r mH

If Q is an arbitrary line bundle, choose in such a way that is very am-

jQ r mH j E

ple, and pick a general curve C . If is a -stable vector bundle constructed

detE O C Qr mH

according to the recipe above with determinant X , then

mH Q 2 E is -stable with determinant and large second Chern class.

Remark 5.2.6 Ð One can in fact choose in the proof of the theorem in such a way that

HomE E K Ext E E n

vanishes. First check that for suf®ciently large,

E E K O X

any homomorphism E can be extended to a homomorphism

E O K

X . Conversely, such a homomorphism leaves invariant, if and only if there

H X K

is a section X such that . It is easy to see that the condition on

O s r

to be traceless requires to factor through a quotient bundle , . As before, X

since the family of such quotients is obviously bounded, for suf®ciently large n and general 2 this will never be the case.

5.3 Examples of Moduli Spaces

Fibred Surfaces. We ®rst show that for certain polarizations on ruled surfaces the moduli space is empty. This will be a consequence of the relation between stability on the surface and stability on the ®bres, which can be formulated for arbitrary ®bred surfaces. The arguments may give a feeling for Bogomolov's restriction theorem proved in Chapter 7. For

simplicity, we only deal with the rank two case, but see Remark 5.3.6.

C X C

Let X be a surface, let be a smooth curve, and let be a surjective morphism.

c c c c f

Fix Chern classes and . As usual, let . By we denote the homology

class of the ®bre of .

H c c

De®nition 5.3.1 Ð A polarization is called -suitable if and only if for any line

M PicX c M c f c M c

bundle with either or

f c M c H c M c

and have the same sign.

C C X Sp ec k

Let be the generic point of and denote the generic ®bre C by

F E X E E F

. If is a coherent sheaf on , let be the restriction of to .

H c c E

Theorem 5.3.2 Ð Let be a -suitable polarization and let be a rank two vector

c E c c E c E H E

bundle with and . If is -semistable (with respect to ), then

E E

is semistable. If is stable, then is -stable.

E M E E M

Proof. Let be -semistable and let be a rank one subbundle such that

is locally free. Then there exists a unique saturated locally free subsheaf M of rank

M M c M c M

one such that . By (5.3) we have . If is destabilizing,

f c M c H c c H c M c

i.e. if then, since is -suitable, also , E contradicting the -semistability of .

132 5 Construction Methods

E M E

Conversely, assume that is stable. If is a saturated subsheaf of rank one, then

f c M degM degE f c

f c M c H c c

Hence . Since is -suitable by assumption and since again

c M c H c M c E

by (5.3), we conclude that . Hence is

H 2

-stable with respect to . E

Recall from Section 2.3 that is semistable or geometrically stable if and only if the

restriction of E to the ®bre is semistable or geometrically stable, respectively, for C

all t in a dense open subset of .

c E f E

If , then, for obvious arithmetical reasons, is geometrically stable if E

and only if is semistable. Hence

c f H c c

Corollary 5.3.3 Ð If and if is -suitable, then a rank two vector

E c E c c E c E

bundle with and is -stable if and only if is stable. Moreover,

E 2

E is -semistable if and only if is -stable.

C E X

Corollary 5.3.4 Ð If X is a ruled surface, then there exists no vector bundle on

c E c E

that is -semistable with respect to a -suitable polarization and that satis®es

c E f

Proof. This is a consequence of the fact that there is no stable rank 2 bundle on P .

c c

Remark 5.3.5 Ð The Hodge Index Theorem shows that for any choice of there

H H H nf

exists a suitable polarization. Indeed, let be any polarization and de®ne n .

H n c c n H f

Then n is ample for and -suitable for . To see this let

c M c M f

for some line bundle and assume that . Since and

f f H f H n

n , the Hodge Index Theorem implies that

f H f H f H f f H H f H

n n n n n

H f

Dividing by n we obtain

H H f

f H f n

H f

Hence either f , or and therefore

H f

f H n

n H f

for suf®ciently large . The last inequality means that n and have the same sign. 5.3 Examples of Moduli Spaces 133

Remark 5.3.6 Ð Theorem 5.3.2 can be easily generalized to the higher rank case. In fact,

H for r one says that a polarization is suitable if it is contained in the chamber close to

the ®bre class f . (For the concept of walls and chambers we refer to Appendix 4.C.) Numer-

PicX

ically, this is described by the condition that for all such that

f H

either f or and have the same sign. The argument of the previous remark

H nf n r H f shows that if H is any polarization then is suitable if (see also Lemma 4.C.5).

K3-Surfaces. In the second part of this section two examples of moduli spaces of sheaves on K3 surfaces are studied. We will see how the techniques introduced in the ®rst two sections of this chapter can be applied to produce sheaves and to describe the global structure of the moduli spaces. We hope that studying the examples the reader may get a feeling for the geometry of these moduli spaces. They will also serve as an introduction for the general results on zero- and two-dimensional moduli spaces on K3 surfaces explained in Section 6.1. Both examples share a common feature. Namely, the canonical bundle of the moduli space of stable sheaves is trivial. This is a phenomenon which will be proved in broader generality in Chapter 8 and Chapter 10.

The canonical bundle of a K3 surface is trivial and the Euler characteristic of the struc-

i i

A B Ext B A

ture sheaf is 2. Hence Serre duality takes the form Ext for any

E homE E

two coherent sheaves A B . Any stable sheaf is simple, i.e. , so that

ext E E homE E homE E M Q

. Thus any moduli space of stable rank 2 sheaves with determinant Q and discriminant is empty or smooth of ex-

pected dimension

dim M Q r O c c

P Example 5.3.7 Ð Let X be a general quartic hypersurface. By the adjunction for-

mula X has trivial canonical bundle, and by the Lefschetz Theorem on hyperplane sections

X X

is trivial ([179] Thm. 7.4). Hence is a K3 surface. Moreover, by the Noether-

Lefschetz Theorem (see [90] or [42]) its Picard group is generated by X , the restriction

X of the tautological line bundle on P to . In particular, there is no doubt about the polar-

ization of X which therefore will be omitted in the notation.

M O c

Consider the moduli spaces X . For any rank two sheaf with determi-

O M O c

nant X -semistability implies -stability. Thus is a smooth pro-

M O c c

jective scheme. If X is not empty then its dimension is (since

c O deg X X

). In particular, if a stable sheaf with these invariants exists

then . This is slightly stronger than the Bogomolov Inequality 3.4.1. The smallest

possible moduli space is at least two-dimensional. In fact

X M M O

Claim: X .

F M

Proof. Since the re¯exive hull of a -stable sheaf is again -stable, any de®nes

F M O c c c F

a point in X with . By the inequality above , i.e.

134 5 Construction Methods

F F M F

is locally free. For any the Hirzebruch-Riemann-Roch formula gives

F homF O h F h F

and hence X (we have because of the

F F O i X

stability of ). Let i , be three linearly independent homomorphisms

F O

and let denote the sum . We claim that ®ts into a short exact X

sequence of the form

F O I

I x X im

where x is the ideal sheaf of a point . If were not injective, then would be

I a V I a O V

of the form V for some codimension two subscheme . Since , one has

a F I a

. On the other hand, as a quotient of the stable sheaf the rank one sheaf V has

non-negative degree. Therefore, a . But then

F I O O

V X

HomF O X and hence the i would only span a one-dimensional subspace of , which con-

tradicts our choice. Therefore is injective. A Chern class calculation shows that its coker-

O rkcoker

nel has determinant X and second Chern class . Since ,it is enough

F F O F

to show that coker is torsion free. If not, let be the saturation of in . Then X

is a rank two vector bundle as well and

detF det F O b O

b F F detF b detF

for some . Since both and are locally free, ; hence

O F I

. Thequotient then is necessarily of the form V for a codimension two subscheme

O V HomO I V F

. But X unless , which then implies that , contradicting

F M

again the linear independence of the i . Eventually, we see that indeed any is part

of a short exact sequence of the form

F O I

F H X F H X O H X I

The stability of implies , so that the map x

Ext F O H X F k

is bijective. Hence X . Inserting this bit of information

homF O

into the Hirzebruch-Riemann-Roch formula above one concludes that X . F This implies that (and hence the short exact sequence) is uniquely determined by (up

to the action of GL).

On the other hand, if we start with a point x and denote the kernel of the evaluation

H X I O I F F

X x x x

map x by , then is locally free and has no global section.

h X F F F

x x

Clearly, x implies that is stable; for any subsheaf possibly destabilizing

O X X

must be isomorphic to X . In order to globalize this construction let denote

I p q X

the diagonal, its ideal sheaf, and let and be the two projections to . De®ne a sheaf

F by means of the exact sequence

F p p I q O I q O

X X

5.3 Examples of Moduli Spaces 135

F p F F j F X M x F F

x x x

x is -¯at and p . Thus de®nes a morphism , .

The considerations above show that this map is surjective, because any F is part of an exact F

sequence of this form, and injective, because is uniquely determined by . Since both

M 2 spaces are smooth, X is an isomorphism.

We will prove that `good' two-dimensional components of the moduli space are always closely related to the K3 surface itself (6.1.14). In many instances the rÃole of the two factors can be interchanged. Let us demonstrate this in our example. It is straightforward to

complete the exact sequence

F H X I O I

x x X x

to the following commutative diagram with exact rows

F H X I O I

x x X x

j H P O O O

X X X

I O k x

x X

Both descriptions

F H X I O I

x X x x (5.8)

and

F j I

x X x

P (5.9)

are equivalent. Back to the proof, we had constructed the exact sequence

F p p I q O I q O

X X

X fxg X

on X . Restricting this sequence to yields (5.8), and restricting it to the ®bre

X fxg I q O q O O

yields (5.9). (Use the exact sequence

p I q O F X X j

to see that P .) Thus the vector bundle on identi®es

each factor as the moduli space of the other.

X P

Example 5.3.8 Ð Let be an elliptic K3 surface with irreducible ®bres. We

P X

furthermore assume that X has a section . By the adjunction formula is

a -curve. For the existence of such surfaces see [22]. If denotes the class of a ®bre,

H f H H mf m then , and more generally, m for , are ample divisors.

This follows from the Nakai-Moishezon Criterion.

c f c m

If , then for ®xed and , the -semistability of a rank two vector

bundle is equivalent to its H -stability (cf. Corollary 5.3.3).

X O f m M M

Claim: If is suf®ciently large, then H . m 136 5 Construction Methods

There are two ways to prove the claim. The ®rst uses Serre's construction, and the second

relies on the existence of a certain universal stable bundle on X , discovered by Friedman and Kametani-Sato, that restricts to a stable bundle on any ®bre. For both approaches one

needs Corollary 5.3.3 which says:

m E Q c Qf

For a bundle with determinant such that is -stable (with H

respect to the polarization m ) if and only if its restriction to the generic ®bre is stable.

c f dim M

Note that , so that . As before M this implies that any -stable sheaf in is locally free.

Proof of the Claim via Serre's construction.

E M E

Let be a closed point. By the Hirzebruch-Riemann-Roch formula

f c h E homE O h E

, and by stability X , so that .

E F s

Since the restriction of to the generic ®bre is stable of degree , a global section

X E H can vanish in codimension two or along a divisor not intersecting the generic

®bre, i.e. a union of ®bres. Hence one always has an exact sequence of the form

O nf E I O n f

X Z X

c E n Z

A comparison of the Chern classes yields the condition . Hence

Z fxg s x X

either n and , i.e. vanishes in codimension two at exactly one point ,

n Z E O f O f X

or and , i.e. is an extension of the line bundle X by .

The following calculations will be useful: essentially because of , there is no

f D

effective divisor D such that for any integer . This means that the groups

h O f h O f h O f

X X

X vanish. On the other hand,

h O f deg O f f m f X

X because of stability:

m m h O f h O f

for . It follows that X

O f

X .

n Let us now take a closer look at the two cases n and :

i) An extension

O E I O f

X x X

x is stable if and only if it is non-split and x . Moreover, for given there is exactly one

non-split extension.

ext I O f O E

X X

Proof. First check that indeed x . Let be the unique non-

E E

trivial extension. is stable if and only if the restriction to the generic ®bre is stable. Let x

F be any smooth ®bre that does not contain . It suf®ces to show that the restriction map

Ext I O f O Ext O F O

x X X F F

X F

E E

is nonzero, for then F is stable and hence is stable. Now is dual to

H F O F H X I f

F x

5.3 Examples of Moduli Spaces 137

H X I f H X I x

Since x , the kernel of is precisely . Finally, since

h O F h I x

F , the homomorphism is nonzero if and only if , i.e.

2 if and only if x .

ii) An extension of the form

O f E O f

X X

is stable if and only if it does not split. The space of non-isomorphic non-trivial extensions

PH O f P

is parametrized by X . Proof. Again, it suf®ces to check that a given nontrivial extension class is not mapped to

zero by the restriction homomorphism

Ext O f O f Ext O F O

X X F F

X F

H O f H O F

X F

for a general ®bre F . Consider the exact sequence

H O f H O f H O F

X X F

of vector spaces of dimensions 1,2 and 1, respectively. Using the Leray Spectral Sequence

H O f H R O O

X X

we can identify P for , which

R O O

implies that P . In this way the problem reduces to showing that

F P s H O

varying the base point , which is the zero locus of a section P ,

H O 2 H O P

leads to essentially different embeddings P , which is obvious.

X X I p q

iii) Let be the diagonal, its ideal sheaf and and the projections to the

two factors. It follows from the Base Change Theorem and our computationsof cohomology

B R p I q O f

groups above that is a line bundle. Similarly, one checks

Ext I q O f p B HomB B C

that . Let

B F I q O f

X X F p F

be the unique nontrivialextension on . Then is -¯at and X parametrizes stable

n M

sheaves. This produces an open embedding X , whose complement is isomorphic

2 to P , by i) and ii). This proves the claim.

Proof of the Claim via elementary transformations.

O f O f Ext X We have seen that X is one-dimensional. Hence there is a unique

non-split extension

O f G O f

X X

det G O c G H

Obviously, X and . By Remark 5.3.5 the polarization

f O f H f H G

is -suitable. Since and , the bundle is - H

stable with respect to .

138 5 Construction Methods

X O f H X O f H F O F

Since H , the map G

is injective for any ®bre F , i.e. the extension de®ning is non-split on any ®bre. More-

G G

over, any stable sheaf E with the same invariants as is isomorphic to . Indeed, by the

O f E

Hirzebruch-Riemann-Roch formula Hom and by stability the cokernel of

any homomorphism is torsion free, hence isomorphic to O .

x X F x x I

Let be any closed point, the ®bre through and F x the ideal F

sheaf of x in . Since the extension

O G O F

F F F

I HomI G H G

F x F F

is non-split, F x . Hence, by Serre duality .

G I G F x

Since F , there is a unique nontrivial homomorphism

G F I

F x up to nonzero scalars. Again, since the extension de®ning is non-split on , the

E detE O

x x

homomorphism must be surjective. Let be the kernel of . Then X

Gj c E F E j f

F x x F

and . Moreover, for the generic ®bre we have , which

E E x X x

implies that x is stable. In this way we get a stable bundle for every point . To

X M

see that indeed , it suf®ces to write down a universal family.

X I X X X

Let P denote the diagonal and the ideal sheaf of as a

X p q X X X

subscheme in P . As before and denote the projections of to the two

factors. The Base Change Theorem and the dimension computations above imply that L

q G p L p I q G

is a line bundle and that the natural homomorphism

I q O E 2

X is surjective. The kernel de®nes a universal family.

As in the previous example one might ask about the symmetry of the situation. Using

E q G I p L q O

one can compute the restriction of E to the ®bres of . We get

E O I L

F x

q x

c E O f c E degI L

X F x

x q x

In particular, q and . To see that

deg L j X X X X F

calculate as follows: Observe that the ideal sheaf of P in

p O f q O f K X X I X

is given by X and that in we have the relation

O p O f q O f O

X X X

X . From this we deduce

L p I q G detp I q G

det p q G detp p O f

q G f detp q G j

O G f f detG O f

X X

degL j

Hence F . 5.3 Examples of Moduli Spaces 139

The situation is not quite so symmetric as in 5.3.7, e.g. the determinant has even inter-

section with the class of the ®bre. Nevertheless, the second factor is a moduli space of the

®rst one. One can check that q is stable and that the dimension of its moduli space is

c c E E x

. In fact, q determines the point uniquely by the condition that

F E I L F is the only ®bre where is not semistable and that its destabilizing quotient is F x . Details are left to the reader. The reader may have noticed that in the second example we made little use of the fact

that the elliptic surface is K3. Especially, the construction of the `unique' bundle G goes

through in broader generality.

P X

Proposition 5.3.9 Ð Let X be a regular elliptic surface with a section . If

H O f M K

H X

is the polarization X , then consists of a single

O f G

reduced point which is given by the unique nontrivial extension X

K f 2

X .

Comments: Ð Theorem 5.1.1 is standard by now (cf. [91],[132]). Ð The existence of stable rank two bundles via Serre correspondence (5.1.3 and 5.1.4) was shown in [16].

Ð We would like to draw the attention of the reader to Gieseker's construction in [79]. Gieseker

c p E detE O

proved that for there exists a -stable rank two vector bundle with X

c E c

and . Note that the bound is purely topological and does not depend on the polarization. As

detE O Corollary 5.3.4 shows, such a bound cannot be expected for X . Ð Proposition 5.1.5 is due to Ballico [12]. The statement about the existence and regularity of the

bundle E in Theorem 5.1.6 was proved by W.-P. Li and Z. Qin [153]. The details of the proof can be found there. The assertion on the image under the Albanese map is a modi®cation of Ballico's argument. Ð Other existence results for higher rank are due to Sorger [239]. Ð Elementary transformations were intensively studied by Maruyama ([163, 167]). Proposition 5.2.4 and Theorem 5.2.5 are due to him. Ð The notion of a suitable polarization was ®rst introduced by Friedman in [67] for elliptic sur-

faces. He also proved Theorem 5.3.2. It seems Brosius observed Corollary 5.3.4 the ®rst time, though

c f

Takemoto in [242] already found that for there exists no rank two vector bundle which

is -stable with respect to every polarization. Suitable polarizations for higher rank vector bundles where considered by O'Grady [209]. He only discusses the case of an elliptic K3 surface, whose Pi- card group is spanned by the ®bre class and the class of a section, but his arguments easily generalize.

Ð With the techniques of Example 5.3.7 one can attack a generic complete intersection of a quadric

M O

and a cubic hypersurface in P . The moduli space is a reduced point. The locally free

O Z Hilb X

part of M is isomorphic to the open subset of , such that the line through

X Z

Z meets exactly in . This isomorphism was described in [189]. The birational correspondence

O Hilb X X between M and re¯ects the projective geometry of . Ð Example 5.3.8 is entirely due to Friedman [68]. He treats it in the more general setting of ellip-

tic surfaces which are not necessarily K3 surfaces. He also gives a complete description of the four-

Hilb X dimensional moduli space M . It turns out that it is isomorphic to , though the iden- 140 5 Construction Methods

ti®cation is fairly involved. The distinguished bundle G was also discovered by Qin [215] and in a broader context by Kametani and Sato [118]. We took the description as the unique extension from

there. Friedman's point of view is that G is the unique bundle which restricts to a stable bundle on any ®bre, even singular ones. For this purpose he generalizes results of Atiyah for vector bundles on singular nodal elliptic curves. 141

6 Moduli Spaces on K3 Surfaces

By de®nition, K3 surfaces are surfaces with vanishing ®rst Betti number and trivial canoni-

cal bundle. Examples of K3 surfaces are provided by smooth complete intersections of type

a n a a P

i n n in with , Kummer surfaces and certain elliptic surfaces. In the Enriques classi®cation K3 surfaces occupy, together with abelian, Enriques and hyperelliptic surfaces, the distinguished position between ruled surfaces and surfaces of positive Kodaira dimension. The geometry of K3 surfaces and of their moduli space is one of the most fascinating topics in surface theory, bringing together complex algebraic geometry, differential geometry and arithmetic. Following the general philosophy that moduli spaces of sheaves re¯ect the geometric structure of the surface it does not come as a surprise that studying moduli spaces of sheaves on K3 surfaces one encounters intriguing geometric structures. We will try to illuminate some aspects of the rich geometry of the situation. We present the material at this early stage in the hope that having explicit examples with a rich geometry in mind will make the more abstract and general results, where the geometry has not yet fully unfolded, easier accessible. At some points we make use of results presented later (Chapter 9, 10). In particular, a fundamental result in the theory, namely the existence of a symplectic structure on the moduli space of stable sheaves, will be discussed only in Chapter 10. Section 6.1 gives an almost complete account of results due to Mukai describing zero- and two-dimensional moduli spaces. The result on the existence of a symplectic structure is in this section only used once (proof of 6.1.14) and there in the rather weak version that the

canonical bundle of the moduli space of stable sheaves is trivial. In Section 6.2 we concentrate on moduli spaces of dimension . We prove that they provide examples of higher dimensional irreducible symplectic (or hyperkÈahler) manifolds. The presentation is based on the work of Beauville, Mukai and O'Grady. Some of the arguments are only sketched.

Finally, the appendix contains a geometric proof of the irreducibility of the Quot-scheme

E E Quot of zero-dimensional quotients of a locally free sheaf .

6.1 Low-Dimensional ...

We begin this section with some technical remarks and the de®nition of the Mukai vector. F

De®nition 6.1.1 Ð If E and are coherent sheaves then the Euler characteristic of the

E F pair is

142 6 Moduli Spaces on K3 Surfaces

E F dim Ext E F

E F E F is bilinear in and and can be expressed in terms of their Chern characters.

But before we can give the formula one more notation needs to be introduced.

ev i i

v v H X Z H X Z v v i

De®nition 6.1.2 Ð If i then .

The de®nition makes also perfect sense in the cohomology with rational or complex coef-

E chE

®cients and in the Chow group. The notation is motivated by the fact that ch

for any locally free sheaf E .

Lemma 6.1.3 Ð If X is smooth and projective, then

E F ch E chF tdX

Proof. If E is locally free this follows directly from the Hirzebruch-Riemann-Roch for-

mula and the multiplicativity of the Chern character:

chE F tdX E F E F

chE chF tdX

ch E chF tdX

E E chE

If E is not locally free we consider a locally free resolution and use

ch E

. 2

E X

De®nition 6.1.4 Ð Let X be a smooth variety and let be a coherent sheaf on . Then

E H X Q E chE tdX

the Mukai vector v of is .

td X tdX

Note that and hence the square root can be de®ned by a power series expansion.

De®nition and Corollary 6.1.5 Ð If X is smooth and projective, then

v w v w

X Q E F

de®nes a bilinear form on H . For any two coherent sheaves and one has

E F v E v F 2

X E X rkE r c E c

Let now be a K3 surface. If is a coherent sheaf on with , ,

c E c v E r c c c r r c c

and , then . Clearly, we can recover , , and

from v .

M r c c M v

Instead of we will use the notation for the moduli space of semistable

v r c c c r v M M v

sheaves, where . If is ®xed we will also write for . We

s denote the open subset parametrizing stable sheaves by M .

6.1 Low-Dimensional ... 143

M r c r c r v v

By 4.5.6 the expected dimension of is , which

is always even, since the intersection form on X is even. The obstruction space (cf. Section

HomE E Ext E E Ext E E

4.5) vanishes, since by Serre duality for

s s

M M

any E . Hence by 4.5.4 the moduli space is smooth. For the following we assume

r .

There are general results, mostly due to Mukai, which give a fairly complete description

M of moduli spaces of low dimensions, i.e. dimension . As is even-dimensional, we

are interested in zero- and two-dimensional examples.

v v M M

Theorem 6.1.6 Ð Suppose . If is not empty, then consists of a single

reduced point which represents a stable locally free sheaf. In particular, M .

F M

Proof. Let F be a semistable sheaf de®ning a point . By 6.1.5 the Euler charac-

F E F F F

teristic depends only on the Chern classes of and not on itself. Since and

F E E E v v

E have the same Chern classes, one has . This implies

F E HomE F

that either Hom or, by using Serre duality, that . The stability

E F E F

of and the semistability of imply in both cases that (see 1.2.7).

G E

It remains to show that E is locally free. For this purpose let be the re¯exive hull

S E G

of E and the quotient of the natural embedding . If the rank is two, one can argue

G c G

as follows. is still -semistable and hence satis®es the Bogomolov Inequality

E S G c E S c c G c G c

. On the other hand, .

S E S x X k x

Hence , i.e. if is not locally free, then where is a pointin . Denote

PG X

by E the ¯at family on de®ned by

E q G O p O

X X PG X

where is the diagonal and is the projection (for details see

SuppE E t t x E

t t

8.1.7). Then t and for some the sheaf is isomorphic

E t PG E E

to . Hence for the generic the sheaf t is stable but not isomorphic to . Since

the moduli space is zero-dimensional, this cannot happen. In fact a similar argument works

G S

in the higher rank case: Here one exploits the fact that Quot is irreducible. This

G S G S

is proved in the appendix (Theorem 6.A.1). Thus can be deformed to t with

SuppS Supp S 2

t .

Remark 6.1.7 Ð Note that the moduli space M might be non-empty even if the expected

v v M O O

dimension of the stable part is negative. Indeed, is a semistable sheaf

v O O v O O with .

We now turn to moduli spaces of dimension two. In general there is no reason to expect

M M

that M or that is irreducible. But as above, whenever there exists a `good' com- s

ponent of M , then both properties hold:

144 6 Moduli Spaces on K3 Surfaces

v v M

Theorem 6.1.8 Ð Assume . If has a complete irreducible component

M M M M M

, then , i.e. is irreducible and all sheaves are stable. In particular, if

M M M , then is smooth and irreducible. Proof. The idea of the proof is a globalization of the proof of Theorem 6.1.6. The Hirze-

bruch-Riemann-Roch formula is replaced by Grothendieck's relative version.

E M X rkE r

Let us ®x a quasi-universal family over and denote the multiplicity

F M

by s (cf. 4.6.2). Let be an arbitrary point in the moduli space represented by a

F t M

semistable sheaf . For any we have

if F E

HomF E

s s

E k if F

and also

E if F

Ext F E HomE F

t t

s s

E k if F

s F E E E

t t

Since t we also have

E if F

Ext F E

s s

E k if F

F M Ext F E t M i

Thus if , then for all and . Therefore

i i

E xt q F E F M E xt q F E i

we have . If , then for and

p p

E xt q F E t k t t F

, where : this is an application of the Base Change

Theorem. In our situation we can make it quite explicit. By [19] there exists a complex P

i i

i H P

of locally free sheaves P of ®nite rank such that the -th cohomology is isomorphic

i i i

E xt q F E H P t P Ext F E

to and t . This complex is bounded above, i.e.

for i . An argument of Mumford shows [193] that one can also assume that for

i Ext F E i P P i

. Since t for , the complex is exact at for . The kernel

kerd i P

i of the -th differential is the kernel of a surjection of a locally free sheaf to a tor-

imd kerd M

sion free sheaf i . Hence is locally free, since is a smooth surface. Replac-

d d

P kerd P P kerd P P

ing by we can assume that . We have seen that

t P

is concentrated in . At the same time, as a subsheaf of , it is torsion free, hence zero,

d kerd P

i.e. is injective. Also is locally free, contains the locally free sheaf and actu-

t P kerd H P

ally equals it on the complement of the point . Hence , i.e.

imd P H P q F E E xt

. For the last statement use which shows

H P t P t imd t H P t k t Ext F E

that t .

On the other hand, the Grothendieck-Riemann-Roch formula computes

a chE xt q F E E xt q F E E xt q F E

p p p

H M Q chq F chE

as an element of and shows that it only depends on and as ele-

H M X Q chF F M chF

ments of . Since is constant for all , in particular

s chE F M a F M

t even for , one gets a contradiction by comparing if

E xt q F E ha tdM M i F M 2

and if . p

6.1 Low-Dimensional ... 145

Remark 6.1.9 Ð The assumption M is satis®ed frequently, e.g. if degree and rank

are coprime any -semistable sheaf is -stable (see 1.2.14).

F M

Note that under the assumption of the theorem any -stable is locally free. In-

M G F

deed, if F is -stable, then is still -stable and thus de®nes a point in

M r c c F .If were not locally free, i.e. , the latter space would have negative dimension.

The following lemma will be needed in the proof of Proposition 6.1.13.

v v M M

Lemma 6.1.10 Ð Suppose that and . Moreover, assume that there is

M X

a universal family E on . Then

if i

p E p E E xt

O i

p M M X

where ij is the projection from to the product of the indicated factors, and

M M

is the diagonal.

t M E E xt q E E

Proof. Step 1. Let be a closed point representing a sheaf . Then

i E xt q E E k t

for and . The ®rst statement was obtained in the proof of p

Theorem 6.1.8 together with a weaker form of the second statement, namely that the rank

q E E E xt t S

of at is 1. It suf®ces therefore to show that for any tangent vector

Sp eck t k t E q E E O E xt O E E E xt

S S S

at one has S ). Here is

p p

E S X E E

the restriction of to . This family ®ts into a short exact sequence S

E E Ext Hom E E

de®ning a class . Applying the functor S to this

i i

E E Ext E E

sequence and using Ext we get

S X X

Ext E E Ext E E Ext E E Ext E E

X X S X X

Ext E E k

Since and since the cup product is non-degenerate by Serre duality, the

E E E E Ext

map Ext is surjective. Hence the restriction homomorphism

X X

E E E E Ext Ext

S is an isomorphism.

X S X

Step 2. It follows from this and the spectral sequence

i j

H M E xt q E E q E E Ext

M X

that

k n

n if

Ext q E E

M X

else

p p M M X

Consider the Leray spectral sequence for the composition

M M

M :

ij j i

p E p E p E p E E xt E xt R

146 6 Moduli Spaces on K3 Surfaces

p E p E xt

As E is (set-theoretically) supported along the diagonal, the spectral se-

j j

p E p E E xt p E p E E xt

quence reduces to an isomorphism . It fol-

lows from the base change theorem and Step 1, that

p E p E t Ext E O E t M E xt

t X

M X

p E p E p E p E E xt E xt

for all j . This implies that and hence are line bun-

M M

dles on M and , respectively. It remains to show that this line bundle is triv-

E xt E E p E p E j E xt E xt

ial. But as base change holds for we have: ,

p p p

tr E xt E E H X O O

X X

and the trace map E (cf. 10.1.3) gives the desired p

isomorphism. 2

After having shown that in many cases the moduli space M is a smooth irreducible pro-

jective surface we go on and identify these surfaces in terms of their weight-two Hodge

n H H R

structures. Recall that a Hodge structure of weight consists of a lattice Z in a real

H H H C

R R

vector space and a direct sum decomposition C such that

pq n

q p

H H Q Q H H R

. A -Hodge structure is a -vector space Q in a real vector space of

H H C

Q Q

the same dimension and a decomposition of C as before. n

Let Y be a compact KÈahler manifold. Then there is a naturally de®ned weight Hodge

pq n n

H Y Y Z H Y C

structure on H Torsion, which is given by . In par-

pq n

H Y Z ticular, Y admits a natural weight-two Hodge structure on the second cohomology

de®ned by

H Y C H Y C H Y C H Y C

H Y Z

If Y is a surface the intersection product de®nes a natural pairing on . The Global

Y Y Torelli Theorem for K3 surfaces states that two K3 surfaces and are isomorphic if

and only if there exists an isomorphism between their Hodge structures respecting the pair-

H Y Z H Y C H Y Z

ing, i.e. there exists an isomorphism which maps to

H Y C

and which is compatible with the pairing. For details see [22], [26]. For sur-

i ev

H Y Z H Y Z Y Z

faces one can also de®ne a Hodge structure on H as i

follows.

H Y Z H Y Q

De®nition 6.1.11 Ð Let Y be a surface. (or ) is the natural weight-two

Y Z H Y C H Y C H Y C H Y C

Hodge structure on H given by , ,

Y C H Y C H Y C H Y C

and H .

H Y Z

Let be endowed with the pairing de®ned in 6.1.5. The restriction of

Y Z Y Z H H Y Z to H equals the intersection product. The inclusion is com-

patible with the Hodge structure.

H Y Z

The Mukai vector v can be considered as an element of of type . The ex-

s v

pected dimension of M is two if and only if is an isotropic vector.

s M

Assume that v is an isotropic vector such that has a complete component. By 6.1.8

M E M X

the last condition is equivalent to M . Let be a quasi-universal family over

r of rank s .

6.1 Low-Dimensional ... 147

H X Q H M Q f H M Q H X Q

De®nition 6.1.12 Ð Let f and

f c p q c f c q p c

be the homomorphisms given by and , where

E s v E s

v and .

E f f

If we want to emphasize the dependence on we write E and . For any locally free

M E p W

sheaf W on the family is also quasi-universal. The corresponding homomor-

f c f cch W rkW

p W

phisms are related as follows: E .

M M

Proposition 6.1.13 Ð Let v be an isotropic vector and assume . Assume there

M X exists a universal family E over . Then:

i) M is a K3 surface.

f f

ii) E .

H M Z H X Z f

iii) E de®nes an isomorphism of Hodge structures which is compatible with the natural pairings.

Proof. Consider the diagram

M M M

p k

p p

M M X M X M

p q

M X X

M M

Then, by the projection formula

p p c f f c p q q p c p p

p c p p p p c p p p p p

p p c p p p p c p

p p

tdM tdM p tdX E p E chp ch p c

It follows from Lemma 6.1.10 and the Grothendieck-Riemann-Roch formula for that

E chi O E p p tdX chE xt E p E chp ch p p

148 6 Moduli Spaces on K3 Surfaces

p p

chi O f f c tdM tdM c

Hence, . Now the Grothendieck-

i chi O tdM M i chO td

Riemann-Roch formula applied to says .

f f c i chO c c f f

Hence, . Therefore, the homomorphisms and are

odd ev en

H H

injective and surjective, respectively. Moreover, f preserves and , because is

M Q an even class. Hence H . By 10.4.3 the moduli space admits a non-degenerate

two-form and, therefore, the canonical bundle of M is trivial (This is the only place where

we need a result of the later chapters). Using the Enriques-classi®cation of algebraic sur-

M b M

faces one concludes that is abelian or K3. Since , it must be a K3 surface. In

H M Q dim H X Q f f

particular, dim . Hence, and are isomorphisms.

H f f f

The isomorphisms f and do respect the Hodge structure . Indeed, and are de-

p p

®ned by the algebraic classes and , which are sums of classes of type . It is straight-

forward to check that this is enough to ensure that f and respect the Hodge type of an

H element c . (Note that the compatibility with the Hodge structure is valid also for the case of a quasi-universal family.)

The compatibility with the pairing is shown by:

a f c ha f c M i ha p q c M i

hp p a q c M i hp a q c M X i

hp a q c M X i hp a q c M X i

hf a c X i f a c

f b f a f b a f f b a b f If c we conclude , i.e. is compatible with

the pairing.

f tdX

To conclude, we have to show that the isomorphisms f and are integral. Since

M chE

and td are integral, it is enough to show that is integral. This goes as fol-

c E ch E H X Z

lows. The ®rst Chern class is certainly integral. Since ,

p c E j q c X M

fxg

it equals M . Since and are K3 surfaces, the intersection form

e ch E c E c E chE

is even. Hence is integral. Writing with

pq p q

H M Q H X Q e e e e e

e this says that the classes , , , , and are

P P

pq pq

E p tdM e e p PD pt PD pt

all integral. Moreover, ch , where de-

q q

e e q chE p tdM

notes the PoincarÂedual of a point. Hence . On

chq E

the other hand, is integral and, by the Grothendieck-Riemann-Rochformula, equals

q chE p tdM e e ch E e

. Hence and are also integral. In particular, is

p chE q tdX e

integral. The same argument applied to shows is integral. Hence,

ch E e e chE 2

is integral. Altogether this proves that is integral.

V v v X Z

The orthogonal complement of in H contains . If we in addition assume that

V Zv Z

v is primitive, i.e. not divisible by any integer , then the quotient is a free -

V Zv

module. Since v is of pure type and isotropic, the quotient inherits the bilinear

X Z form and the Hodge structure of H .

6.1 Low-Dimensional ... 149

s M

Theorem 6.1.14 Ð If v is isotropic and primitive and has a complete component (i.e.

M M f

), then E de®nes an isomorphism of Hodge structures

H M Z V Z v

compatible with the natural pairing and independent of the quasi-universal family E .

f V Q H X Q H M Q H M

Proof. We ®rst check that E has no -

M H X v v c v c

component. Indeed, the H component of is and and

H M f c v c c V

hence the component of E is , which vanishes for . Since

f c f cch W rkW

E p W E

H f c c V E

the -component of E for is independent of . Thus we obtain a well-de®ned

V H M Q

(i.e. independent of the quasi-universal family) map f . The following

f v H M

computation shows that E has trivial -component:

s f v p q v

p p p

tdM p tdX q tdX ch E q chE

tdM chE xt E q E

tdM chk t s

s tdM PD pt

s PD pt

t M f V Zv H M Q

where . Hence de®nes a homomorphism . If a universal fam-

ily exists then this map takes values in the integral cohomology of M (Proposition 6.1.13).

V Zv H M Z Hence . The general case is proved by deformation theory. The basic idea is to use the moduli space of polarized K3 surfaces and the relative moduli space of

semistable sheaves. It is then not dif®cult to see that the moduli space M is a deformation

of a ®ne moduli space on another nearby K3 surface. (For the complete argument see the proof of 6.2.5.) Since the map f is de®ned by means of the locally constant class , it is

enough to prove the assertion for one ®bre. 2

M M

Corollary 6.1.15 Ð Suppose that v is an isotropic vector and that . Then there

H X Q H M Q exists an isomorphism of rational Hodge structures which is compatible with the intersection pairing.

Proof. This follows from the theorem and the easy observation that

H X Q H X Q

w w c w r

H X Q Q V Zv Q induces an isomorphism of -Hodge structures compatible with

the pairing. The assumption that v be primitive is unnecessary, because we are only inter- 2 ested in Q -Hodge structures. 150 6 Moduli Spaces on K3 Surfaces

6.2 ... and Higher-Dimensional Moduli Spaces

The aim of this section is to show that moduli spaces of sheaves on K3 surfaces have a very special geometric structure. They are Ricci ¯at and even hyperkÈahler. In fact, almost all known examples of hyperkÈahler manifolds are closely related to them. Thus the study of these moduli spaces sheds some light on the geometry of hyperkÈahler manifolds in general.

Let us begin with the de®nition of hyperkÈahler and irreducible symplectic manifolds.

M g

De®nition 6.2.1 Ð A hyperkÈahler manifold is a Riemannian manifold which admits

J I J J I g

two complex structures I and such that and such that is a KÈahler

J X

metric with respect to I and . A complex manifold is called irreducible symplectic if

X H X H X

is compact KÈahler, simply connected and is spanned by an X

everywhere non-degenerate two-form w .

T X

Recall, that a two-form is non-degenerate if the associated homomorphism X

M g I J

is isomorphic. If is hyperkÈahler and and are the two complex structures then

I J g g

K is also a complex structure making to a KÈahler metric. If is a hyperkÈahler

M g Spm dim M m M g

metric then the holonomyof is contained in where R . is

Sp m

J K

called irreducible hyperkÈahler if the holonomy equals .If I , , and denote the

w i K

corresponding KÈahler forms, then the linear combination J de®nes an element

X X M I w

in H , where is the complex manifold . Obviously, is everywhere non- X

degenerate.

M g X

Theorem 6.2.2 Ð If is an irreducible compact hyperkÈahler manifold, then

M I is irreducible symplectic. Conversely, if X is an irreducible symplectic manifold,

then the underlying real manifold M admits a hyperkÈahler metric with prescribed KÈahler

class I .

Proof. [25] 2

Even if one is primarily interested in hyperkÈahler metrics, this theorem allows one to work in the realm of complex geometry. In the sequel some examples of irreducible symplectic manifolds will be described, but the hyperkÈahler metric remains unknown, for Theorem

6.2.2 is a pure existence result based on Yau's solution of the Calabi conjecture.

O X w K X

X Remark 6.2.3 ÐIf admits a non-degenerate two-form , then X .If is com-

pact, this implies that the Kodaira dimension of X is zero. In dimension two, according to the Enriques-Kodaira classi®cation, a surface is irreducible symplectic if and only if it is a K3 surface. On the other hand, due to a result of Siu, any K3 surface admits a KÈahler metric,

hence is irreducible symplectic.

Hilb X Theorem 6.2.4 Ð Let X be an algebraic K3 surface. Then is irreducible symplectic.

6.2 ... and Higher-Dimensional Moduli Spaces 151

Hilb X M Proof. M is smooth, projective and irreducible (cf. 4.5.10). Moreover,

can be identi®ed with the moduli space of stable rank one sheaves with second Chern num- w

ber n and therefore admits an everywhere non-degenerate two-form (cf. 10.4.3). In order M

to prove that M is irreducible symplectic it therefore suf®ces to show that is simply con-

H M

nected and that dim .

X X X

For the second statement consider the complement U of the

fx x jx x for some i j g

n i j

`big diagonal' . There is a natural mor-

U M x x U Z fx x g M

n n

phism mapping to . This

U S

morphism identi®es the quotient of by the action of the symmetric group n with an

H U H V M H M

open subset V of . Then , where the lat-

U V M

H X S U H U n

ter is the space of n -invariant two-forms on . But and

X U

n S S

n n

H X H U co dim H X

n , since . Since , we have

X U X

n n S

H X H X H X

n . Together with the isomorphism

X X X

H M C H M H X C

this yields . A similar argument shows

M M X

b M M fg H X

, which immediately gives . Inorderto show we

n n

X U X fg

argue as follows. The real codimension of in is , so that

M n V M j V

is an isomorphism. And since has real codimension in , the map

M pr U V S V

is surjective. The projection induces an isomorphism n

x x X x x

n n

which is described as follows: Choose distinct points and take

fx x g U V S

n n

and as base points in and , respectively. For each choose a path

U x x x x

in connecting and . Then is a closed path in

V fx x g M fg

with base point . In order to prove that it suf®ces to show that

j M S n

is null-homotopic in . Since is generated by transpositions, it suf®ces to con-

x x

sider the special case . We may assume that and are contained some open

W X W x x X n W B C

set (in the classical topology) such that and .

x x C B

Then a path can be described by rotating and in a complex line around a

x x x x fx x g

point . Now let and collide within this complex line to , i.e. converges

C B Z X Supp Z x m m T M

Z x

to with and . Then is in freely

Z fx x g j 2

n homotopic to the constant path . Hence .

For the higher rank case we again use the Mukai vector

v v v v r c c c r

M r c c M v dim M v v v

and denote the moduli space H by . Recall, ,

M v v

where is the moduli space of stable sheaves. The component of the Mukai vector

H X Z

v is called primitive if it is indivisible as a cohomology class in . Recall that from

v c

one recovers the ®rst Chern class and the discriminant . We therefore have a chamber

structure on the ample cone with respect to v (see 4.C).

v H

Theorem 6.2.5 Ð If is primitive and is contained in an open chamber with respect

v M v

to , then H is an irreducible symplectic manifold. 152 6 Moduli Spaces on K3 Surfaces

The proof of the theorem consists of two steps. We ®rst prove it for a particular example

(Corollary 6.2.7). The general result is obtained by a deformation argument.

X P

We take up the setting of Example 5.3.8. Let be an elliptic K3 surface with

H X Z X

®bre class f and a section the class of which is also denoted by

PicX Z O Z O f v X Z

H . Assume . Let be a Mukai vector such that

v f H v mf

and let be suitable with respect to (cf. 5.3.1). Note that is ample

m r

for m and suitable if (cf. 5.3.6).

M v

Proposition 6.2.6 Ð Under the above assumptions the moduli space H is birational

X to Hilb .

Proof. The proof is postponed until Chapter 11, (Theorem 11.3.2), where the proposition

is treated as an example for the birational description of a moduli space. 2

M v

Corollary 6.2.7 Ð Under the assumptions of the proposition H is irreducible sym-

plectic.

X X

Proof. We consider the following general situation. Let f be a birational map

between an irreducible symplectic manifold X and a compact manifold admitting a non-

U X f

degenerate two-form w . Let be the maximal open subset of -regular points, i.e.

f j co dimX n U C w H X H U

U is a morphism. Then . Hence .

X U

f H X C w H X H U

Moreover, is injective and thus .

X X U

f w j w j C w U

We can write U for some . Since is non-degenerate every-

f j f

where, U is an embedding. The same arguments apply for the inverse birational map

X X U X f j

. One concludes that there exists an open set such that U is regular,

U co dimX n U f U X U

, and . Moreover, this also implies

X U X fg

. Hence is irreducible symplectic as well. Now, apply the argu-

Hilb X M v 2

ment to the birational correspondence between and H postulated in 6.2.6.

The proof of Theorem 6.2.5 relies on the fact that any K3 surface can be deformed to an

elliptic K3 surface. To make this rigorous one introduces the following functor:

K S chC S ets

De®nition 6.2.8 Ð Let d be a positive integer. Then is the functor

f X Y L

that maps a scheme Y to the set of all equivalence classes of pairs such

f X Y t Y L

that is a smooth family of K3 surfaces and for any the restriction t

L d L X f t c t

of to the ®bre t is an ample primitive line bundle with . Two

f X Y L f X Y L Y

pairs , are equivalent if there exists an -isomorphism

L f N g X X N Y g L and a line bundle on such that .

This is a very special case of the moduli functor of polarized varieties. The next theorem

is an application of a more general result.

K K

Theorem 6.2.9 Ð The functor is corepresented by a coarse moduli space d which is d a quasi-projective scheme. 6.2 ... and Higher-Dimensional Moduli Spaces 153

Proof. [258] 2 K

Similar to moduli spaces of sheaves, the moduli space d is not ®ne, i.e. there is no uni-

K K PGLN d

versal family parametrized by d . But, as for moduli space of sheaves, is a -

H K H Hilb P P n

d d

quotient d of an open subset of a certain Hilbert scheme ,

n n d

where P . The universal family over the Hilbert scheme provides a smooth

X H L X t K d

projective morphism d and a line bundle on , such that corre-

X L t

sponds to the polarized K3 surface t . K

An alternative construction of d can be given by using the Torelli Theorem for K3 sur-

faces. This approach immediately yields K

Theorem 6.2.10 Ð The moduli space d of primitively polarized K3 surfaces is an irreducible variety.

Proof. [26] 2

H K d

Using an irreducible component of d , which dominates , the theorem shows that any

X H X H H H

two primitively polarized K3 surfaces and with are deforma-

H K d tion equivalent. (In fact, d itself is irreducible.) More is known about the structure of

and the polarized K3 surfaces parametrized by it. We will need the following results: For

X H K PicX Z H

the general polarized K3 surface d one has . `General' here

X H K

means for in the complement of a countable union of closed subsets of d . In fact

X H K X K d the countable union of polarized K3 surfaces d with is dense in .

For the proof of these facts we refer to [26, 22]. K

It is the irreducibility of d which enables us to compare moduli spaces on different K3

surfaces:

v v Z

Proposition 6.2.11 Ð Let and . Then there exists a relative moduli

M H

space d of semistable sheaves on the ®bres of such that: i) is projective,

v t H t M L

ii) for any d the ®bre is canonically isomorphic to the moduli space

X v v c L v

t of semistable sheaves on t , where , and iii) is smooth at all points corresponding to stable sheaves.

(Don't get confused by the extra sign . It is thrown in for purely technical reasons which will be become clear later.) Proof. i) and ii) follow from the general existence theorem for moduli spaces 4.3.7. As-

sertion iii) follows from the relative smoothness criterion 2.2.7: By Serre duality, we have

E E Hom E E Ext E X

for any stable sheaf on the ®bre t . By Theorem

t X

M H R

4.3.7 the relative moduli space d is a relative quotient of an open subset of an

P m

Quot R M L P

appropriate Quot-scheme . Since is a ®bre bundleover

X H

M H E M

the stable sheaves, the morphism d is smooth at a point if and only if

P m

R H q L K q E R t

d is smooth at over it. Let be the kernel of . Since

Ext K E Ext E E T R T H

q t d , the tangent map is surjective by 2.2.7,

154 6 Moduli Spaces on K3 Surfaces

M H

2.A.8 and 4.5.4. Hence d is smooth at all points corresponding to stable sheaves

on the ®bres. 2

X H X H H

Corollary 6.2.12 Ð Let and be two polarized K3 surfaces with

v v H v v v H v E M v H

and let , . Assume that every sheaf in

v v M M v M

H H and in H is stable. Then is irreducible symplectic if and only if is

irreducible symplectic.

H H

Proof. Let d denote the dense open subset of regular values of . Then there

X H L t t H X L X H X

t t t

exist points such that t and . Hence

the restriction of to is a smooth projective family over a connected base with the two

v M v M t t H moduli spaces H and occurring as ®bres over and , respectively. As for any smooth proper morphism over a connected base the fundamental groups and Betti numbers

of all ®bres of are equal. On the other hand, the Hodge numbers of the ®bres are upper- semicontinuous. Since the Hodge spectral sequence degenerates on any ®bre and hence the sum of the Hodge numbers equals the sum of the Betti numbers, the Hodge numbers of the

®bres of stay also constant.

M v M v H

Therefore, if H is irreducible symplectic, then is simply connected and

v v h M M H

H . Since by 10.4.3 the moduli space admits a non-degenerate

v M 2

symplectic structure, H is irreducible as well.

X X

Proof of Theorem 6.2.5. Step 1. We ®rst reduce to the case that . If ,

v H H PicX

then , where is the ample generator of . As we have mentioned above,

X H K X

the set of polarized K3 surfaces d with is a countable union of

K X H

closed subsets which is dense in d . On the other hand, the set of K3 surfaces

v v M M K

H H d such that is not smooth is a proper closed subset. Indeed, is smooth at

stable points and the set of properly semistable sheaves is closed in M and does not dom-

H X H K X H d

inate d . Thus we can ®nd such that and is generic with

v M v

respect to . By 6.2.12 the moduli space H is irreducible symplectic if and only if

M v

H is irreducible symplectic.

Step 2. We may assume . Let us show that one can further reduce to the case

r H v

that H . By assumption is contained in an open chamber with respect to .

Hence there exists a polarization H in the same chamber which is not linearly equivalent

v mH v v M v m v mH M M

H H

to . Then we get H for any . Here is the

v E O mH E M

Mukai vector of , where H , and the second isomorphism is given

E E O mH m m m

by mapping to . For any there exists an integer such that

v mH v r mH H

is ample, contained in the chamber of , and primitive. Hence

v mH M v M v r mH

v r mH

H . Clearly, can be made arbitrarily large for

X H H r X

Step 3. Assume now that is a polarized K3 surface with . Let be

X Z f Z H

an elliptic K3 surface with Pic as in Proposition 6.2.6. Then

H f v v H v is ample and suitable with respect to by 5.3.6. Moreover,

6.2 ... and Higher-Dimensional Moduli Spaces 155

H H d X H X H K M v H

. Thus d . Hence is irreducible symplectic if

v M 2

and only if H is irreducible symplectic which was the content of Proposition 6.2.6.

In Section 6.1 we ®rst established that any two-dimensional moduli space is a K3 surface, i.e. irreducible symplectic, and then determined its Hodge structure. Here we proceed along the same line. After having achieved the ®rst half we now go on and study the Hodge structure of the moduli space.

A weight-two Hodge structure on any compact KÈahler manifold is given by the Hodge

X C H X H X H X decomposition H . For an irreducible symplectic

manifold the full information about the decomposition is encoded in the inclusion of the

X H X C PH X C

one-dimensional space H , i.e. a point in . This point

is called the period point of X . Next we introduce an auxiliary quadratic form, by means

of which one can recover the whole weight-two Hodge structure from the period point in

H X C

P .

De®nition and Theorem 6.2.13 Ð If X is irreducible symplectic of dimension , then

q b X H X Z

there exists a canonical integral form of index on given by

w w q aw bw ab n

w w H X C w H X

where , with and is a positive scalar. X

Moreover,

n n n

n X

Proof. [25], [73] 2

Note that for K3 surfaces this is just the intersection pairing. For the higher dimensional examples constructed above one can identify the weight-two Hodge structure endowed with this pairing. We begin with the Hilbert scheme. The higher

rank case is based on this computation.

n Theorem 6.2.14 Ð Let X be a K3 surface and . Then there exists an isomorphism

of weight-two Hodge structures compatible with the canonical integral forms

H Hilb X Z H X Z Z

where on the right hand side is a class of type and the integral form is the direct

sum of the intersection pairing on X and the integral form given by . The

constant in 6.2.13 is .

Proof. [25] 2

For the higher rank case, we recall and slightly modify the de®nition of the map

156 6 Moduli Spaces on K3 Surfaces

f H X Q H M v Q

E M v X

introduced in the proof of 6.1.14. If is a quasi-universal family over H of

m r c H X Z f c p fq cg m

rank and , then , where this time

p p

E q tdX tdM

ch . This differs from the original de®nition by the factor . As be-

v H X Q

fore, denote by V the orthogonal complement of endowed with the quadratic

dimM v

form and the induced Hodge structure. Note that under our assumption H

v v V f M v

the vector is no longer isotropic, i.e. . Apriori, need not be integral even if H admits a universal family.

Theorem 6.2.15 Ð Under the assumptions of 6.2.5 the homomorphism f de®nes an iso-

H M v Z V

morphism of integral Hodge structures H compatible with the quadratic forms.

Proof. [209] 2

This theorem due to O'Grady nicely generalizes Beauville's result for the Hilbert scheme

and Mukai's computations in the two-dimensional case. Indeed, if v , then

M v Hilb X V fa b an ja Z b H X Zg

H and . And with its

H X Z Z

induced Hodge structure V is isomorphic to the direct sum of and where for

Z q a a an a an a n a one has .

We conclude this section by stating a result which indicates that although the moduli

M v

spaces H provide examples of higher dimensional compact hyperkÈahler manifolds,

they do not furnish completely new examples.

v H

Theorem 6.2.16 Ð Let be primitive and contained in an open chamber with respect

v M v Hilb X n v v to . Then H and are deformation equivalent, where

Proof. [113] 2 6.A The Irreducibility of the Quot-scheme 157 Appendix to Chapter 6

6.A The Irreducibility of the Quot-scheme

In the appendix we prove that the Quot-scheme Quot of zero-dimensional quotients E of length of a ®xed locally free sheaf is irreducible. This result was used at several oc-

casions in this chapter but it is also interesting for its own sake.

x k x

In the following the socle of a zero-dimensional sheaf T at a point is the -vector

t T O X x space of all elements x which are annihilated by the maximal ideal of . This usage

of the word 'socle' differs from that in Section 1.5. Exercise: In what sense are they related?

E QuotE Hilb X If rk , then is isomorphic to the Hilbert scheme . The latter

was shown to be smooth and irreducible (4.5.10).

Theorem 6.A.1 Ð Let X be a smooth surface, a locally free sheaf and an integer.

E rkE

Then Quot is an irreducible variety of dimension .

QuotE PE Proof. The assertion is proved by induction over . If , then ,

which is clearly irreducible.

N O E T Y

Let Quot be the universal quotient family over

QuotE X s x Y N x s

. For any point and a nontrivial homomorphism

k we can form the push-out diagram

k x T T

x x

N E T

s s

s x hi s x PN Y Thus sending to de®nes a morphism . We want to use

the diagram

PN Y Y

s E T x Y is x homk x T

for an induction argument. For each s let

T x i

denote the dimension of the socle of s at . Then is an upper semi-continuous function

Y Y i i

on : let denote the stratum of points of socle dimension . It is not dif®cult to see that

hi y hi hi PN jiy iy j if y and for some , then . This shows

that

Y Y

i j

(6.1)

jij j

158 6 Moduli Spaces on K3 Surfaces

is x

s x PSo cleT s x PN x P

Check that , and s

dim N x

T P

. Moreover, using a minimal projective resolution of s over the local ring

O dim N x rkE is x s

X x , one shows: . Using this relation, the information about

co dimY Y i

the ®bre dimension, and the relation (6.1) one proves by induction that i

A B N N

for all i and . Let be a locally free resolution of . Then

B rkA rkE PN PB

rk , and is the vanishing locus of the homomorphism

A B O PB Y Y

P , denoting the projection. In particular, as

N rkA rkB rkE PB

is irreducible, and P is locally cut out by equations in ,

PN dim Y rkE

every irreducible component of has dimension . Now it is

Y Y

easy to see that is irreducible and has the expected dimension and that

is too small for all i to contribute other components. Hence is irreducible, and

PN Y QuotE Y

as the composition is surjective, is irreducible as

well. 2

Comments: Ð 6.1.6, 6.1.8, 6.1.14 are contained in Mukai's impressive article [188]. He also applies the results to show the algebraicity of certain cycles in the cohomology of the product of two K3 surfaces. Ð For a more detailed study of rigid bundles, i.e. zero-dimensional moduli spaces see Kuleshov's article [133]. Ð The relation between irreducible symplectic and hyperkÈahler manifolds was made explicit in Beauville's paper [25]. He also proved Theorem 6.2.4 for all K3 surfaces provided the Hilbert scheme is KÈahler. That this holds in general follows from a result of Varouchas [256]. Furthermore, Beauville described another series of examples of irreducible symplectic manifolds, so called generalized Kum- mer varieties, starting with a torus. Ð The main ingredient for 6.2.5, namely the existence of the symplectic structure is due to Mukai. His result will be discussed in detail in Section 10. The irreducibility and 1-connectedness was ®rst shown in the rank two case by GÈottsche and Huybrechts in [87] and for arbitrary rank by O'Grady in [209]. The proof we presented follows [209]. Ð The calculation of the Hodge structure of the Hilbert scheme (Theorem 6.2.14) is due to Beau- ville [25]. Note that the Hodge structures (without metric) of any weight of the Hilbert scheme of an arbitrary surface can be computed. This was done by GÈottsche and Soergel [86]. Ð The description in the higher rank case (Theorem 6.2.15) is due to O'Grady. The proof relies on the proof of 6.2.5, but a more careful description of the birational correspondence between moduli

space and Hilbert scheme on an elliptic surface is needed. Once the assertion is settled in this case, K the general case follows immediately by using the irreducibility of the moduli space d . This part is analogous to an argument of Mukai's. Ð In [87] GÈottsche and Huybrechts computed all the Hodge numbers of the moduli space of rank two sheaves. They coincide with the Hodge numbers of the Hilbert scheme of the same dimension. But this is not surprising after having established 6.2.16. Ð Theorem 6.2.16 was proved by Huybrechts [113]. The proof is based on the fact that any two birational symplectic manifolds are deformation equivalent. Note that in [113] the proof is given only for the rank two case, but the techniques can now be extended to cover the general case as well. Ð The results of [113] (and their generalizations) show that all known examples of irreducible symplectic manifolds, i.e. compact irreducible hyperkÈahler manifolds, are deformation equivalent either 6.A The Irreducibility of the Quot-scheme 159

to the Hilbert scheme of a K3 surface or to a generalized Kummer variety. In particular, all examples

have second Betti number either , , or .

E rkE

Ð The irreducibility of Quot (Theorem 6.A.1) was obtained by J. Li [148] for

and by Gieseker and Li [82] for rk . The proof sketched in the appendix is due to Ellingsrud

E S X

and Lehn [58]. They also show that the ®bres of the natural morphism Quot are

E irreducible. Note that in the case rk Theorem 6.A.1 reduces to the theorem of Fogarty that the Hilbert scheme of points on an irreducible smooth surface is again irreducible (cf. 4.5.10). 160

7 Restriction of Sheaves to Curves

In this chapter we take up a problem already discussed in Section 3.1. We try to understand

how -(semi)stable sheaves behave under restriction to hypersurfaces. At present, there are three quite different approaches to this question, and we will treat them in separate sections. None of these methods covers the results of the others completely.

The theorems of Mehta and Ramanathan 7.2.1 and 7.2.8 show that the restriction of a - stable or -semistable sheaf to a general hypersurface of suf®ciently high degree is again -

stable or -semistable, respectively. It has the disadvantage that it is not effective, i.e. there is no control of the degree of the hypersurface, which could, a priori, depend on the sheaf itself. However, such a bound, depending only on the rank of the sheaf and the degree of the variety, is provided by Flenner's Theorem 7.1.1. Since it is based on a careful exploitation of the Grauert-MÈulich Theorem in the re®ned form 3.1.5, it works only in characteristic zero

and for -semistable sheaves. In that respect, Bogomolov's Theorem 7.3.5 is the strongest,

though one has to restrict to the case of smooth surfaces. It says that the restriction of a -

stable vector bundle on a surface to any curve of suf®ciently high degree is again -stable, whereas the theorems mentioned before provide information for general hypersurfacesonly. Moreover, the bound in Bogomolov's theorem depends on the invariants of the bundle only. This result provides an important tool for the investigation of the geometry of moduli spaces in the following chapters.

7.1 Flenner's Theorem n

Let X be a normal projective variety of dimension over an algebraically closed ®eld of

X Z

characteristic zero and let O be a very ample line bundle on . Furthermore, let

X jO aj X D D

be the incidence variety of complete intersections

D jO aj q Z X X

with i . (For the notation compare Section 3.1.) Recall that is a

X PicZ q PicX

product of projective bundles over (cf. Section 2.1) and therefore

Pic Z p . The same holds true for any open subset of containing all points of codimen-

sion one.

Z Z X

y y

s jO aj

7.1 Flenner's Theorem 161

E Z p s

If is -semistable, then for a general complete intersection s one has (The-

orem 3.1.5):

g T j E j maxf E j E j

min Z i Z i Z Z

s s s s

Roughly, the proof of the Grauert-MÈulich Theorem was based on this inequality and the

a deg X T j

min Z

upper bound ZX . The Theorem of Flenner combines the

T j

min Z

inequality for with a better bound for ZX . The new bound allows one to

E j a

conclude that , i.e. Z -semistable, for . Note that in the following theorem

s a only the rank of E enters the condition on .

Theorem 7.1.1 Ð Assume

degX maxf g

E r E j

If is a -semistable sheaf of rank , then the restriction D to a general complete

D jO aj

intersection with i is -semistable.

Proof. The proof is divided into several steps. We eventually reduce the assertion to the

case that X is a projective space. Step 1. We claim that it suf®ces to show

T j degX

min Z

ZX (7.1)

E r E j s

Assume is of rank and the restriction Z is not -semistable for general . Let

F F F q E j

j Z

be the relative Harder-Narasimhan ®ltration with

Z i

respect to the family p . Then for some

degF F j deg F F j

i i Z i i Z

s s

E j

i i Z

rkF F rkF F

i i i i

lcm rkF F rkF F

i i i i

detF F Q PicX M Pic q Q p M

Indeed, since i , where and , one

deg Q a deg Qj deg F F j

Z i Z

has i . Using the inequality

s s

g lcm rkF F rkF F maxf

i i i i

and (7.1) we obtain

a a

T j E j deg X

min Z Z

s s

g maxf

a which immediately contradicts the assumption

162 7 Restriction of Sheaves to Curves

g deg X maxf

P PV jO j

Step 2. Since O is very ample, one ®nds a linear system such that

X P T j

V min Z

is a ®nite surjective morphism. Since ZX can only decrease when

Z D D

specializing s , it is enough to show (7.1) for a general complete intersection

D PS V jO aj Z X

with i . Moreover, we may replace the incidence variety

Z Z PS V T j T j

V V Z Z

by , where : Since ZX and are related

s s

i Z X

via the exact sequence

O T j T j

Z Z

s Z s

Z X

T j m h X O a dim S V T j

min Z min Z

( ), one has ZX . Thus it

s s Z X

suf®ces to show

T j degX

min Z

Z X

with V .

Z jO aj P P P V

Step 3. If P denotes the incidence variety on , then

Z Z X T T

. Hence Z . Using the above exact sequence this yields

Z X

T j T j

min Z min Z

Z P

s s

Z X

T j deg

min Z

Z P

T j deg X

min Z

Z X

This completes the reduction to the case X .

T j X P PV

min Z

Step 4. We now prove ZX for the case . To

S V N

shorten notation we introduce A and .

Z P v P PA

Let V be the incidence variety and let be the Veronese embed-

ding. Then Z is the pull-back of the incidence variety

PA PAjx H g fH H x

PT PT

A PA PA PA

which is canonically isomorphic to P . Hence

Z P Pv T Pv T

P P

A PA is as a -scheme isomorphic to P , and the

relative Euler sequence takes the form

O p O q v T

PA ZP

v j T j

min Z min Z P

Therefore, Z . Using the Euler sequence on

s s

P :

A O O

PA PA

7.1 Flenner's Theorem 163

v A O O a

the pull-back P can naturally be identi®ed with the kernel of .

a K

According to the notation of Section 1.4 this is K , which will be abbreviated by . We a

conclude the proof by showing

K j

min Z

a a

Recall from 1.4.5 that K is semistable and by the exact sequence

K A O O a

one has .

Y Z D D D PA

If s is a general complete intersection with , then the

Koszul complex takes form

B a B a B a O O

P Y

A Y where B is the subspace spanned by the sections cutting out . Splitting the Koszul

complex into short exact sequences we obtain

E B j a E

j (7.2)

E E O K

with and . From the dual of the short exact sequence de®ning ,

O a A O K

one gets short exact sequences for the exterior powers of K :

q q q

K a A O K

n H P A b

For b and the cohomology groups vanish. This gives isomor-

phisms

q q q

H K b H K b a H K b a

n K

for all b and . By Lemma 1.4.5 and Corollary 3.2.10 the sheaf is -

q q

P K b b K b q K

semistable, so that H as soon as ,

q a

which is equivalent to b . By tensorizing the sequences (7.2) with and pass-

ing to cohomology we get the exact sequences

j j p j p j p

B H K b j a H E K b H E K b

j j

b p aN The term on the left vanishes for all j as soon as . For

such b one gets

p p p

H Y K O b H E K b H E K b

H Y K O b b

Hence, Y for .

164 7 Restriction of Sheaves to Curves

K F p

If p denotes the minimal destabilizing quotient with respect to the family

q O b p M detF M Pic Z

V , then with . Hence the surjection

K j F j H Y K O b q rkF Y

Y de®nes a non-trivial element in , where and

therefore

q a

b b

On the other hand, , for is a negative integer, and thus

N a

q a

Both inequalities together imply

q a a b a

a K j

min Y

q N q N a 2

7.2 The Theorems of Mehta and Ramanathan

In this section we work over an algebraically closed ®eld of arbitrary characteristic.

n O

Theorem 7.2.1 Ð Let X be a smooth projective variety of dimension and let

E a

be a very ample line bundle. Let bea -semistable sheaf. Then there is an integer such

a a U jO aj D U

a a

that for all there is a dense open subset such that for all the

D E j divisor is smooth and D is again -semistable.

Proof. Let a be a positive integer and let as before

X Z

jO aj a

be the universal family of hypersurface sections.

E a D

The -semistable sheaf is torsion free and for any and general a the restriction

E j q E a

D is again torsion free (Lemma 1.1.13). Moreover, is ¯at over , since, indepen-

D E j P E j m

D D

dently of a , the restriction has the same Hilbert polynomial

E m P E m a

P . According to the theorem on the relative Harder-Narasimhan ®l-

F V q E j

a a Z

tration (cf. 2.3.2). there is a dense open subset a and a quotient that

E j D V Q a

restricts to the minimal destabilizing quotient of D for all . Let be an extension

detF Z Q a

of a to some line bundle on all of . Then can be uniquely decomposed as

Q q L p M

L PicX M Pic a with a and .

7.2 The Theorems of Mehta and Ramanathan 165

L j a L L X L j

Lemma 7.2.2 Ð Let . If and are line bundles on such that D

L D L

for all in a dense subset of a , then .

L L L h Lj h L j D D

Proof. Let . Then D for all in a dense subset

O h Lj h L j D Lj

D D D a D

of a . By semi-continuity, for all . Thus if

D B a

is integral. Now the set a of all integral divisors in is open and its complement has

codimension at least 2 (Use Bertini's theorem and the assumption a ). Therefore, there

N p q Lj N PicB Pic

B a a

is an isomorphism for some line bundle and an

p N q L p B Z L O

a X

isomorphism on a , hence on the whole of . This implies

N O 2

and .

U V D V D E j

a a D Let a denote the dense open set of points such that is smooth and

torsion free (cf. 1.1.13).

a a a a D U

i i a

Lemma 7.2.3 Ð Let be positive integers, , and let be divi-

D D

sors such that i is a divisor with normal crossings. Then there is a smooth locally

C D C n fD g U

a a

closed curve a containing the point such that and such that

Z Z C

C is smooth in codimension .

D U D U i

a i a

Remark 7.2.4 Ð If is given, one can always ®nd for such that

D D

i is a divisor with normal crossings.

L D

Proof of the lemma. A general line a through the closed point will not be

U L n U D C a

contained in the complement of a . Then is a ®nite set containing . Let

L U fD g C H

a . The curve is completely determined by the choice of a hyperplane

D H Z C

in the cotangent space . We must choose in such a way that is smooth in

z D D D a codimension . Let i be a closed point in the ®bre over . The homo-

morphism

z D p

a a z

z D D j is injective if and only if is not contained in any of the intersections i , and the kernel

is 1-dimensional otherwise. Choose H such that the corresponding projective subspace does

not contain any of the images of the maps

D z kerp D D P

i j

a z

z H z D

Then Z is injective for all points outside a closed subset of codimension

Z Z C

2, and C is smooth in these points. This means that the set of points where fails to be

Z 2

smooth has codimension at least three in C .

a r a

Let and denote the slope and the rank of the minimal destabilizing quotient of

E j D r a rkE a

D for a general point . Then and

deg L Z a

a r a rkE

166 7 Restriction of Sheaves to Curves

P P

a a a a a a

j i i

Lemma 7.2.5 Ð Let be positive integers, . Then ,

r a min fr a g

and in case of equality i .

a a

Corollary 7.2.6 Ð r and are constant for .

Proof. The function a takes values in a discrete subset of and is bounded

E N

from above by . Therefore it attains its maximum value on any subset of . Suppose

the maximum on the set of integers is attained at and the maximum on all positive

b b b

integers coprime to is attained at . If and are any positive integers and

b b

, then the lemma says

b b b b b

b b b b

Hence and also for all that can be written as a positive linear

b b b b

b b b b b

combination of and , hence in particular for all . A similar argument shows

b 2

that r is eventually constant. D

Proof of the lemma. Let i be divisors satisfying the requirements of Lemma 7.2.3 and

C V

let be a curve with the properties of 7.2.3. Over a there exists the minimal destabilizing

F q E j V C C a

quotient Z . Its restriction to can uniquely be extended to a ¯at quotient

F q E j F P F j P F c C n fD g

C C C D Cc

Z . The ¯atness of implies that for all .

rkF j r a F j a

D C D

Hence C and .

F F j T F j rkF j rkF rkF j r a a

C D C D C D

Let . Then D i and

F F j F D

C . Moreover, since is pure, the sequence

M M

F j F j F

D D D

i j i

ij i

n is exact modulo sheaves of dimension n . Computing the coef®cients of degree in the Hilbert polynomials of these sheaves (use the Hirzebruch-Riemann-Roch formula), we

get the equation:

D D K H r a F

n n

H r a F j D D K H

D i i X

D D H rkF j

i j D D

i j ij 7.2 The Theorems of Mehta and Ramanathan 167

Cancelling super¯uous terms one gets

X X

rkF j

D D

i j

F j F a a

D i j

r a

j i

T F j F F j

D D

Let i . Then

i i

rkF j

D D

i j

a a F F j

i j i D

j i

r a

rkF j

D D

i j

F j a a

D i j

j i

r a

P P P

a F a a F a

i i

It follows that i . Moreover, if we

i i i

D r a rkF j F a

j i i

must have equality everywhere. In particular, D and .

F r a rkF r a 2

i i

Since i has the minimal possible slope, .

aa a a r a r a i F j

i i C D

Supplement to the proof: if i and for all , then

E j n

differs from the minimal destabilizing quotient of D only in dimension , in particular

i D

their determinant line bundles as sheaves on i are equal. This can be used to prove:

L L PicX L a

Lemma 7.2.7 Ð There is a line bundle such that a for all .

d r a a d

Proof. Let be an integer such that and are constant for all . Let

a d d a d D U D U

d d

and let . Choose arbitrary and let such that

D D D C

is a normal crossing divisor. Let be a curve as in the previous lemma and

r eg

F q E j detF j C A

C Z C

consider the quotient as above. Extend Z to a -¯at sheaf on

L j Z Aj D C n fD g F j

a D D C D

C . Then for all and, since differs from the minimal

L n Aj

destabilizing quotient only in dimension , D outside a set of codimension 2

i i

D i L j Aj

a D D

in i for . By semi-continuity there exist non-trivial homomorphisms

j L Aj L j L j

D d a D a D

and D . Hence there exists a non-trivial homomorphism for

i i i

some i. Since both line bundles are of the same degree, it is an isomorphism. Which in turn

L j D

implies that also on the other component there is such an isomorphism. Hence a

L j L i D L

d D d a

for . Since was arbitrary, Lemma 7.2.2 implies that for all

i i

a d 2

We can now ®nish the proof of Theorem 7.2.1: suppose the theorem were false, i.e. we

Lr E r rkE r r a a a

had deg and , where for . Let be suf®ciently

D U E j F

D D

large, let a , and let be the minimal destabilizing quotient. There is a

D D F j r D

large open subscheme such that D is locally free of rank . This induces a

E j Lj D

D D

homomorphism D which is surjective over and morphisms

D GrassE r P E

Consider the exact sequence

r r r r

Hom E La Hom E L Hom E j Lj Ext E La

D D

168 7 Restriction of Sheaves to Curves

By Serre's theorem and Serre duality, one has for i

r ni r

Ext E La H X E L a

a n a

for all , since by assumption . Hence if is suf®ciently large, D extends

E L

uniquely to a homomorphism . The support of the cokernel of this homo-

morphism meets the ample divisor D in a subset of codimension . Hence is surjective

X D X D

on a large open subset X with . We want the induced morphism

X P E GrassE r GrassE r

i to factorize through . The ideal sheaf of in

r r

P E I S E i

is generated by ®nitely many sheaves , . The morphism

E r

factors through Grass if and only if the composite maps

I S E L

vanish. But we know already that the restriction of to vanishes, so that we can consider

HomI L a a

as elements in . Clearly, these groups vanish for . This proves

F F E j r detF

X X X

that D extends to a quotient of which is locally free of rank with

X . Hence

deg L

F E

r This contradicts the assumption that E is -semistable and, thus, concludes the proof of

Theorem 7.2.1 2

We now turn to the restriction of -stable sheaves.

n O

Theorem 7.2.8 Ð Let X be a smooth projective variety of dimension and let

E a

be a very ample line bundle. Let bea -stable sheaf. Then there is an integer such that

a a W jO aj D W

a a

for all there is a dense open subset such that for all the

D E j

divisor is smooth and D is -stable.

The techniques to prove the theorem are quite similar to the ones encountered before. The

main dif®culty is the fact that a destabilizing subsheaf of a -semistable sheaf is not unique. By 1.5.9 a -semistable sheaf which is simple but not -stable has a proper extended socle. Thus we ®rst show that the restriction is simple and then use the extended socle (rather its

quotient) as a replacement for the minimal destabilizing quotient.

a D jO aj E j

Lemma 7.2.9 Ð For and general the restriction D is simple.

F E a D jO aj F j

Proof. Let be the double dual of . Then for arbitrary and general , D

E j E E j F F j

D D is thedoubledual of D (cf. Section 1.1). Since and are torsion free and and

are re¯exive (cf. 1.1.13), there are injective homomorphisms

EndE EndF EndE j EndF j D and D

7.2 The Theorems of Mehta and Ramanathan 169

F j a D E

Therefore, it suf®ces to show that D is simple for and general. But if is - F

stable, then so is F . In particular, is simple. Consider the exact sequence

HomF F a EndF EndF j Ext F F a

i j

H X E xt F F a Ext F F a X

Recall the spectral sequence X . For

suf®ciently large a we get

Ext F F a Ext F F a H X E xt F F a

X X

xt F F F dhF n

But E , since is re¯exive and thus (cf. Section 1.1).

a EndF EndF j 2

Hence for suf®ciently large, D is surjective.

a a a D jO aj

Let be an integer such that for all and general the restriction

E j E j D E j

D D

D is -semistable and simple. Suppose is not -stable for a general . Then

jO aj

is geometrically -unstable for the generic point , i.e. the pull-back to some ex-

tension of k is not -stable. This follows from the openness of stability (cf. 2.3.1). Since

E j E j D

D is simple, the sheaf is stable if and only if it is geometrically stable (Lemma

E j E j D

1.5.10). Hence D is not -stable. In fact, the extended socle of is a proper destabi-

lizing subsheaf (1.5.9). Extend the corresponding quotient sheaf to a coherent quotient

q E F Z W D jO aj

a a

a over all of . Let denote the dense open subset of points such

D F W E j F j

a D D

that is smooth and a is ¯at over . Then is a destabilizing quotient for

D W

all a .

E j D W a a E j

a D

Lemma 7.2.10 Ð If D is -stable for some , , then is -stable

D W a a

for all a and all .

D W D D D

a a

Proof. Choose such that is a normal crossing divisor, and

jO a j

let C be a curve as in the proof of Lemma 7.2.3. Then the destabilizing quotient

F jZ F q E j F j E j

a C Z C D D

nfD g

C can be extended to a ¯at quotient of . Then destabilizes

C i i

in contradiction to the assumptions. 2

E j a a D

Assume now the theorem is false. Then D is unstable for all and general

L PicX det F j L j W

a a D a D

a . As before there are line bundles such that for all

D W a a a a

a and all . The same argument as in Lemma 7.2.7 shows: if are

P P

a a a D W D D

i i a i

integers and , and if are points such that is a normal

L j D

crossing divisor, then a is the determinant line bundle of some destabilizing quotient

E j

of D .

D E

Lemma 7.2.11 Ð If is a smooth projective variety, and if D is a -semistable sheaf,

T E D

then the set D of determinant bundles of destabilizing quotients of is ®nite and its car-

rkE D

dinality is bounded by .

170 7 Restriction of Sheaves to Curves

L L rkE

Proof. Let , , be the determinant bundles of the factors of some

E T D

Jordan HÈolder ®ltration of D . Then is contained in the set of line bundles of the form

L I f g 2

i , where .

a a D W L j T

a a D D

Let , and let be an arbitrary point. We saw that . In fact,

we get a function

N T

a D

D W

N a a

Let be the equivalence relation on generated by: if the set of points

rkE

s W as a s W a

a with is dense in . Then there are at most distinct

equivalence classes, and in particular, there is at least one in®nite class N . For assume that

rkE

N N a N

i i

there are distinct classes , . Choose representatives . For

D W

®xed a , we have

a D a D T

jT j

Since D , at least two of these elements must be equal. In this way we can pick for

D W i j

any a a pair of indices . But the set of all these pairs is ®nite. Hence their is at

i j W

least one pair which is associated to all points in a dense subset of a . But by de®nition

a a N N

j i j

this means i , hence , a contradiction.

L PicX L L a N

Lemma 7.2.12 Ð There is a line bundle such that a for all .

j L a a W L j D

D a a D

Proof. If equals on a dense subset of a then for all in a

L jO aj L 2

dense subset of , so that Lemma 7.2.2 implies a .

N N F r

Finally, let be an in®nite subset such that a has the same rank, say , for all

N L X r

a . Summing up, we have: there is a line bundle on and an integer

rkE degL r E a N D W

such that and such that for all and general a

E j F rkF r detF Lj

D D D D there is a destabilizing quotient D with and . But the

arguments at the end of the proof of the previous theorem show that this suf®ces to construct

E F a

a destabilizing quotient X for suf®ciently large . This contradicts the assumptions

of the theorem. 2

7.3 Bogomolov's Theorems

This section is devoted to a number of results due to Bogomolov. The original references are

[28, 29, 31]. In our presentation we use the fact that tensor powers of -semistable sheaves

are again -semistable (in characteristic zero), i.e. we build on the Grauert-MÈulich Theorem and Maruyama's results, discussed in Chapter 3. In this we deviate from Bogomolov's line 7.3 Bogomolov's Theorems 171

of argument, which is independent of the before-mentioned theorems. However, the essen-

tial ideas are all due to Bogomolov. In the following let X be a smooth projective surface over an algebraically closed ®eld of characteristic zero.

Recall that Bogomolov's inequality 3.4.1 states that the discriminant of any -semistable torsion free sheaf is nonnegative. Before we begin to improve upon this result, we give a short elegant variant of the proof of 3.4.1, say as a warm-up for calculations with discriminants, following an argument of Le Potier. Using one of the restriction theorems of the pre-

vious sections one can generalize the inequality to sheaves on higher dimensional varieties.

n H

Theorem 7.3.1 Ð Let X be a smooth projective variety of dimension and an ample

divisor on X . If is a -semistable torsion free sheaf, then

F H

Proof. By 7.1.1 or 7.2.1 the restriction of F to a general complete intersection

D D D jaH j a

with i and is again -semistable and torsion free.

n n

a F H F j

Since X , we may reduce to the case of a -semistable sheaf

X F

on a surface. Thus, let H be an ample divisor on a surface and let be a torsion free F

-semistable sheaf. As in the earlier proof we may assume that is locally free and has

trivial determinant. By Theorem 3.1.4, the vector bundles F are all -semistable. They

r r n

have trivial determinant and their ranks and discriminants are given by n and

F H

nr . Replacing by some large multiple, it follows from the restriction theo-

F j F j C

rem of Flenner or Mehta-Ramanathan, that C ± and hence also Ð is semistable

jH j

for a general curve C . In particular, it follows from Lemma 3.3.2 that there is a pos-

X h F r

itive constant , depending only on , such that n . By Serre duality, and

n n n

h F r F r

enlarging if necessary, we also get n , and therefore . On the other hand, the Hirzebruch-Riemann-Roch formula for bundles with vanishing ®rst

Chern class says:

n n n

F r O r O r F

n X X

n n

F r F 2

If n goes to in®nity, this contradicts , unless .

Corollary 7.3.2 Ð Let F be a torsion free sheaf. If is -semistable with respect to an

ample divisor H , then the discriminants of the -Jordan-HÈolder factors of satisfy the in-

F F g r i

equality for all .

F F r i

Proof. Assume ®rst that an arbitrary ®ltration of with torsion free factors i of rank

P P

r rkF c F r

i i

and ®rst Chern classes i is given. Let and .

i i

r ch c

Recall that the Chern character and the discriminant are related by . The additivity of the Chern character in short exact sequences therefore provides the ®rst equality in the following identity and a direct calculation gives the second:

172 7 Restriction of Sheaves to Curves

X X X

F F

i i j

r r j

i (7.3)

r r r r r r r

i i i j

i i ij

F F r r H

i i j j

Now if the factors i arise from a Jordan-HÈolder ®ltration of , then

r r i j

i j j

, and therefore i for all , by the Hodge Index Theorem. Since

i by Bogomolov's Inequality, we get

F F F

j i

r r r

i j

j i

which is even stronger than the assertion of the corollary. 2

We can rephrase the Bogomolov Inequality as follows: if for some torsion free

F H X sheaf F , then must be -unstable with respect to all all polarizations on . Indeed, the next theorem implies that one can ®nd a single subsheaf which is destabilizing for all

polarizations. Before stating the theorem, we introduce some notations: let Num denote the

PicX

free Z-module , where means numerical equivalence. Its rank is called the Num

Picard number of X . The intersection product de®nes an integral quadratic form on ,

Num K

whose real extension to R is of type by the Hodge Index Theorem. Let

denote the open cone

K fD Num jD D H H g

R for all ample divisors

Note that the second condition is added only to pick one of the two connected components

of the set of all D with . This cone contains the cone of ample divisors and in turn

is contained in the cone of effective divisors. K satis®es the following property:

K D L L K n fg

D for all (7.4)

For any pair of sheaves G G with nonzero ranks let

c G rkG c GrkG Num

G G R

Theorem 7.3.3 Ð Let F be a torsion free coherent sheaf with . Then there is a non-

F K F F

trivial saturated subsheaf with F . Equivalently, if is a torsion free sheaf

K F

which is -semistable with respect to a divisor in , then .

Before we prove the theorem the reader may check the following identities: let

F be a short exact sequence of non-trivial torsion free coherent sheaves. If

G is a non-trivial subsheaf, then

F F GF

GF (7.5)

F s G F

And if G is a proper subsheaf of rank and the kernel of the surjection

G F , then

7.3 Bogomolov's Theorems 173

s r r s

F F G F

GF (7.6)

r sr r s

r r F F F where, of course, r , and are the ranks of , and , respectively. Note that in both

cases the coef®cients in the linear combinations are positive numbers.

Proof of the theorem. If , the claim follows directly from the previous theorem: any

Num

saturated destabilizing subsheaf suf®ces. So assume that . For any nonzero R

fD K jD g K

let C denote the open subcone . Property (7.4) says that is in

K n fg

if and only if C .

r F L I L I

If , then Z , where is a line bundle and the ideal sheaf of a zero-

X F Z F

dimensional subscheme Z , and . Now assume that .

H F

Let F be a saturated destabilizing subsheaf with respect to some polarization , and let

be the quotient F F . Then writing the identity (7.3) in the form

r r

F F

r r r r

we see that either , and we are done, or that or are negative. In this case we

F F

G F K

can assume by induction that there is either a saturated subsheaf with GF

K G F G F

or a saturated subsheaf with G . In the latter case, let be the kernel of

F F G

F F

. In any case, GF is a positive linear combination of and some element

K K C

F F F

by (3.4) and (7.3). Now by assumption, F is not in and therefore

K n fg C F

is a proper subcone of . But is strictly positive on the closure of F in

K n fg C C

F F

. Thus GF contains this closure and a fortiori as proper subcones.

F G C F

Hence replacing by strictly enlarges the cone F . Repeating this process we

K n fg

get a sequence of strictly increasing subcones of until at some point .

F F

H H

All we are left with is to prove that this process must terminate: let be ample

Num R C G

F F

divisors whose classes in R form an -basis and are contained in . Let be

Num F C C

F F GF

any subsheaf of with GF . Then is contained in the lattice

and satis®es the relations

H H

j j

F F H

GF j

max

j Num 2

for all . That is, is contained in a bounded discrete and hence ®nite subset of R.

F F K F

Having found a subsheaf with F the next step is to improve the

theorem in a quantitative direction by giving a lower bound for the positive square :

F F

Theorem 7.3.4 Ð Let F be a torsion free coherent sheaf with . Then there is a sat-

F K F

urated subsheaf with F satisfying the inequality

F F

r r

174 7 Restriction of Sheaves to Curves

F F K F Proof. If is a saturated subsheaf with F , then the Hodge Index The-

orem implies

H H

H H F F H

F F

F F max

H F

for any ample divisor . In particular, the numbers for varying are bounded from

F F

above. Let be such that attains its maximum value. As before, let be the quotient

F F

K F F G F

. Suppose now that and let be a saturated subsheaf with GF .

F F Since GF and are both elements in the positive cone , the Hodge Index Theorem

shows that

j j j j j j j j j j

F F F F F F GF GF GF

j j

contradicting the maximality of F . Here we have used the notation . Hence

. Assume now that

r r

F F r

Using the additivity relation (7.3) again, we get

r r r r r r r r

(7.7)

F F F F F F

r r r r r

Arguing by induction on the rank, we can now apply the theorem to F . As before let

F be a destabilizing subsheaf of rank satisfying the relation

F F G F

r r r

F F G For the last inequality use (7.7). Let G denote the kernel of . Using (7.6) we

have

r r s s

j j j j j j

GF F F G F

r sr r s

s r r r s

j j j j j j

F F F F F F

r sr r s r

2 Again this contradicts the maximality of F , and therefore proves the theorem.

We are now prepared to prove Bogomolov's effective restriction theorem. Let r be an

r r

integer greater than 1, and let R be the maximumof the numbers for all .

c b

(Certainly the maximum is attained for .)

r F

Theorem 7.3.5 Ð Let F be a locally free sheaf of rank . Assume is -stable with

K N um C X C nH

respect to an ample class H . Let be a smooth curve with .

n F j F

If , then C is a stable sheaf. r

7.3 Bogomolov's Theorems 175

C F j

Proof. Suppose satis®es the conditions of the theorem and C has a destabilizing quo-

s s

tient E of rank . By taking exterior powers we wish to reduce the proof to the case .

s s s s

F E E F

Then is still destabilizing and is a line bundle, but need not be

-stable. At this point we evoke the Theorem 3.2.11, that powers of -stable bundles are - polystable. Recall, that the proof, which we only sketched, relies on the Kobayashi-Hitchin

correspondence (or the interpretation of stable bundles on a curve in terms of unitary repre-

n F

sentations). We replace the numerical assumption by the two inequalities

n F (7.8)

and

H C F

n (7.9)

r r r

s s

F F

Indeed, (7.8) follows from rk and . The second

r s s s

inequality is a consequence of the ®rst and : slightly improving (7.8) by using

s s s

F n H n F F

integrality we get n . Hence

s s s

F F j E L L

Next consider the exterior power C , where is a line

s s

L E F j F j F F

C i

bundle with C , and the decomposition ,

F F F F j L

where the bundles i are -stable with slope . We may assume that

L F L

is not trivial. Replacing by the image of , which has even smaller degree, and

F F F

using by 7.3.2, we obtain a -stable bundle with a destabilizing line

F j L n F C F rkF

bundle such that and . The case

rkF can be excluded by a lemma stated after the proof. If we have concluded our

reduction to the case of a rank one destabilizing line bundle, i.e. we may assume that F is

-stable of rank with

n F (7.10)

and

F F r

C (7.11)

F j E

and that C is a destabilizing quotient of rank one.

G F F j E c G

Let be the kernel of the composite homomorphism C . Then

G F deg E g C c F C

and . Expressing the Euler characteristic G

of F and in terms of their discriminants we get (cf. 5.2.2)

G F deg F jC r degE r C

deg F jC r degE

Since E is destabilizing, .

G F r C

176 7 Restriction of Sheaves to Curves

G because of (7.11). By the previous theorem there is a saturated subsheaf G of rank,

say, t with

G and

G G

r r

C F C

G G F G F

Then G . The stability of implies that , and since the r

intersection product on Num takes integral values,

n n

H C

G G

r t r

D K DD D D

For any two divisors D and in the inequality holds. Apply this

C G

to and G and get

G n n

n H C H

G G

r r r r t

G F r n

Using the estimate H and cancelling common factors we get

F n

H H

r t t

hence

F F n

r tH

which contradicts (7.10). 2

In the proof we made use of the following lemma:

F M

Lemma 7.3.6 Ð Let F be a -semistable vector bundle and be a rank one

s s

F M F j M C C

torsion free quotient with . If the restriction C to a curve

s F j E s F M

is the -th exterior power of a locally free quotient C of rank , then

E s F

is induced by a torsion free quotient F of rank . In particular, if is -stable, then

rkF s .

Proof. The technique to prove this was already used twice in Section 7.2. Let

GrassF s P F

be the PlÈucker embedding of the relative Grassmannian. Its ideal sheaf is generated by

S . In fact, it is generated by the PlÈucker relations which are the image of a homo-

s s s s

F F S F F M

morphism . The quotient corresponds to a section

P F j U U X n SuppM M

of U , where . The image of this section is contained

F s

in Grass if and only if the composite maps

I S F S M 7.3 Bogomolov's Theorems 177

vanish or, equivalently, if the composition

s s s

F F S F S M

s s

F F sF S M

vanishes. Standard calculations show and

sF F j E C U GrassF s

. The existence of C implies that the curve is mapped to

F j M j U

by the cross section that corresponds to the homomorphism U . Hence is an

s s s

F F S M C F

element in Hom . Using the -semistability of

s s s

F F F S M C U

(cf. 3.2.10) and , this yields , i.e.

F s 2 maps to Grass .

Remark 7.3.7 Ð Of course, the theorem remains valid if F is only torsion free but the

F H

curve C avoids the singularities of . One can also weaken the assumption on the class :

K Num F

let H be an arbitrary class in and let be a -stable vector bundle with respect to

H . If we furthermore assume that also all exterior powers of are -stable with respect to

rkF

H , which is automatically satis®ed if , then the conclusion of the theorem holds

H K true. In Chapter 11 this will be applied to minimal surfaces of general type and X which is only big and nef.

Comments: Ð We wish to emphasize that the results in Section 7.1 and 7.3 assume that the characteristic of our base ®eld is zero. The restriction theorems of Mehta and Ramanathan are valid in positive characteristic as well. Unfortunately, it is not effective. In fact, an effective restriction theorem would settle the open question whether families of semistable sheaves with ®xed topological data are bounded in positive characteristic. Ð The proof of Flenner's Theorem 7.1.1 follows quite closely the original presentation in [63], though we avoided the use of spectral sequences. Since its proof relies on the Grauert-MÈulich Theo-

rem, and hence the Harder-Narasimhan ®ltration, it does not generalize to the case of -stable sheaves. Ð The references for the theorems of Mehta and Ramanathan (7.2.1, 7.2.8) are of course [175] and [176]. Also see [174]. The complete argument for the fact, used in Lemma 7.2.2, that the complement of the integral divisors has codimension at least two can be found in [175]. Ð Tyurin generalized their arguments (cf. [250] and for a more detailed proof [111]) and showed

that a family of -stable rank two bundles on a surface restricts stably to a general ample curve of high degree.

Ð One should also be aware of the following result due to Maruyama ([164], also [174]):

If X is smooth and projective and is very ample, then the restriction of a -semistable sheaf of

dimX rank to the generic hypersurface is again -semistable. The proof of it is rather easy, but as it has obviously no application to sheaves on surfaces, we omitted the proof.

Ð Proofs of Theorem 7.3.5 for rank two bundles for special cases can be found at various places.

X Z

O'Grady in [205] treats the case Pic and Friedman and Morgan give a proof for the case

[71]. The complete proof is in Bogomolov's papers [29] and [31]. Ð In special cases one can improve the results. Hein [101] and Anghel [1] deal with the case of rank two bundles on K3 surfaces and on abelian surfaces, respectively. 178

8 Line Bundles on the Moduli Space

This chapter is devoted to the study of line bundles on the moduli space. In Sections 8.1 and 8.2 we ®rst discuss a general method for associating to a ¯at family of coherent sheaves a determinant line bundle on the base of this family. The next step is to construct such de-

terminant bundles on the moduli space of semistable sheaves even if there is no universal

L L

family. Having done this we study the properties of two particular line bundles and on

L L

the moduli space of semistable torsion free sheaves on a smooth surface. Whereas

X m jL j

is ample relative to Pic for suf®ciently large , the linear system contracts certain parts of the moduli space and in fact de®nes a morphism from the Gieseker-Maruyama moduli space of semistable sheaves to the Donaldson-Uhlenbeck compacti®cation of the

moduli space of -stable vector bundles. The presentation of the material is based on the work of J. Le Potier and J. Li.

In the ®nal section we compare the canonical bundle of the good part of the moduli space L with the line bundle . This is an application of the Grothendieck-Riemann-Rochformula.

8.1 Construction of Determinant Line Bundles

n K X

Let X be a smooth projective variety of dimension . The Grothendieck group of

X O F F

coherent sheaves on becomes a commutative ring with X by putting

F F F F u u K X

for locally free sheaves and . Two classes and in are said to be

numerically equivalent: u , if their difference is contained in the radical of the quadratic

a b a b K X K X S K X

form . Let num . If is any subset, let

K X S S be the subset of all elements orthogonal to with respect to this quadratic

form. By the Hirzebruch-Riemann-Roch formula we have

a b chachbtdX

a K X rka

Thus the numerical behaviour of num is determined by its associated rank

c a

and Chern classes i .

X S E

A ¯at family E of coherent sheaves on parametrized by de®nes an element

p S X S S X

K , and as the projection is a smooth morphism, there is a well

p K S X K S

de®ned homomorphism (cf. 2.1.11).

K X PicS

De®nition 8.1.1 Ð Let E be the composition of the homomorphisms:

q p

det

K X K S X K S X K S PicS 8.1 Construction of Determinant Line Bundles 179

Here is a list of some easily veri®ed properties of this construction:

E E E S

Lemma 8.1.2 Ð i) If is a short exact sequence of -¯at families

E E

of coherent sheaves then E .

S f S S u K X

ii) If E is an -¯at family and a morphism then for any one has

u f u

E E

f .

S G E G S

iii) If G is an algebraic group, a scheme with a -action and a -linearized -¯at fam-

X Pic S

ily of coherent sheaves on , then E factors through the group of isomorphism S

classes of G-linearized line bundles on .

E S c K X N

iv) Let be an -¯at family of coherent sheaves of class num and let be a

rkN cu

O u u detN

S E p N locally free -sheaf. Then E

Proof. The last assertion follows from the projection formula for direct image sheaves:

i i rk B

R p E p N R p E N detAB detA

, and the general isomorphism

rk A

2 B

det for arbitrary locally free sheaves.

x X u O

Examples 8.1.3 Ð i) Let be a smooth point and x the class of the structure

x E S X E E sheaf of . Let be an -¯at family of sheaves on and a ®nite locally free

resolution. Then by 8.1.2 i)

O O

i i

det R p E O u u

i x E E

i i

detR p E O det p E j p det E j

i x i i

fxg S fxg

Now S . Hence

det E j p detE j u p

i E

S fxg S fxg

H X h O K X

ii) Let be a very ample divisor and let H be its class in . Then

O h h h

X . In Section 4.3 we used the line bun-

det p F q O QuotH P X

dles on the quotient scheme in the construction of the

moduli spaces. These bundles are very ample for . Using the -formalism above we

can express them as follows:

detp F q O O

X X

h h

e e e

F F F

detp F q O X In particular, does not, in general, depend linearly on and projective

embeddings given by multiples of this line bundle might be quite different for different .

s M

iii) Let E be a universal family parametrized by the moduli space . As above we ®nd

n n

O h h

X E

that the dominant term in the -expansion of E is . Now

degX

O x x n

where deg are the intersection points of general hyperplanes.

j According to the example in i) we can write

180 8 Line Bundles on the Moduli Space

degX

n n

detE j h

M fP g

E p L L M

If E is replaced by for some line bundle on , the expression on the right hand

rkE degX

L L O

L X

side changes by . Thus, if is very negative, E becomes

very negative for .

c K X cm c O m P c X

For any class in num , we write and denote by the

c m cm F S

associated Hilbert polynomial P . If is an -¯at family of coherent

P c s S F c

sheaves with Hilbert polynomial the points such that s is of class form an F

open and closed subscheme of S . This follows from the fact that for a ¯at family the Euler

s F

characteristic s is a locally constant function. As a consequence the moduli space

M P M c c i

decomposes into ®nitely many open and closed subschemes i , where runs

P c P E M c X

through the set of classes with i . A universal family on is well-

de®ned only up to tensorizing with a line bundle on M . Part iv) of the lemma shows

u c u u c

that E is independent of this ambiguity, if , i.e. if is orthogonal to . We

therefore de®ne:

c K X

De®nition 8.1.4 Ð For a given class num let

K c K c f h h h g cH

c and

The following theorem says that the condition c is also suf®cient to get a well-de®ned

c u

determinant line bundle on M by means of . More precisely:

c K X

Theorem 8.1.5 Ð Let be a class in num . Then there are group homomorphisms

s s

K PicM c K PicM c cH

c and with the following properties:

K K c

1. and commute with the inclusion cH and the restriction homomorphism

M c PicM c

Pic .

c X S

2. If E is a ¯at family of semistable sheaves of class on parametrized by , and if

S M c K X PicS E

E is the classifying morphism, then and com-

K K X PicM c

mute with the inclusion cH and the homomorphism

Pic .

c X S

3. If E is a ¯at family of stable sheaves of class on parametrized by , then and

K X PicS K K X c

E commute with the inclusion and the homomor-

PicM c PicS

phism . E

8.1 Construction of Determinant Line Bundles 181

In order to prove the theorem we have to recall the set-up of the construction of M in

V P c m

Section 4.3: we choose a very large integer m, ®x a vector space of dimension

H V O m R c QuotH P

and let X . Let denote the open subscheme of

q H F F c q

those quotients for which is a semistable sheaf of class and induces an

V H F m O H F m

isomorphism . There is a universal family R . If was

R c R c

chosen large enough and , is the set of semistable points in with respect

O SLV M c X

to the action of and the canonical linearization of e . Moreover,

R cSLV F

. The determinant bundle det of the universal family induces a morphism

det R c PicX detF det P P p A

such that where is the PoincarÂeline

X X A R c det R c PicX

bundle on Pic and some line bundle on . (Of course,

c deg P c dimX can be the constant morphism, for example if dim .) We

®x these notations for the rest of this section.

u K X

Proof of the theorem. Let num be an arbitrary class and consider the line bundle

F u L R c L GLV

e on . inherits a -linearization from . We want to know whether

M c M c L descends to a line bundle on or .

According to the criterion of Theorem 4.2.15 we must control the action of the stabi-

V q H F

lizer subgroup in GL of points in closed orbits. The orbit of a point

c F

R is closed if and only if is a polystable sheaf, i.e. if it is isomorphic to a direct sum

F W F k W q

i k i i

with distinct stable sheaves i and -vector spaces . The stabilizer of

Aut F GL W A A A GLW

i i then is isomorphic to i , and an element , ,

acts on the ®bre

dimW F u

i i

det H F u detW Lq

i i

detA

via multiplication with the number i (cf. the remarks following 2.1.11).

c F r r F F

i i i Let i , and let and be the multiplicities of and , respectively. By construc-

tion, we have for all :

r c O r P F r P F r c O

i X i i i X

r c r c h r c r c f h h g

i i i

This is equivalent to: i for all , i.e. .

F Aut F G u K

Now distinguish two cases: if is in fact stable, so that m , and if ,

A G k A A F

then m acts by . If on the other hand is not stable but

u K r c r c u r c r c f h h g

i i i i

cH , then we have , since .

c u c u

Therefore i . Thus, again, any element in the stabilizer subgroup

u K u K u cH

acts trivially. It follows that c or are suf®cient conditions on to let the

s s

u M c u M c line bundle L descend to bundles on , or on , respectively.

It remains to check the commutativity relations. Part 1 of the theorem is trivial. To get the S

universal properties 2 and 3 proceed as follows: suppose E is an -¯at family of semistable

c S IsomV p E O m S X

sheaves of class . Let be the frame bundle (cf.

S R c p E O m

E X

4.2.3) associated to the locally free sheaf , and let be

V O E

the classifying morphism for the quotient e which is the composition

S X

182 8 Line Bundles on the Moduli Space

GLV

of the tautological trivialization (4.2.6) and the evaluation map. E is a -equivariant

S M

E E morphism, and E , where is the classifying morphism for the

family E .

S R c

S M

We obtain the following sequence of GL -equivariant isomorphisms

e e

u u u u u u

E E

e e

E E E

GLV

PicS Pic S Assertions 2 and 3 follow from this and the fact that is injec-

tive. (cf. 4.2.16). 2

Before we describe natural line bundles in the image of , we want to raise the question of how many line bundles one can construct this way. The best result in this direction is due to J. Li. Unfortunately, the techniques developed here are not suf®cient to cover his result. In particular, we have not explained the relation to gauge theory essential for its proof.We only state the following special case of the result in [151] which can be conveniently formulated

in the language introduced above.

q X

Theorem 8.1.6 Ð If X is a regular surface, i.e. , then

K X Q PicM Q c Q

c 2

is surjective for . u

Let X be a surface. We will see in Example 8.1.8 ii) below that only the degree of classes

CH X u M r Q c

in matters for the restriction of to the moduli space of semistable

K X Z PicX

sheaves with ®xed determinant Q, i.e. we can reduce from to the group

u rku det u u

Z, sending to the triple . Moreover, the condition to be orthogonal u

to c imposes a linear condition on . c

The above theorem gives reason to expect that for large second Chern number the Pi-

M r Q c

card group of the moduli space contains a subgroup which is (roughly) of the

X Z

form Pic . More evidence is given by the following example:

r Q

Example 8.1.7 Ð Let E be a -stable locally free sheaf of rank , determinant and sec-

c E c S PE X

ond Chern class , and let be its projectivization. Let

S S X F

be the graph of and let be the kernel of the surjective homomorphism

q E q E j E O

S S Then F is an -¯at family with ®bres

8.1 Construction of Determinant Line Bundles 183

F kerE E x k x

s s PE x x S E

for x and . As is -stable, the same is true for

F F E F F

s s s

s . Moreover, there is a unique way of embedding into . Hence and are non-

s s S F S M r Q c

isomorphic for all in , and induces an injective morphism .

Applying to the short exact sequence

F q E O

we get according to 8.1.2:

u u u

F q E

As is the graph of ,

u detO u O rku detu

u O rku detu

Hence F We will see in the next chapter that sheaves

E c PicM r Q c s

as above always exist for large . The calculations then show that

PicX 2

contains Z .

K X

In the following we investigate some particular classes in cH and their associated

line bundles.

n H

Examples 8.1.8 Ð Let X be a smooth variety of dimension , a very ample divisor and

c K X

a class in num .

j i i j

i j n v c c h h c h h

i) For any pair of integers , the class ij K

is an element in cH , as is rather obvious.

D D K X

ii) Let be the classes of zero-dimensional sheaves of the same length,

D D D D D K X

and let . Then , so that is in particular an element in . More-

det M D M PicX det M c PicX over, for some line bundle on , where

is the determinant morphism. It clearly suf®ces to prove this assertion for the special case

D x i

that i is the structure sheaf of a closed point . Then we have the following isomorphisms

of line bundles on R :

D p detF j A det P j

Rfx g PicX fx g

i i

PicX X

where as before P is the PoincarÂeline bundle on . Thus

D det P j P j

PicX fx g PicX fx g

n i i n

v c v c c h h c h h in

iii) Observe that in the expression i the

n n

c h rkc deg X h deg X

®rst coef®cient equals , whereas is represented by points

on X. Choose a ®xed base point x and de®ne

i i

u c r h c h O

i x

184 8 Line Bundles on the Moduli Space

i n

v c degX u c c h h deg X O

i x Then i . It follows from part ii)

that

deg X

v c u c det M

i i

PicX 2

for some line bundle M on .

u c

The line bundles i play an important rÃole in the geometry of the moduli spaces.

We therefore de®ne:

i i

x X u c r h c h O x

De®nition 8.1.9 Ð Let be a closed point, let i ,

i L PicM c

, and let i be the line bundle

L u c

i i

i L det Q

for . The restriction of the line bundles i to the ®bres of the determinant

M c PicX x

det is independent of the choice of .

M c M cm

Proposition 8.1.10 Ð Let m be the isomorphism which is induced

F F O m

by X . Then

L L

O m O h h K X x

Proof. Recall that X , and of course

for i . Hence

u cm O mH

i X

P P

m mj

i i j

r h c h O h

i i

r h c h O

u c

Applying we get the isomorphism of the proposition.

X H c

Theorem 8.1.11 Ð Let be a smooth polarized projective variety, and let be the

r m L M cm

class of a torsion free sheaf of rank . For the line bundle on is

M cm PicX

relatively ample with respect to the determinant morphism det .

q H F R c

Proof. Recall that in the general set-up explained above for all points

m V H F m

the sheaf F is regular and is an isomorphim. Hence the universal

e e

F m V O detV O p det p F m

c Rc

family yields isomorphisms R and .

u cm r O P m O

X x

Since , we get

r P m

u cm det V det F j

Rcfxg

F m

r P m

det V A det P j

PicX fxg

8.2 A Moduli Space for -Semistable Sheaves 185

PicX A

Theorem 4.A.1 says that A is ample relative to and that some tensor power of

c PicX

descends to a line bundle on M which is again ample relative to . This shows

L M cm PicX 2

that the line bundle on is ample relative to .

X L

Remark 8.1.12 Ð i) If is of dimension one, only is non-trivial and the theorem says

L PicX L L

that is ample relative to . For the case of a surface only and are non-trivial

m L L M c

and for the line bundle on the moduli space is ample on the ®bres

M c PicX of det .

ii) For later use we point out that the argument above shows that on any ®bre of the mor-

u cm det A SLV

phism ` ' the line bundles and e are isomorphic as -linearized

F m line bundles.

8.2 A Moduli Space for -Semistable Sheaves

X H c K X

Let be a smooth projective surface with an ample divisor . Fix a class num

r c c Q c Q c

with rank and Chern classes and , and a line bundle with . Proposition

L L M r Q c

8.1.10 and Theorem 8.1.11 show that the line bundle is ample on for

m L L suf®ciently large . What can be said about itself? It is clear that the class of must

be contained in the closure of the ample cone. It will be shown that for suf®ciently large

m jL j M r Q c

the linear system is base point free and leads to a morphism from to

the Donaldson-Uhlenbeck compacti®cation of the moduli space of -stable vector bundles

as de®ned in gauge theory. In fact the main purpose of this section is to construct a moduli

ss ss

M M r Q c

space for -semistable sheaves. The assertions about the linear

j jL M r Q c

system on will follow from this.

m j

In order to demonstrate some properties of the linear system jL , we study the line bun-

dle in the following examples for two particular families. These provide strong hints

which sheaves in M cannot possibly be separated and which on the contrary should be ex-

pected to be separable.

r X

Example 8.2.1 Ð Let E be a torsion free sheaf of rank on . For consider the

E E scheme Quot that parametrizes zero-dimensional quotients of of length . There is

a universal exact sequence

F O E T

Quot

X QuotE c F s S

of families on parametrized by . Let be the class of s for some

u u c u

and let . From the short exact sequence one gets an isomorphism

u u u

q E T

T . Recall that any zero-dimensional sheaf is semistable, so

T QuotE M S X

that induces a morphism T from to the moduli space , cf. u

4.3.6. Since is orthogonal to any zero-dimensional sheaf we can apply Theorem 8.1.5

186 8 Line Bundles on the Moduli Space

u u u

and conclude that F . We claim that is an ample line bundle on

X X S X

S . To see this consider the quotient map for the action of the sym-

pr X X i O

metric group. Let i denote the projection to the -th factor and the structure

pr X X E O

sheaf of the diagonal . Then is an equivariant ¯at family

of sheaves on X of length and is the classifying morphism for . Clearly

O O

pr O r H pr det u u

X E

i i

X u u

is an ample line bundle on . On the other hand . Since is ®nite,

u T

is ample as well. Since the ®bres of are connected, we conclude that for suf®-

n j u j s s QuotE

ciently large the complete linear system F separates points and in

T T E s

ifandonlyif s . Note that if is -semistable or -stable then the same holds

F s QuotE 2

for all s , .

F X r r

Example 8.2.2 Ð Let F and be coherent sheaves on of rank and , respectively.

PExt F F k F F

The projective space P , parametrizes all extensions of by ,

including the trivial one, F , and there is a tautological family

q F p O F q F

X u K X F

on P . Let be orthogonal to . Then

F u

O O u u u

P P q F F

q F p O

F u

since by assumption. This applies in particular to the following situation:

F c u u c F F

Let be a -semistable sheaf of class and let . If is -destabilizing,

F u c u F

then . The argument above shows that no power of can separate and

F F 2

F .

We begin with the construction of M : the family of -semistable sheaves of class is

m R

bounded (cf. 3.3.7), so that for suf®ciently large m all of them are -regular. Let

H P q H F F

Quot be the locally closed subscheme of all quotients such that is

r Q c q

-semistable of rank , determinant and second Chern class and such that induces

H F m SLV R

an isomorphism V . The group acts on by composition. The

H F q O

universal quotient R allows to construct a line bundle

N u c

F ss

on R .

Proposition 8.2.3 Ð There is an integer such that the line bundle is generated

V by SL -invariant global sections. 8.2 A Moduli Space for -Semistable Sheaves 187

The main technique to prove the proposition consist in the following: if S parametrizes

C jaH j a

a family F of -semistable sheaves, and if is a general smooth curve and ,

S C C

then restricting F to produces a family of generically -semistable sheaves on

S M S M C

(cf. Chapter 7) and therefore a rational map C from to the moduli space of

C L M C

semistable sheaves on the curve . The ample line bundle on pulls back to a power

u c

of F , and in this manner we can produce sections in the latter line bundle. In detail:

C X jaH j w K X

Let i be a smooth curve in the linear system . For any class ,

w j i w K C cj C

let C be the induced class in . In particular, is completely determined

r Qj P P cj P n C

by its rank and the restriction C . Clearly, is also given by

P m

m H O m P c n P c n a

. Let be a large positive integer, C , and

Q Quot H P Qj C

let C be the closed subset of quotients with determinant . More-

H F O L

over, let Q be the universal quotient and consider the line bundle

Q m u cj

C C

e on . If is suf®ciently large the following holds:

q H E Q

1. Given a point C , the following assertions are equivalent :

V H E m

1.1. E is a (semi)stable sheaf and is an isomorphism.

q Q SLP m

1.2. is a (semi)stable point in C for the action of with respect to

the canonical linearization of L .

SLP m L

1.3. There is an integer and a -invariant section in such that

q H E i i

2. Two points i , are separated by invariant sections in some

L E E

tensor power of , if and only if either both are semistable points but and are

not S-equivalent or one of them is semistable but the other is not.

S X

Suppose now that F is an -¯at family of -semistable torsion free sheaves on . The as-

F s S F F j

S C sumption that s is torsion free for all implies that the restriction is

still S -¯at (Lemma 2.1.4) and that there is an exact sequence

F O a F F

X (8.1)

Increasing m if necessary we can assume that in addition to the assertions 1 and 2 above

we also have:

F m s S

3. s is -regular for all .

m p F m O P S S S

Then is a locally free -sheaf of rank . Let be the associated

O H F O

projective frame bundle. It parametrizes a quotient e which in turn

SLP m S Q G C

induces a -invariant morphism F .If is an algebraic group acting

S F G

on S and if carries a linearization with respect to this action, then inherits a -action

SL G SL

which commutes with the -action such that and F are both equivariant for .

Before we go on, we need to compare certain determinant line bundles. Consider the fol-

K X

lowing element in num :

188 8 Line Bundles on the Moduli Space

w c hO c O h

C C

O ah h K X

As C we have

w w a w O c hO O c O hO

C C C C C

a ch h c h h

a v c a deg X u c

and

w j cj hj cj hj v cj a degX u cj

C C C C C C C

From the short exact sequence (8.1) we get

w a w w

F F F

and

a degX a degX

w w a u c u cj

F F C F (8.2)

Returning to the situation

Q S

above we get:

degC

L v cj v cj

C C

F O

by 8.1.2 ii)

F F

v cj

F C

by 8.1.2 iv)

v cj

F by 8.1.2 ii)

a deg X

u c

F by (8.2)

degC

SL L G SL

Assume now that is an -invariant section in . Then is a -

a degX

G u c invariant section and therefore descends to a -invariant section in F .In

this way we get a linear map

G SL

a degX degC

S u c H Q L s H

F C F

We conclude (cf. Theorem 4.3.3 and De®nition 4.2.9):

Lemma 8.2.4 Ð

s S F j C

1. If is a point such that s is semistable then there is an integer and a

G u c s -invariant section in F such that .

8.2 A Moduli Space for -Semistable Sheaves 189

j j s s S F F

C C s s

2. If and are two points in such that either and are both semistable

but not S-equivalent or one of them is semistable and the other is not, then there are

G v c s s 2

-equivariant sections in some tensor power of F that separate and .

Proposition 8.2.3 now follows trivially from the ®rst part of the lemma: Just apply it to

R G SLV 2

the case S and .

If N is generated by invariant sections, we can also ®nd a ®nite dimensional subspace

ss SLV ss

W W H R N N R PW W

that generates . Let be the

P m

induced SL -invariant morphism. We claim that

M R

W W

is a projective scheme. In fact, one has the following general result:

k R T Proposition 8.2.5 Ð If T is a separated scheme of ®nite type over , and if

is any invariant morphism, then the image of is proper.

t R

Proof. This is a direct consequence of Langton's Theorem: let be a closed

K f

point. Then there is a discrete valuation ring A with quotient ®eld and a morphism

Sp ecA T t t

that maps the closed point to and the generic point to a point in the

y t k t k y

image of . Let be a closed point in the ®bre, then is a ®nite

K K k y K

extension, and there is a ®nite extension ®eld of and a homomorphism

such that

K k y

K k t

K A

commutes. Let A be a discrete valuation ring that dominates . Geometrically,

k y K g Sp ecK Sp eck y R

corresponds to a morphism and

K H F q K

thus to a quotient K .

Sp ec K R

j j

j Sp ecA j

Sp ecK Sp ec A T

F A F A

According to Langton's Theorem 2.B.1, the family K extends to an -¯at family of

A p F m A A

-semistable sheaves. Since is a local ring and therefore a free -module

A H F P m q f Sp ecA R A

of rank , there is a quotient A . Let

F K F K q q

K A K

be the induced morphism. Since A , the quotients and differ

SLV K f j

by an element in . But is an invariant morphism, so that Sp ec

g f j Sp ec A Sp ecA T

Sp ec , where is the natural projection. Since

f f Sp ecA

is separated, we have . Thus if is the closed point in , we see that

190 8 Line Bundles on the Moduli Space

t f f f

and is therefore contained in the image of .

N W

Proposition 8.2.6 Ð There is an integer such that N is a ®nitely generated

graded ring.

Proof. Let be an integer such that is generated by a ®nite dimensional subspace

W W d W W W W d

. For let be the image of the multiplication map ,

d d

W W W W W

and let d be a ®nite dimensional space containing . Then and generate

M N M

W W W W

W W W

and there is a ®nite morphism W such that

d O O

M M

and . Moreover, there are inclusions

W W

O W H M O d H M

d W W

H M O d H M O

W W W

and W is an isomorphism, if and only if . Clearly,

fM g R

W W

the projective system W has a limit since it is dominated by . If the limit

M W W H M O k W k 2

N W k N is isomorphic to, say, W with , then for all .

De®nition 8.2.7 Ð Suppose that N is a positive integer as in the proposition above. Let

ss ss

M c

M be the projective scheme

ss k N SLP m

Pro j H R N

ss ss

R M and let be the canonically induced morphism.

This resembles very much the GIT construction of Chapter 4. The main difference is that

ss ss

M R

N is not ample. And indeed, will in general not be a categorical quotient of .

ss ss M

Still, M has a certain universal property. Namely, let denote the functor which as-

sociates to S the set of isomorphism classes of -¯at families of torsion free -semistable

ss ss

X M M

sheaves of class c on . It is easy to construct a natural transformation

with the property that for any S -¯at family of -semistable sheaves and classifying mor-

ss N

S M O u c

M F F

phism F the pull-back of via is isomorphic to .

M O N

Furthermore, the triple is uniquely characterized by this property up to

O N O d dN

unique isomorphism and replacing by some multiple . In particular,

ss m the construction of M does not depend on the choice of the integer . We omit the details.

De®nition and Theorem 8.2.8 Ð Because of the universal property of M the functor mor-

M phism M induces a morphism

8.2 A Moduli Space for -Semistable Sheaves 191

M M

L O 2

such that .

In order to understand the geometry of M and the morphism better, we need to study

ss ss ss

R M R the morphism in greater detail and see which points in are separated

by and which are not. The ultimate aim of this section is to show that at least pointwise ss

M can be identi®ed with the Donaldson-Uhlenbeck compacti®cation. See also Remark

8.2.17.

M O Hilb X

Example 8.2.9 Ð Recall that X . According to the calculations

g L L g Hilb X S X

in Example 8.2.1, there is an isomorphism , where

S X

is the morphism constructed in 4.3.6 and L is an ample line bundle on . It follows

Hilb X L H S X L

from Zariski's Main Theorem that H . This leads to a

M O S X

complete description of the morphism in this particular case: X

and .

X g r F

De®nition 8.2.10 Ð Let F be a -semistable torsion free sheaf on . Let be the F

graded object associated to a -Jordan-HÈolder®ltration of with torsion free factors. Then

F F g r F

g r is torsion free. Let denote the double dual of : it is a -polystable locally

l X N x F g r F

free sheaf, and let F be the function , which can be con-

S X l c F c F F

sidered as an element in the symmetric product with . Both

l F

and F are well-de®ned invariants of , i.e. do not depend on the choice of the -Jordan-

HÈolder ®ltration (cf. 1.6.10).

F F r

Theorem 8.2.11 Ð Let and be two -semistable sheaves of rank and ®xed Chern

c c H X F F M

classes . Then and de®ne the same closed point in if and only

F l F l

if and F .

g r F

Proof. One direction is easy to prove: if F is -semistable, and if is the torsion F

free graded object associated to an appropriate -Jordan HÈolder ®ltration of , then we can

g r F F F P F F

construct a ¯at family parametrized by such that and for all

t P M P

. Hence the induced classifying morphism F maps to a single

F g r F M

point. This means that in . We may therefore restrict ourselves to -

E F

polystable sheaves: let F be -polystable torsion free, and let be its double dual.

F y QuotE c F

Then is (non-uniquely)represented by a closed point in , where

F c E F F

. Any other -polystable torsion free sheaf satis®es the conditions

l l F y QuotE y F

and F if and only if is represented by a closed point in , such that

QuotE S X and y lie in the same ®bre of the morphism . But any such ®bre

is connected, and as we saw in Example 8.2.1, the restriction of N to a ®bre is trivial. This

j QuotE

means that any ®bre of is contracted to a single point by the morphism

ss F M associated to the family . This proves the `if'± direction of the theorem. The `only if'± direction is done in two steps:

192 8 Line Bundles on the Moduli Space

F F X a

Lemma 8.2.12 Ð Let and be -semistable sheaves on . If is a suf®ciently large

C jaH j F j F j

C C

integer and a general smooth curve, then and are S-equivalent if and

F F

only if .

g r F F Proof. Let be the graded object of a -Jordan-HÈolder ®ltration of with tor-

sion free factors. Using the theorems of Mehta-Ramanathan 7.2.8 or Bogomolov 7.3.5 we

can choose a so large that the restriction of any summand of to any smooth curve in

aH j C

j is stable again. Now choose in such a way that it avoids the ®nite set of all singu-

F j g r F g r F j

C C

lar points of . Then is the graded object of a Jordan-HÈolder

F j C F j

C C

®ltration of . This shows that for a general curve of suf®ciently high degree

F j F j F j a i

C C C

and are S-equivalent if and only if . For and we have

F F C F F

Ext (and the same with the rÃoles of and exchanged), so that

Hom F F F j F j Hom F j F j

X C C C C

C . This means that if and only

F 2

if F .

F R F F F

In particular, if then any two points in representing and can

be separated by invariant sections in some tensor power of N by the second part of Lemma

F F F F

8.2.4. The most dif®cult case therefore is that of two sheaves and with

E l l c F c E x l

F F i

but . Let F . We have already seen that the

E S X j QuotE

®bres of the morphism Quot are contracted to points by

ss ss

M S X j j S X M

. As is normal, Quot factors through a morphism . r ed

Clearly, the proof of the theorem is complete if we can show the following proposition:

S X M Proposition 8.2.13 Ð The morphism is a closed immersion.

Without further effort, just using what we have proved so far, we can at least state the

O following: as M is ample by Example 8.2.2, must be ®nite. Moreover, using

Lemma 8.2.4 and Bogomolov's Restriction Theorem 7.3.5, one can show that j separates

QuotE T T

points s s if the corresponding zero-dimensional sheaves have set- theoretically distinct support. Hence is, generically, an embedding. This does not quite

suf®ce to prove the proposition. The path to the proof begins with a detour:

pr X X i L

Let i be the projection onto the -th factor. If is an arbitrary line bundle

X pr L S

on , then i has a natural linearization for the action of the symmetric group and

L S X descends to a line bundle on . If are global sections, we can form the

symmetrized tensor

C L L C

which descends to a section of .If is a curve de®ned by a section in , let

denote the Cartier divisor on S given by . It is easy to see that if runs through

H S X L

an open subset of section in L then the corresponding sections span .

L Furthermore, if L is ample, then is ample as well.

8.2 A Moduli Space for -Semistable Sheaves 193

S X

Lemma 8.2.14 Ð i) Let T be an -¯at family of zero-dimensional sheaves on of length

S S X C X

, inducing a classifying morphism T . Let be a smooth curve and

p S X S T O C T T j

let be the projection. The exact sequence S

p T C p T induces a homomorphism between locally free sheaves of rank .

Then

det g C

f (8.3)

S T C s S

ii) Moreover, if is integral, and if s for some (and hence general) , then

p T j S

S C

is a torsion sheaf on of projective dimension 1. If

A B p T j

S C B is any resolution by locally free sheaves A and of necessarily the same rank, then (8.3)

holds for .

Drap S

Proof. i) Let denote the relative ¯ag scheme (cf. 2.A.1) of all full ¯ags

F T F T T s S

The factors of the universal ¯ag parametrized by Drap have length one and induce a mor-

Drap X

phism T so that the diagram

X Drap

S S X

Drap S

commutes. As S is the scheme-theoretic image of , it suf®ces to prove

(8.3) for instead of . Now has diagonal form with respect to the ®ltra-

p F T C p F T p T C p T i

tions and of and , respectively. Hence if ,

det det

, are the induced maps on the factors, we have

P Q

pr C C det i

i . As , it suf®ces to show (8.3) for each instead of

i i

S X

, i.e. for the case . But this case can immediately be reduced to the case ,

T O

, when the assertion is obvious.

p T j

S C

ii) It is clear that under the given assumptions is a torsion sheaf. Hence, the

p T C p T

homomorphism is generically isomorphic and therefore injective every-

p T j

S C

where, so that indeed has projective dimension 1. It is a matter of local commuta-

tive algebra to see that the Cartier divisor de®ned by det is independent of the resolution.

Approaching our original goal, let E be a locally free sheaf and consider the variety

E F T

Quot parametrizing a tautological families and that ®t into an exact sequence

F O E T S

194 8 Line Bundles on the Moduli Space

QuotE S X T C jaH j

Let T be the morphism associated to . Let be an

G X H F

arbitrary smooth curve, and let be a locally free sheaf on with the property that s

Gj H E Gj s S C

C for all . Then there is a short exact sequence

p F Gj p O E Gj p T Gj

C S C C

As the conditions of part ii) of the lemma are satis®ed, we get

div det C rkG C

GT T

K X Now locally free sheaves G of the type above span . Hence by linearity we get the

following result:

w K X F O

Lemma 8.2.15 Ð For any the following holds: the homomorphism S

E induces a rational homomorphism

w O w

O E j F j

S C S C

rkw C rkw 2

with div , i.e. has zeros or poles depending on the sign of .

r Q

Proof of Proposition 8.2.13. E is now a -polystable sheaf of rank and determinant ,

c E c c a C jaH j

and . If is suf®ciently large, and if is an arbitrary smooth

E j

curve, then C is again polystable by Bogomolov's Restriction Theorem 7.3.5. The two

F E O E S QuotE S

families and S on induce homomorphisms

a degX a degX

H S O u E S H Q L H s

S E C E

S S

and

a degX a degX

S u c Q L H s H

F C F

C S

On the complement U of in the two line bundles on the right hand side are iso-

j s j s

U F U

morphic and E for any invariant section . Moreover, the rational ho-

s s E j

E C

momorphism maps F to . Since is polystable, there is an integer and

a degX SL

H Q L s s

C E F

a section such that . Therefore, must

have zeros of precisely the same order as the poles of up to an additional factor

a deg X s n r

. Hence, Lemma 8.2.15 says that the vanishing divisor of F equals

C O M

. We ®nally conclude: induces a section in some tensor power of

C S X

such that the vanishing divisor associated to on is a multiple of . But we

have seen before hat these divisors span a very ample linear system as C runs through all

aH j a 2

smooth curves in the linear system j for suf®ciently large . Hence is an embedding.

ss lf

M M M M

Corollary 8.2.16 Ð embeds the open subscheme of -stable

dim M dimM

locally free sheaves. In particular, d and

h M L

8.3 The Canonical Class of the Moduli Space 195

j lf

Proof. Theorem 8.2.11 implies that M is injective. But in fact, the proof of Lemma

C C

8.2.12 shows that M embeds into the moduli space of stable sheaves on , where is

jaH j C j lf

any smooth curve in for suf®ciently large , which implies that M is an embed-

O 2

ding. The second assertion is clear, as M is ample.

p oly

M r Q c M r Q c Remark 8.2.17 Ð Let denote the subset representing

-polystable locally free sheaves. The previous results can be interpreted as follows: set-

theoretically, there is a strati®cation

ss p oly

M r Q c M r Q c S X

We will brie¯y indicate how this is related to gauge theory: In order to study differentiable

structures on a simply connected real 4-dimensional smooth manifold N , Donaldson in-

asd

M SU c

troduced moduli spaces of irreducible antiselfdual -connections in a

C c N

-complex vector bundle with second Chern class on , equipped with a Rieman-

C nian metric. He proved that if N is the underlying -manifold of a smooth complex pro-

jective surface X with the Hodge metric, then there is an analytic isomorphism between

asd

c M c M X

and the moduli space of -stable locally free sheaves on

X asd

of rank 2 and the given Chern classes. In general, the space M is not compact. As Don-

aldson pointed out, results of Uhlenbeck can be interpreted as follows: the disjoint union

asd

M c S N

can be given a natural topology which makes the disjoint union a compact space and induces asd

the given topology on each stratum. The closure of M in this union is called the Donald-

son-Uhlenbeck compacti®cation. Li [148] and Morgan [180] show that there is a homeo-

lf asd

asd

M M M M morphism extending the analytic isomorphism constructed by Donaldson. For more information on the relation to gauge theory, see the books of Donaldson and Kronheimer [46] and Friedman and Morgan [71] and the references given there.

8.3 The Canonical Class of the Moduli Space

M M r Q c F r

Let be the open subscheme of stable sheaves with rank , determi-

Q c Ext F F M

nant , second Chern class and . According to Theorem 4.5.4, is

r O M

smooth of expected dimension X . We will later see that is dense in

M r Q c

for suf®ciently large discriminant and that the complement has large codi- M

mension. The purpose of this section is to relate the canonical bundle of to the line bun- L

dle studied in the last section. The main technical tool here is the Grothendieck-Riemann-

K X S Roch formula. It states that for any class one has

196 8 Line Bundles on the Moduli Space

chp p ch q tdX CH S

S X S q S X X

(Recall that p and are the projections.)

S X F Let F be an -¯at family of sheaves on . Then there is a bounded complex of locally

free sheaves which is quasi-isomorphic to F , and

i i

E xt F F E xt F F p F F

K S CH S i is an element in . If Q , let denote the homogeneous component of

of degree i.

S X r Q

Proposition 8.3.1 Ð Let F be an -¯at family of sheaves on of rank , determinant

c c

and Chern classes and . Let

F r c F r c F CH S X

F CH S

denote the discriminant of the family . Then the following equations hold in Q .

fp F q K g c E xt F F

X p

i) .

c u fp F q H g

ii) .

Proof. Both results are direct applications of the Grothendieck-Riemann-Roch formula.

i) By Grothendieck-Riemann-Roch we have

chF q tdX F p chF c E xt F F c p F

As these Chern class calculations are purely formal, we can use the identity (3.4) on page

72 and write

F r F chF r c F chF

where the dots indicate terms of degree . On the other hand

K c X c X tdX

chF q tdX chF F q K

Hence the only term of degree 3 in is X , and terms

of other degrees do not contribute to the left hand side of the equation in i).

r r

H H K u r h c h O c h c H

X x

ii) By de®nition, . Since

one gets

chu r chh c h chO

r r

r H H c H H H K

r H c H H K

8.3 The Canonical Class of the Moduli Space 197

chu tdX r H c H F

and therefore . The assumption that the family has ®bre-

detF p S q Q S S

wise determinant Q implies that for some line bundle on , so that

c F p c S q c Q p s q c

. Now

c u c p F q u fp chF q chu tdX g

chF q chu tdX F q H

After expansion of and cancellation of most

terms the only thing left is

c F c F q c q H p s q c q H p s q H

Integration of this term along the ®bres of p gives 0, as asserted.

K H

As an immediate consequence of the proposition we see that if X and are linearly

Q K H PicX Q

dependent over , i.e. if X , then, under the hypotheses of the

c E xt F F c u

p F proposition, one also has . We can reduce to the following

cases:

K X

1. X is ample, i.e. is a Del Pezzo surface.

2. is a minimal surface of Kodaira dimension 0.

K X

3. X is ample, i.e. is a minimal surface of general type without -

curves.

M M M r Q c F F

Let be the open subset of points where is a stable sheaf

Ext F F R M

with , and let be the pre-image of under the quotient morphism

F R M R X

. Moreover, let denote the universal family on . Then

e e

detE xt F F K

Theorem 8.3.2 Ð M .

This is a direct consequence of Theorem 10.2.1. Here, we must appeal to the patience of

the reader. 2

X H K H

Theorem 8.3.3 Ð Let be a polarized projective surface with X ,

L K

or . Then M modulo torsion line bundles.

Proof. It follows from the discussion above and the theorem, that

e e

detE xt F F u u L K

p M

modulo torsion line bundles on . As is injective (4.2.16), the assertion of the theorem

follows. 2

Note that we can state the isomorphism of the theorem only up to torsion line bundles,

CH R Q because the Chern class computations above were carried out in . 198 8 Line Bundles on the Moduli Space

Combining Theorem 8.3.3 and Corollary 8.2.16 we see that the canonical bundle on the

moduli space of -stable vector bundles is anti-ample for Del Pezzo surfaces, is torsion for minimal surfaces of Kodaira dimension zero, and is ample for surfaces with ample canonical bundle. This gives strong evidence that the moduli spaces of higher rank sheaves detect the place of the surface in the Enriques classi®cation.

Comments:

Ð The homomorphism was introduced by Le Potier [144]. Theorem 8.1.5 is taken from that

L N

paper. Le Potier also shows that is globally generated for suf®cient divisible , and that the in-

duced morphism N separates the open part of -stable locally free sheaves from its complement

jL j in the moduli space. His approach is a generalization of [52]. The comparison with the Donaldson-

Uhlenbeck compacti®cation in the case of rank two sheaves with trivial determinant was done by J. L

Li [148]. The line bundle used by Li can be compared to by the following lemma:

g c

c r C jk H j Pic C C

Lemma 8.3.4 Ð If and , then for any smooth curve and

K X E L u

one has as classes in . In particular, if a universal family exists, then

detp E q C .

Ð The construction of the `moduli space' of -semistable sheaves is essentially contained in J.

Li's paper [148]. The proof of 8.2.11 is a mixture of methods from [144] and [148], though in order to

jaH j

prove an equivalent of 8.2.13, Li varies the curve C and uses relative moduli spaces for one-

F QuotE dimensional families of curves, instead of varying in as we did in the proof presented

in these notes. One should also mention that both approaches of Le Potier and Li were motivated by

M M Donaldson's non-vanishing result [47]. Li also shows in [148] that the image of is homeomorphic to the Donaldson-Uhlenbeck compacti®cation in gauge theory. For this see also the

work of Morgan [180].

q X Ð The surjectivity of the map (Theorem 8.1.6) for can be deduced from Li's results in

[151] in the case of rank two sheaves. He developes a more general technique to produce line bundles

X X by starting with the K -group of the product . His proof relies on the computation of the second cohomology of the moduli space via gauge theory [150]. It would be nice to have an algebraic argument of this part. In the description of the Picard group of the moduli space of curves the same problem arises. In order to show the surjectivity of a natural map to the Picard group one uses transcendental information about the second cohomology. Ð For information related to Lemma 8.2.14 see the paper of Knudson and Mumford [126]. Ð For details about the Grothendieck-Riemann-Roch formula see the book of Fulton [73]. Ð The identi®cation (8.3.1, 8.3.3) of the canonical class of the `good' part of the moduli space in the rank two case was done in [149] and [112]. 199

9 Irreducibility and Smoothness

For small discriminants, moduli spaces of semistable sheaves can look rather wild: their dimension need not be the expected one, they need not be irreducible nor need they be reduced let alone non-singular. This changes if the discriminant increases: the moduli spaces become irreducible, if we ®x the determinant, normal, of expected dimension, and the codimension

of the locus of points which are singular or represent -unstable sheaves increases. This behaviour is the subject of the present chapter. The results in this chapter are due to Gieseker,

Li and O'Grady. Our main source for the presentation is O'Grady's article [208].

H X K

Let X be a smooth projective surface, a very ample divisor on , and a canonical

O O H X divisor. We write X for the corresponding line bundle.

9.1 Preparations

r Q PicX c c Q c

Fix a rank , a line bundle and Chern classes . Let

r c r c P

and be the associated discriminant and Hilbert polynomial, respec-

M M M r Q c

tively. Let be short for the moduli space X . By the Bogomolov

Inequality 3.4.1 M is empty, unless , as we will assume from now on. Recall

some elements of the construction of M in Section 4.3: there is an integer such

P m

H k O m R Quot H P

that the following holds: Let X and let be the

H F F

locally closed subscheme consisting of those quotients q where is semistable,

H H m H F m detF Q

is an isomorphism, and . Then there is a morphism

R M M SLP m R such that is agoodquotientforthe -action on . (These notations

differ slightly from those in Chapter 4 as we have ®xed the determinant!)

R e R H

Let e be a nonnegative real number. Let be the closed subset in of quotients

F e e

, where F is -unstable. (For -stability see 3.A). This set is certainly invariant under the

e R e

group action, so that M is closed as well.

B r H X

Theorem 9.1.1 Ð There is a constant B such that

dim R e de endH

dim M e de r

r KH

r e e B e

with d .

r jH j

200 9 Irreducibility and Smoothness

F F R e H

Proof. Let be a semistable sheaf with . Then there is a ®ltration

F H H e F H H H

such that is an -destabilizing submodule of

H H F F ejH jrkF H H H H

, i.e. , and such that is the

F F y

Harder-Narasimhan ®ltration of . This ®ltration de®nes a point in the ¯ag-scheme

Y DrapH P P P H H i

i i i

, with for . There is a natural morphism

f Y R H R e

given by forgetting all of the ¯ag except , and is theunionoftheimages

of all Y appearing in this way. By Grothendieck's Lemma 1.7.9 the number of such ¯ag-

schemes is ®nite. In order to bound the dimension of R it is therefore enough to bound

Y Ext the dimension of . By Proposition 2.A.12 and the de®nition of the groups in 2.A.3

one has

dimY ext H H endH ext H H

R e R

The estimate for dim follows from this and Proposition 3.A.2. Any ®bre of

endH M contains a closed orbit whose dimension is given by the difference of and the

dimension of the stabilizer of a polystable sheaf of rank r . The dimension of this stabilizer

F M dim F endH r

is bounded by r . Hence for any point one has ,

M e dim R e end H r 2

and therefore dim . This proves the second claim.

F M

Recall (cf. 4.5.8) that there is a number such that for any point one has

dimension bounds

r O dim M r O

X X

Using the theorem above we can, at least for suf®ciently large discriminant , exclude the

possibility of irreducible components in M which parametrize semistable sheaves which

R M R are not -stable. Let and denote the open subschemes of -stable sheaves in and

M , respectively.

B r O R M

Theorem 9.1.2 Ð If X , then and are dense

R M dim Z r O

in and , respectively. In particular, X for all irreducible

Z M co dimM nM M r O B

components of . Moreover, X .

R R

Proof. By de®nition, R . The assumption of the theorem and the dimension

bound for R of Theorem 9.1.1 give:

dim R d endH endH B

r O endH

By Proposition 4.5.9 for any point one has

r O endH dim R

R R

Therefore the -unstable locus in is of smaller dimension than any componentof , which

R M M

means that R is dense in . Hence is dense in , too, and the remaining two esti-

dim M exp dim M r O

mates of the theorem follow from X and

B r M 2

dim .

r 9.2 The Boundary 201

9.2 The Boundary

r X S Let F be a ¯at family of torsion free sheaves of rank on parametrized by a scheme .

The boundary of S by de®nition is the set

S fs S jF g

s is not locally free

S S co dim S S r

Lemma 9.2.1 Ð S is a closed subset of , and if , then .

L F L S X

Proof. Choose an epimorphism with a locally free sheaf on of constant

L p p F q O n q O n n

rank . For example, for would do. Then the

L S S X r

kernel is -¯at and ®brewise locally free, hence locally free on of rank .

L L F x X F

If denotes the homomorphism , then: is locally free at is locally

s x s x Y s x F

free at has rank . Hence the set of points where is not locally

free can be endowed with a closed subscheme structure given by the -minors of ,

Y S X r and by the dimension bounds for determinantal varieties one gets co dim

(cf. [4] Ch. II). Since ®brewise F is torsion free and therefore locally free outside a zero-

S S dimensional subscheme, the projection Y is ®nite with set-theoretic image . This

proves the lemma. 2

We want to extend the de®nition of the boundary to subsets of M , though in general there

is no universal family which could be used. Consider the good quotient morphism

M M M Z Z U U M

. If Z is a locally closed subset, say for some open ,

Z U Z Z

then is closed in , and is an invariant closed subset in

U Z Z Z

and . This implies that is a closed subset in . Consider the

fF Z jF g Z Z

boundary of the open subset Z is -stable . If , then

Z Z r R M

co dim . For is a principal bundle at stable points, so that

the estimate of the lemma carries over to Z .

r r r r r

In the following we will need the polynomial . The particular values of its coef®cients are not really interesting unless one wants to do speci®c

calculations in which case they could most likely be improved.

A C C r O H H K

Theorem 9.2.2 Ð There are constants , , depending on , and

K A Z M

such that if and if is an irreducible closed subset with

dim Z C C

then Z .

The proof of this theorem will be given in Section 9.5. 202 9 Irreducibility and Smoothness

9.3 Generic Smoothness

For any coherent sheaf F let

F ext F F homF F K

where the subscript 0 indicates the subspace of traceless extension classes and homomor-

Z F Z minf F js Z g

phisms, respectively. If parametrizes a family , let s , which

Z Z F

is the generic value of on if is irreducible. If is -semistable torsion free then we

have the uniform bound (cf. 4.5.8).

F F

De®nition 9.3.1 Ð A sheaf F is good, if is -stable and . It is clear from Corollary 4.5.4 that at good points the moduli space M is smooth of the expected dimension. If we want to bound the dimension of the locus of sheaves which are not good, then half of the problem is solved by Theorem 9.1.1. For the other half consider

the closed set

W fF M j F g R

(As before one ought to de®ne W as the image of the corresponding closed subset in ).

C C r X H

Theorem 9.3.2 Ð There is a constant depending on , such that for all

C dim W C

Again we postpone the proof to a later section (see 9.6) and derive some consequences

®rst: Suppose that satis®es the following conditions:

r O B r

2. X

r O C C

r Then we can apply Theorems 9.1.2 and 9.3.2 and conclude that the points in M which are not good form a closed subset of codimension at least 2. This leads to the following

result:

A r X H A

Theorem 9.3.3 Ð There is a constant depending on such that if then 1. Every irreducible component of M contains good points. In particular,it is generically smooth and has the expected dimension.

9.4 Irreducibility 203

2. M is normal and is a local complete intersection.

A A

Proof. Choose such that for the conditions (1)±(3) are simultaneously satis- R ®ed. Then the good points are dense in R . Hence is generically smooth and has the ex-

pected dimension. By Proposition 2.2.8 R is a local complete intersection. Moreover, the

R S

singular points have large codimension. Hence satis®es the condition and is normal M

by the Serre-Criterion ([98] II 8.23). As a GIT-quotient of R , is normal, too. It follows

from Luna's Etale Slice Theorem 4.2.12, that M is a local complete intersection if this 2 holds for R .

9.4 Irreducibility

A F M F

Assume now that , and let be a good point. Let be the kernel

k p p X F k p

of any surjection F , where is a point at which is locally free and

F HomF F K HomF F

is the structure sheaf of p. Then is -stable and

F F

K , implying that is again good. In particular, is contained in a single irreducible

r F

component of M , and this component does not change if or the

k p

morphism F vary in connected families. This proves the following lemma:

Lemma 9.4.1 Ð If denotes the set of irreducible components of , then sending

F F 2

to induces a well-de®ned map .

Our aim is to show that for suf®ciently large the map is surjective and that this implies

that consists of a single point.

A A

Theorem 9.4.2 Ð There is a constant such that for all the following holds:

F 1. Every irreducible component of M contains a point which represents a good lo-

cally free sheaf F .

F F F

2. Every irreducible component of M contains a point such that both and

F F

are good and .

This is a re®nement of Theorem 9.3.3. Its proof uses the same techniques as the proof of Theorem 9.3.2 and will also be given in Section 9.6.

Now we have collected enough machinery to prove

A A M Theorem 9.4.3 Ð There is a constant such that for all the moduli space is irreducible.

204 9 Irreducibility and Smoothness

Proof. Clearly part 2 of Theorem 9.4.2 implies that the map is surjec-

A Z F Z F F

tive for . For if , pick a good point with such that

F Z

F is good. Then the component containing is mapped to .

E E

Hence it suf®ces to show, that if are any two good locally free sheaves, then

some power will map their components to the same point in . This, together with

the surjectivity of and the ®niteness of , implies that contains only one point for

E m E m

suf®ciently large . Now and are globally generated for suf®ciently large

m. Choosing generic global sections one ®nds exact sequences (cf. 5.0.1)

O m E Q I

i Z

Q Q O r m I Z X I I

Z Z Z

with and zero-dimensionalsubschemes i . Let

F E i

and de®ne sheaves i by

O m E Q I

i Z

O m F Q I

i Z

F F M r Z Z

and are good points in for and determine the images

E E

of the components of and under the map . The open subset in

PExt Q I O m

F F

that parametrizes good points is nonempty, for it contains the extensions de®ning and ,

F F M

and is certainly irreducible. This forces and to lie in the same component of

2 r .

9.5 Proof of Theorem 9.2.2

C jnH j M

Proposition 9.5.1 Ð Let be a smooth curve and let C be the moduli space of

C r Qj Z M

semistable sheaves on of rank and determinant C . Let be a closed irreducible

Z dim Z dim M F Z F j C subvariety with . If C , then there is a point such that

is not stable.

F j F Z

Proof. Assume to the contrary that C is stable for all . Then the restriction

F F j Z M C

C de®nes a morphism . By equation 8.2 in Section 8.2 we know

n degX

L j L L M

, where is an ample line bundle on C (cf. 8.1.12).

L M n M

Moreover, sections in some high power of de®ne an embedding of (cf. 8.2.16).

Z M n M L j Z

By assumption . Hence the line bundle is ample. Thus is ®nite and

dimZ dim M 2 C

9.5 Proof of Theorem 9.2.2 205

g F

Proposition 9.5.2 Ð Let C be a smooth connected curve of genus , and let bea¯at

C k S family of locally free sheaves of rank r on parametrized by a -scheme of ®nite type.

Then the closed set

S fs S jF g

s is not geometrically stable

g S

is empty, or has codimension in .

As the moduli space need not be ®ne, the proposition cannot be applied to the moduli M

space C itself. Indeed, the codimension of the subset parametrizing properly semistable

sheaves can be larger than predicted by the proposition.

F F s S m regF

Proof. Suppose s is not stable for some closed point . Let be

F G O m H F m F

the regularity of and C the evaluation map. There

s S U Quot G P F s

is an open neighbourhood U of in and a morphism mapping

F j

to such that U is the pullback of the universal quotient. Hence, it suf®ces to prove the

proposition for the following `universal example': F is the universal family parametrized by

QuotG P F G F F

the open subset S corresponding to all points such that is

H Gm H F m

an m-regular locally free sheaf and induces an isomorphism .

d F d r r r r r r

Let be the degree of . For any pair of integers with let ,

d d d P m r m d r g

i i i i

and let denote the corresponding Hilbert

P P G P F D d r

polynomial. Finally, let . The relative Quot-scheme

DrapG P P P QuotF P

is an open subset of the ¯ag-scheme . Consider the can-

D d r S

onical projection .Theimageof is precisely the closed subset of points

s S F d r y D d r

in such that s has a submodule of degree and rank . A point cor-

G G G G s y

responds to a ®ltration , and then corresponds to the

G GG F F G G F

i i

quotient . Let be the induced ®ltration of . The smooth-

S Ext G F HomG G HomG F

ness obstruction for is contained in . As and

Ext G F i S Ext G F Ext F F

for by de®nition of , we have , S

since C is a curve. Thus is smooth. Moreover, there is an exact sequence

i i i i

Ext F F Ext G F Ext G G Ext F F

(9.1)

(We leave it as an exercise to the reader to establish this sequence. Recall the de®nition Ext

of the groups in Appendix 2.A and write down an appropriate short exact sequence

G F i

which leads to the desired sequence. Cf. [51]). Because of Ext for , we

i i

F F G G Ext Ext i

get for (Use the spectral sequences 2.A.4 and

G G dimC Ext D d r

). Now is the obstruction space for the smoothness of

D d r (cf. Proposition 2.A.12). Hence is smooth as well. By Proposition 2.2.7 there is

an exact sequence

HomF F F T D d r T S Ext F F F

y s

(9.2)

i i

Ext F F F Ext F F

In fact, it follows from , the long exact sequence

206 9 Irreducibility and Smoothness

i i i i

Ext F F Ext G G Ext G F Ext F F

(again, we leave it to the reader to establish this sequence) and the vanishing results listed above, that the last homomorphism in (9.2) is surjective. Using Riemann-Roch, this implies

that

d d

homF F co dim D d r S ext F F F r r g

r r

S D d r d r d r dr

Now is the union of all , where satis®es . Note that by the Grothendieck Lemma 1.7.9 there are only ®nitely many such ¯ag varieties which are

nonempty.

S V

Let V be an irreducible component of . Assume ®rst that a general point of corre-

F V D d r d r

sponds to a semistable sheaf . Then is the image of for a pair with

d r dr d r

. Hence

g co dimV r r g homF F F r r g

F F homF F F r r

Here we used that is also semistable and therefore . F

Assume now that a general point of V corresponds to a sheaf which is not semistable

d r F

and let denote degree and rank of the maximal destabilizing subsheaf of . Then

HomF F F D d r S

. Therefore, is, generically, a closed immersion with

d d

V co dimV S r r g

image of codimension . In case

r r

d r r d

r r r r

g co dimV B r r g r r

we get , and we are done. Hence it suf®ces

to show that the alternative relation

d r r d

r r r r

d r d r

is impossible. Otherwise, the slightly stronger inequality must

hold, since the involved degrees and ranks are integers.

F F k P P C

This means that the kernel of any surjection , , is still destabilizing.

D D d r V

Hence there is a component which surjects onto . The ®bre dimen-

D dim V F F F

sion of this morphism is greater than or equal to dim

F F F r r

by the Riemann-Roch formula. But the tangent space to the ®bre

V F F F D

of D at a point is given by

HomF F F HomF F F HomF k P

homF F F r

In order to get a contradiction it suf®ces to show . But this is equiv-

HomF F Ext O F F

alent to the claim that is not surjective, and, by Serre

HomF F F HomF F O

C P C

duality, that is not trivial. But this is

F k P 2

certainly true for an appropriate choice of .

9.5 Proof of Theorem 9.2.2 207

jnH j e

Proposition 9.5.3 Ð Let C be a smooth curve, let be a rational number and

e Z M

let d be the quantity de®ned in Theorem 9.1.1. Let be a closed irreducible subset

g C de r Z dim Z dim M dim Z

with , and suppose that C and . Then

F Z F e F j

there is a point such that is -stable and C is unstable.

dim Z dim M F Z

Proof. Since C , by Proposition 9.5.1 there is a point in such that

F j Z R Z

C is unstable. Let be an irreducible component of the pre-image under

R M Z

the morphism which maps onto . Then

g c dim Z de endH

Z Let denote the (nonempty!)closed subset corresponding to sheaves whose restriction

to C is unstable. Then by Proposition 9.5.2 and 9.1.1

dimZ de endH dim R e

H F Z F j F e

This implies that there is a point such that C is unstable and is -stable.

M C jnH j

Proposition 9.5.4 Ð Let Z be closed and irreducible. Let be a smooth

r Z F F e

curve and e . Suppose that contains a point such that is -stable but

jH j

r r

C C K F j dim Z exp dim M Z

C is unstable. If , then .

Proof. Assume to the contrary that Z so that all sheaves corresponding to points in

Z F j

are locally free. By assumption C is unstable, i.e. there is an exact sequence

F F j F

O F F F F E with locally free C -modules and with . Let be the kernel of the

composite homomorphism

F F j F

F X C E F

Since F and are locally free on and , respectively, is locally free, too. can be

E q E C j F O C

C C

recovered from and the homomorphism :

F E C F C

q Quot E C j P F C

corresponds to a closed point in the quotient scheme C . C Using Corollary 2.2.9, together with the notations introduced there, we can give a lower

bound for the dimension of :

dim F F C r r C g C

r r C C C K C C K

208 9 Irreducibility and Smoothness

F F q E C j G

(Recall that !) Let C be the universal quotient

C F X

on and de®ne a family of sheaves on by the exact sequence

F q E C i G

F i C X F F

where is the inclusion. Then is -¯at, q by construction and

F A F rkA r is -stable for all points . To see this let be a subsheaf of rank .

There are inclusions

A F E C F C

Hence the e-stability of implies:

r rkA ejH j

F CH F F A F CH

rkA rkA

The family F induces a morphism . Let be an irreducible component of the

®bre product M . Then

dim dim Z dim dim M

r r

C C K exp dim M dim M dim

Since Z , must also be empty. But for any point corresponding to a short

exact sequence

F E C G

F X G C

the sheaf is locally free on if and only if is locally free on . Hence, parame-

C C

trizes locally free sheaves of rank r on and thus induces a -morphism

C GrassE C j r

r rkF GrassE C j r C

where as above. C is a locally trivial ®bre bundle over with

r r

k r r r r dim

®bres isomorphic to Grass and of dimension . Since ,

for a ®xed point c the morphism

c GrassE C c r

cannot be ®nite. Let be a componentof a ®bre of of dimension . Then

j C GrassE C j r

C C

fcg

contracts the ®bre . The Rigidity Lemma (cf. [194] Prop. 6.1, p. 115) then forces

to contract all ®bres and to factorize through the projection onto . But this would mean

2 that all points in parametrize the same quotient, which is absurd.

9.5 Proof of Theorem 9.2.2 209

jnH j n We summarize the results: suppose C is a smooth curve in the linear system , .

Then

g C C C K n H nK H

and, by Corollary 4.5.5,

dim M r g C n H nK H

M What we have proved so far is the following: suppose Z is an irreducible closed subset

such that

dim Z n H nK H n

dim Z r r r H n

r r KH n r B n

r r

H n KH n dim Z

r O n

e r

(here , and are the constants of Propositions 9.5.3 and 9.5.4 for ,

jH j

expressed as functions of n). Then 9.5.3 and 9.5.4 together imply that . We need to

n n

analyze the growth relations between , and . First observe that for

all . Next, consider the `leading terms' of and :

r r r H n n

n H n

x x

Then the equation has the positive solution

r H

r r r r r x x

where . The quadratic polynomials and attain

their minimum and maximum value at

r r KH

r r r H

KH HK

x A r H x

and , respectively. Hence, if , then

H H

x x n bx c n

. Let . Then and

n x n x

210 9 Irreducibility and Smoothness

n A

as is in the range where is increasing and is decreasing. We conclude: if

dim Z maxf x x g Z x x

and then . Now express and

in terms of , using the de®nition of , and check that their maximum is not greater than

the constant given in Theorem 9.2.2. Therefore, if Z satis®es the assumptions of Theorem

Z Z R

9.2.2, then Z . Under the same assumptions let be an irreducible

component that dominates Z . Then and

Z dim Z r

dim by 9.2.1

dim Z end H r r

endH r r

B endH r r

B endH

dim R

by 9.1.1

Z Z 2 Hence, , and thus . This ®nishes the proof of the theorem.

9.6 Proof of Theorem 9.3.2

r X T F F T

Let F be a torsion free sheaf of rank on and let . Since is zero-dimensio-

F F F

nal, F and have the same rank and slope, is -stable if and only if is -stable;

F F r T E xt F O E xt T O

and . Note that is zero-dimensional

T F S

of length . Consider now a ¯at family of torsion free sheaves parametrized by and

T F F

s s

let s .

s T S T s Lemma 9.6.1 Ð The function s is semicontinuous. If is reduced and is

constant then forming the double dual commutes with base change and F is locally free.

L L F Proof. Choose a locally free resolution . Dualizing yields an

exact sequence

E xt F O L H omF O L

F E xt F O

This shows that commutes with base change and proves the semicon-

S T E xt F O S F

tinuity. If is reduced and s is constant then is -¯at. But then

H omF O 2 is also S -¯at and forming the dual commutes with base change.

The double-dual strati®cation of S by de®nition is given by the subsets

S fs S jT g

s 9.6 Proof of Theorem 9.3.2 211

These are closed according to the lemma.

M Z

Let Z be a closed irreducible subset and assume that is nonempty and

. De®ne a sequence of triples

Y Z M M i n

i i i i

Z Z Y Z

by the following procedure: , , is an irreducible component of

the maximal open stratum of the double-dual strati®cation of . Let i be the constant

Y F F Y M M

i i i

value of on i . Then sending to de®nes a morphism ,

r Z

i i i

where i . Finally, let be the closure of the image of this morphism.

n Y

This process breaks off, say at the index , when n . (It must certainly come to an end,

as i for all by the Bogomolov Inequality).

Remark 9.6.2 Ð Strictly speaking we have de®ned the double dual strati®cation only for

Z R Z schemes which parametrize ¯at families, i.e. on M rather than on itself. But

obviously the strati®cation is invariant under the group action on R and therefore projects

Z M Y R M

to a strati®cation on . Similarly, the morphism to etc. is de®ned ®rst on 2

but factors naturally through Y .

dim Z Z E Z

i i

How do i and change? Let be a general point. Then by construction

E is a -stable locally free sheaf. There is a classifying morphism

QuotE M

E T ker F

sending to . This is easily seen to be an injective morphism. If

Y Z F QuotF Y

i i i

i , then the ®bre of over is contained in the image of .

QuotE r i

But by Theorem 6.A.1 i is irreducible of dimension . In particular,

dim Z dim Y r

i i i

dim Z r r

i by 9.2.1

dim Z r

i i

and summing up:

N dim Z dim Z r i

n (9.3)

where N is the number of times that equality holds in

dim Z dim Z r

i i

i (9.4)

N E Z F Y F E

Togetaboundon , consider a general sheaf i and a sheaf with .

F F K HomE E K

Then Hom so that

Z E F Z

i i (9.5)

212 9 Irreducibility and Smoothness

QuotE red

What happens if equality holds in (9.4)? In this case i must be contained in

i , since it is an irreducible scheme. We claim that in this situation the strict inequality

F E F ker E T QuotE

holds for , when i is a general point:

E E K x

namely, let be a nontrivial traceless homomorphism. Then cannot be

x x X E T F

a multiple of the identity on E for all . Thus for a general the kernel

HomF F K F E

is not preserved by . Thus and . This argument shows

Z Z N that we can sharpen (9.5) each time that equality holds in (9.4): we get n .

This can be used to give a bound for N :

N N Z Z

(9.6)

r i

Recall that n . Using this, and the inequalities (9.3) and (9.6) we get:

dim Z dim Z

n (9.7)

r r

We are now ready to prove Theorem 9.3.2: de®ne

r O g C maxfC

W W

If Theorem 9.3.2 were false, let Z be an irreducible component of with

dim Z C C

W Z C C

By the de®nition of we also have . Since , Theorem 9.2.2 can be

applied to Z , so that . The procedure above leads to the construction of a closed

Z M Z n

irreducible subset n such that and such that estimate (9.7) holds. It

C Z n

suf®ces to show that was chosen large enough so that still satis®es the conditions of

Z Z

Theorem 9.2.2 and therefore provides the contradiction . Firstly, n parametrizes, n

generically, -stable sheaves. Hence

dim Z exp dim M r O

n n n X

This and (9.7) give:

r O r dim Z

X n n n

dim Z r O

n X

C C r O

A r

since . Secondly, again using the estimate (9.7):

9.6 Proof of Theorem 9.3.2 213

dim Z C C

n n n

C C dim Z

n n

C C

where all the terms on the right hand side are nonnegative by the assumption on dim and

C 2

the de®nition of .

Proof of Theorem 9.4.2

A A r Z M A

Let , and let be an irreducible component of for .

A A A A A

By the choice of the constants i we have . Thus, for Theorem

9.3.3 applies and says that Z has the expected dimension. Moreover, the conditions 1. ± 3. on

page 202 are satis®ed. Hence, Theorem 9.2.2 applies, and we can conclude that Z .

Let Y be an irreducible component of the maximal open stratum of the double-dual strati®-

Z T s Y Z

cation of , the constant value s , , and let be the closure of the image of the

M M r dim Z dim Y r morphism Y , , as above. Then . Distinguish the following two cases:

Case 1. Suppose Z contains no points corresponding to good locally free sheaves. Then Z

Y is an open dense subset of and it follows:

Z exp dim M r

dim (9.8)

exp dim M r

(9.9)

r A M dim Z

Either , then is generically good by Theorem 9.3.3, hence

exp dim M r A r A A A

, a contradiction; or , then

r r

so that and

dim Z exp dim M

again a contradiction. This proves part 1 of the theorem.

Case 2. Suppose Z is a proper subset of . We must show that . In this case

Y dim Z r

dim by 9.2.1, so that instead of (9.9) we get

dim Z exp dim M r r exp dim M r

2 The very same arguments as in Case 1 lead to a contradiction, unless as asserted. 214 9 Irreducibility and Smoothness

Comments: Ð The main references for this chapter are articles of O'Grady, Li and Gieseker-Li. The general

outline of our presentation follows the article [208] of O'Grady. The bounds O'Grady gives for are all explicit. Moreover, he can further improve these bounds in the rank 2 case. Our presentation is less ambitious: even though all bounds could easily be made explicit we tried to keep the arguments as simple as possible. As a result, some of the coef®cients in the statements are worse than those in the O'Grady's paper. Ð Generic smoothness was ®rst proved by Donaldson [47] for sheaves of rank 2 and trivial determinant, and by Zuo [261] and Friedman for general determinants. Their methods did not give effective bounds. Ð Asymptotic irreducibility was obtained by a very different and also very interesting method by Gieseker and Li for rank 2 sheaves in [81] and for arbitrary rank in [82]. Ð Asymptotic normality was proved by J. Li [149]. 215

10 Symplectic Structures

A symplectic structure on a non-singular variety M is by de®nition a non-trivial regular

w H M

two-form, i.e. a global section . Any such two-form de®nes a ho-

T w

M M

momorphism M which we will also denote by . This homomorphism

w w satis®es w , i.e. is alternating. Conversely, any such alternating map de®nes a

symplectic structure.

w w T M

The symplectic structure is called (generically) non-degenerate if M is

(generically) bijective. The symplectic structure is closed if dw . Sometimes we will also call a regular two-form on a singular variety a symplectic structure. Note that our de®- nition of a symplectic structure is rather weak. Usually one requires a symplectic structure

to be closed and non-degenerate.

K w K

Any non-degenerate symplectic structure de®nes an isomorphism M ,

n dim M K O

where . In particular, M . Using the Pfaf®an one can in fact show that

O K

M . Any generically non-degenerate symplectic structure is non-degenerate on the

w H M K

complement of the divisor de®ned by .

By de®nition a compact surface admits a symplectic structure if and only if g . Going through the classi®cation one checks that the only surfaces with a non-degenerate symplectic structure are K3 and abelian surfaces. The general philosophy that moduli spaces of sheaves on a surface inherit properties from the surface suggests that on a symplectic surface the moduli space should carry a similar structure. That this is indeed the case will be shown in this lecture. We will also discuss how holomorphic one-forms on the surface give rise to one-forms on the moduli space. In Section 10.2 we give a description of the tangent bundle of the good partof the moduli space in terms of the Kodaira-Spencer map. In Section 10.3 one- and two-formsonthemod- uli space are constructed using the Atiyah class of a (quasi)-universal family. The question under which hypotheses these forms are non-degenerate is studied in the ®nal Section 10.4. We begin with a discussion of the technical tools for the investigations in this chapter.

10.1 Trace Map, Atiyah Class and Kodaira-Spencer Map

In this section we recall the de®nition of the cup product (or Yoneda pairing) for Ext-groups of sheaves and complexes of sheaves, the trace map and the Atiyah class of a complex. These

are the technical ingredients for the geometric results of the following sections. k In the following let Y be a -scheme of ®nite type.

216 10 Symplectic Structures

10.1.1 The cup product Ð Let E and be ®nite complexes of locally free sheaves.

om E F

H is the complex with

n i in

H om E F H omE F i

and differential

deg

d d d E

F (10.1)

If G is another ®nite complex of locally free sheaves, composition yields a homomorphism

om F G H om E F H om E G

H (10.2)

deg

d d

such that d for homogeneous elements and .

B X

For any two ®nite complexes A and of coherent sheaves on there is a cup product

i j i j

H A H B H A B

U fU g

most conveniently de®ned via Cech cohomology: let i be an open af®ne cov-

U Y I U

i i

ering of , indexed by a well ordered set . The intersection i is again

j p

fi i g I F p

af®ne for any ®nite (ordered) subset . For any sheaf consider the

F U k

complex C of -vector spaces with homogeneous components

F U C F U

i i

and differential

j d

U i i

i i p

i i i

p j p

C F U

If F is a ®nite complex, we can form the double complex with anticommuting

p q p q p p q

d d C F U C F U d d C F U

differentials and F

p q

C F U F U

C . The cohomology of the total complex associated to computes

H . Now de®ne a cup product

p q p q pp q q

C A U C B U C A B U

q p

j j

i i U U i i i i

p i i i i p

pp pp

Thus composition induces a product

i j ij

Ext F G Ext E F Ext E G

om E E

In particular, H has the structure of a sheaf of differential graded algebras and

E E k Ext E F

its cohomology Ext inherits a -algebra structure. If we interpret

Hom E F i D

as D , where is the derived category of quasi-coherent sheaves, then the

cup product for Ext-groups is simply given by composition.

10.1 Trace Map, Atiyah Class and Kodaira-Spencer Map 217

E tr E ndE O Y

10.1.2 The trace map Ð For any locally free sheaf let E denote

E the trace map, which can be de®ned locally after trivializing E . More generally, if is a

®nite complex of locally free sheaves, de®ne a trace

H om E E O tr

Y E

i j i

j j tr i j tr

E E

omE E E ndE

by setting H , except in the case , when we put

E . Also let

i O H om E E

E Y

id O Y

be the -linear homomorphism that maps E . Clearly,

i i

rkE rkE i tr

E E

H om E E

If and are homogeneous local sections in , then

deg deg

tr tr E

E (10.3)

i j

H omE E

This relation can be easily seen as follows: we may assume that and

m n

H omE E tr tr j m i n E

. Then E and are zero unless and .

i j

tr tr E

Moreover, E . Hence

i j ij

i j

tr tr

E E

i j deg deg deg deg mo d d

and . Let E denote the differential in

the complex E . It follows from this and (10.1) that

degd deg

tr d tr d tr d

E E E E E

i tr O

E Y This shows that both E and are chain homomorphisms (where is a complex con-

centrated in degree 0) and induce homomorphisms

i i

i H Y O Ext E E tr Ext E E H Y O Y

Y and tr

Lemma 10.1.3 Ð i and have the following properties:

i rkE id

i) tr

deg deg

ii) tr for any two homomogeneous elements

Ext F F

i tr E

Proof. The ®rst assertion clearly follows from the equivalent assertion for E and .

B T A B B A

As for the second, suppose A , are chain complexes and let and

degadegb

H A H B H B H A a b b a

T be the twist operator b for any homogeneous elements a and . Then the diagram

218 10 Symplectic Structures

H T

H A B H B A

x x

H A H B H B H A

B H om E E m commutes. Specialize to the situation A and let denote the

composition

H om E E H om E E H om E E O

m T

Then (10.3) can be expressed by saying that m . Thus

tr H m H m H T H m T tr T

which is the second assertion of the lemma. 2

An easy modi®cation of the construction leads to homomorphisms

i i

H Y N Ext E E N tr Ext E E N H Y N

i and Y for any coherent sheaf N on which satisfy relations analogous to i) and ii) in the lemma.

De®nition 10.1.4 Ð Let F be a coherent sheaf that admits a ®nite locally free resolution

i i

Ext F F N F F Ext F F N N

. Then for any locally free sheaf . Let

i i

Ext F F N ker tr Ext F F N H Y N

p p Y Y Y

10.1.5 The Atiyah class Ð Let be the projections to the two factors.

I Y Y O O I

Y Y

Let be the ideal sheaf of the diagonal and let denote the

O p

structure sheaf of the ®rst in®nitesimal neighbourhood of . As is ¯at, the sequence

I I O O

p F F Y p

remains exact when tensorized with for any locally free sheaf on . Applying ,

we get an extension

F p p F O F

AF Ext F F F

whose extension class Y is called the Atiyah class of . Note that

F O Y U p p F O U p F U p

k s O r s

provides a -linear splitting of the extension. If is an Y -linear splitting, then

p F F O F r

Y is an algebraic connection on , i.e. satis®es the Leibniz rule

r f d f rf

10.1 Trace Map, Atiyah Class and Kodaira-Spencer Map 219

O f F r s

for any local sections Y and . Conversely, if is a connection, then

O AF r p

is Y -linear. Thus the Atiyah class is the obstruction for the existence of

an algebraic connection on F .

More generally, if F is a ®nite complex of locally free sheaves, one gets a short exact

sequence

F p p F O F

AF Hom F F Ext F F

Y Y

de®ning a class D .

A quasi-isomorphism F of ®nite complexes of locally free sheaves induces an

Ext F F AF Ext G G

isomorphism Y which identi®es and

G F F

A . In particular, if is a coherent sheaf that admits a ®nite locally free resolution

AF F , then is independent of the resolution and can be considered as the Atiyah class

of F .

The class A can be expressed in terms of Cech cocyles: choose an open af®ne cov-

U fU ji I g

ering i such that the restriction of the sequence

q q q

F p p F O F

q q

F j U q i r F j

Y U U

to i splits for all and . Thus there are local connections .

i i i

(Note that the difference of two (local) connections is an O -linear map.) De®ne cochains

C H om F F U C H om F F U Y Y and

as follows:

q q q q q q

d r r d r r j j

F F U

U and

i i i i

i i i i i i i

d F

where F is the differential of the complex . Since

d j j d d d

i i U U F F F

i i i i

i i i i i i i i

the element is a cocyle in the total complex associated to the double complex

C H om F F U AF

Y . The cohomology class of is .

This provides an easy way to identify the Atiyah class of the tensor product of two com-

E F r r F

plexes and : check that if E and are (local) connections in locally free sheaves

E F r r id id r E F

F E F E F and , then E is a (local) connection on . Whence

one deduces that

AE F AE id id AF

F E

10.1.6 Newton polynomials Ð Assume again that F is a ®nite complex of locally free

i i

F Ext F F i AF

sheaves. Let A be the image of the -fold composition

i i i

AF Ext F F

under the homomorphism induced by and

Y Y

F de®ne the i-th Newton polynomial of by

220 10 Symplectic Structures

i i i i

F tr AF H Y

i F These classes differ by a factor i from the -th component of the Chern character of .

As the trace map does not see anything from a CechÏ cocycle in

p q i

C H om F F U

pq i

i q F

except the components with p , , it follows that depends only on the -part

n n

F AF F

of the cocycle that gives . In particular, .

i j i

d k d H Y

The k -linear differential induces -linear maps

Y Y

j i i

H Y F d d F

. If F is a locally free sheaf, then is -closed, i.e. for all

i: as is additive in short exact sequences, we can reduce to the case of line bundles using

F f O U ij

the splitting principle. If is a line bundle given by transition functions ij ,

df d log f f AF d ij

then ij is a Cech cocycle for that clearly vanishes under .

ij S

10.1.7 Relative versions Ð Let X be a smooth projective surface, a base scheme of

p S X S q S X X

®nite type over k and let and be the projections. Any

F F F S -¯at family of coherent sheaves admits a ®nite locally free resolution so that

we can apply the above machinery to F .

E xt F H om F p H omF

Recall that are the derived functors of p . It is

F G xt

easy to see that E is the shea®®cation of the presheaf

U Ext F j Gj

U X U X

j j

F F F Ext F F Ext F F

If is a ®nite locally free resolution of , then . Thus

tr F

shea®fying the cup product and the maps i and de®ned for , we get maps

j j j j

E xt F F E xt F F E xt F F

p p p

j j j

O H X O tr E xt F F R p O

S k X S X p

and

j j

i O H X O E xt F F

S k X p

satisfying the relations

deg deg

i rkF id tr tr

tr and S

10.1.8 The Kodaira-Spencer map Ð Let F be an -¯at family on a smooth projective

F F

surface X . Choosing a locally free resolution we can de®ne the Atiyah class

AF AF Ext F F

X S and consider the induced section under the

global-local map

Ext F F H S E xt F F

S X S X p

10.1 Trace Map, Atiyah Class and Kodaira-Spencer Map 221

i j

S E xt Ext

coming from the spectral sequence H . The direct sum decomposi-

S X

p q AF AF AF

X S X tion S leads to an analogous decomposition .

By de®nition, the Kodaira-Spencer map associated to the family F is the composition

KS E xt F F p

S S S

E xt F F E xt F F p

S S

p p

S Sp eck

Example 10.1.9 Ð Let X be a smooth surface as above and . Let

v F F

F be a short exact sequence representing an extension class

F S F E E

Ext . We can think of as an -¯at family by letting act on as the homomor-

AF AF AF

phism i . Decompose the Atiyah class according to the splitting

Ext F F Ext F F p Ext F F q

S X S X

S X S X S X

k d F S

Since S , and since is -¯at, we have

Ext F F Ext F F Ext F F p

X S X S X

F v We want to show that under these isomorphisms A is mapped to . According to the

de®nition of the Atiyah class we ®rst consider the short exact sequence of coherent sheaves

Sp ec k X

over

F G F

(10.4)

F i F

where and act trivially on and by on , and

G k F F F F

with actions and . Now is precisely the extension

k O X class of (10.4), considered as a sequence of -modules. But it is easy to see that

there is a pull-back diagram

F F F

x x

F G F

F v 2 which shows that A . 222 10 Symplectic Structures

10.2 The Tangent Bundle

s M

Let X be a smooth projective surface and let be the moduli space of stable sheaves

X r c c M M

on of rank and Chern classes and . The open subset of points

F Ext F F

such that vanishes is smooth according to Theorem 4.5.4. Suppose there

E M X E

exists a universal family on . The Kodaira-Spencer map associated to is a sheaf

homomorphism

E xt E E KS T

It is the goal of this section to show that this map is an isomorphism. In fact one can make

E xt sense of the map KS and the -sheaf on the right hand side even if a universal family does not exist. We will prove this ®rst. There are two ways to deal with the problem that a universal family need not exist: either

one uses an Âetale cover of the moduli space, over which a family exists. Or one works on s the Quot-scheme that arises in the construction of M and shows that all constructions are

equivariant and descend. We will follow this approach.

s s s

QuotH P R M

Let R be the open subset as de®ned in 4.3 so that is a

O H F

geometric quotient. If R is the universal quotient, we can form the sheaves

e e

F F xt GLV

E . These inherit a natural action of `by conjugation'. In particular, the

centre of GL acts trivially. Moreover, both the cup product and the trace map are equiv-

g e e

E xt F F xt

ariant. By descent theory, E and these two maps descend to a coherent sheaf

p p s

on M and homomorphisms

i j ij i

g g g g

E xt E xt E xt E xt H X O O

k M

and X

p p p p

E E p A F

Suppose, a universal family exists. Then for some appropriately lin-

s R

earized line bundle A on . Therefore

i i i i

e e e e

E xt E E E xt E E E xt F F E ndA E xt F F

p p p p

E xt E E E xt E

Thus in the presence of a universal family we have . For this reason

p p

E E xt E

we give in to the temptation to use the notation E even if a universal family itself p does not exist.

Theorem 10.2.1 Ð There are natural isomorphisms

i i

T H X O O E E j E E j E xt E xt i

M X k M M

M and for

p p

(The theorem immediately implies Theorem 8.3.2.)

R M M M

Proof. and are smooth by Theorem 4.5.4. By the de®nition of

Ext F F F M

we have for all , and this implies that the homomorphisms

e e e e

F F E xt F F H X O O i H X O O tr E xt

X R R

X and

p p 10.3 Forms on the Moduli Space 223

are isomorphisms on the ®bres over closed points, and hence are surjective as homomor-

i r id

phisms of sheaves. Since tr , both maps are in fact isomorphisms. Similarly, since

e e

HomF F E xt F F O

for stable sheaves, the same argument shows that R .

From the ®rst isomorphism one deduces that E commutes with base change, and from

e e

xt F F

the second that E is locally free.

e e

E xt F F KS T H F

Now consider the Kodaira-Spencer map R . Let

p R

be a closed point. It follows from Example 10.1.9 and Appendix 2.A that the following

diagram commutes:

T R T M

y y

Hom ker F Ext F F

X X

In the diagram the vertical isomorphisms come from deformation theory (cf. 2.A), and

is the coboundary operator. We conclude that the Kodaira-Spencer map factors through an

e e

E xt F F T

isomorphism M . Since the Atiyah class is invariant, this isomorphism

E xt E E T 2

is equivariant and descends to an isomorphism M .

10.3 Forms on the Moduli Space

We are now going to describe natural one- and two-forms on the moduli space as announced

in the introduction.

S X

Let F be an -¯at family of sheaves on a smooth projective surface .The Newton poly-

i i i i

F F H

nomials are independent of the choice of a ®nite locally

S X

F F p q

X S X X free resolution . Since S , and since is locally free, there

is a KÈunneth decomposition

nj j

ni i n n

H X H S H S X

S X

X S

F F F H S H X O

Let and denote the components of in X and

H S H X X

S , respectively.

De®nition 10.3.1 Ð Let F and be the homomorphisms given by

H X K H X O H S

F X X

and

H S H X H X

S F X X

224 10 Symplectic Structures

(Here is Serre duality.)

H X K H X

Proposition 10.3.2 Ð For any X or the associated two-form

S F

F or one-form , respectively, on is closed.

d d d

X s X Proof. The decomposition S induces similar splittings for the

KÈunneth components:

j nj

i ni

d d d H H

S X S X

S X

j nj j nj

i ni i ni

H H H H

S X S X

d F d d

X S X S X

Since S , one has as well. Write

H H O

for elements and . Then

X X

d d d

S X S X

X d H

Since is a smooth projective variety, one has X for any element

and therefore S . Hence

X X

d d d

S F S S

d 2 F

Similarly one shows S

S H X K

Lemma 10.3.3 Ð Suppose that is smooth. Then for each X the two-form

on is the composition of the maps:

KS KS

T S T S Ext F F Ext F F Ext F F

s s s s s s s s

X X X

H X O H X K k

X X

Proof. This follows readily from the de®nitions. 2 s

In order to de®ne forms on M we use quasi-universal families, which always exist by Proposition 4.6.2. The following lemma implies that the construction is independent of the

choice of the quasi-universal family:

S X B

Lemma 10.3.4 Ð Let F be an -¯at family of sheaves on and let be a locally free

F p B rkB F F p B rkB F

sheaf on S . Then and .

AF p B AF id id p AB B

Proof. We have B . The de®nition of

AF id H S H X

shows, that only B contributes to the component of

S X

F p B tr AF id rkB tr AF

, which is relevant for and . Since B ,

the assertion follows. 2

10.3 Forms on the Moduli Space 225

M X

De®nition and Theorem 10.3.5 Ð Let E be a quasi-universal family on . Then

H X K H M

E X

rkE

and

H X H M

E X M

rkE

are independent of E .

Proof.If E and are any two quasi-universal families, then there are locally free sheaves

E p B B B M E p B and on such that . The assertion then follows from Lemma

10.3.4. 2

In order to make use of the differential forms and in the birational classi®cation of

M M r c c

moduli spaces, it is important to extend them from to the compacti®cation

M r Q c

(or ). The case of the one-form is less involved and provides an alternative def-

inition:

AlbX

Recall that for a smooth projective variety X the Albanese variety is de®ned as

H X H X Z

X (cf. Section 5.1). This leads to a canonical isomorphism of the

H Alb X H X

spaces Alb and . Under this identi®cation the differential

H X H Alb X A X AlbX

X Alb of the Albanese morphism

equals the identity map.

M r Q c x X

Let M and ®x a point .

M AlbX E

Proposition 10.3.6 Ð There is a natural morphism , which maps

Ac E c E E CH X

to , where is the second Chern class of in the Chow-group . If

H Alb X H X M

Alb is identi®ed with , then on .

S X S AlbX t S

Proof. Any family F on de®nes a morphism by mapping

Ac F M M t

to . Since corepresents the moduli functor, we get a morphism

X Alb . The assertion on is more complicated. For a smooth basis representing locally free sheaves, the proof can be found in [89]. But one can give an algebraic proof for

the general case as well, which we omit. 2

Note that this provides an alternative de®nition of and immediately shows that is closed and extends to the complete moduli space. In particular, we obtain a one-form on any

smooth model of M .

Not much is known about the morphism in general. For the rank one case, i.e.

Hilb , there is the following theorem due to M. Huibregtse [110].

Hilb X AlbX

Theorem 10.3.7 Ð For c , the morphism is surjective and

c h X O c

all the ®bres are irreducible of dimension X . If , the morphism is smooth

X AlbX 2 if and only if A is smooth. 226 10 Symplectic Structures

Proposition 5.1.5 and Theorem 5.1.6 in Section 5.1 give a ®rst hint for the higher rank

case.

r M r Q c AlbX 2

Corollary 10.3.8 Ð If c and , then is surjective.

This can also be used to show the non-degeneracy of the one-forms . In fact, 10.3.8

c implies that for any and the one-form is not trivial. The consequences for the birational geometry of the moduli space will be discussed in

Chapter 11.

We certainly cannot expect to have an analogous situation for the two-forms . Since

the dimension of the moduli space grows with and, at least in special examples, is

generically non-degenerate, cannot be the pull-back of a two-form on a ®xed ®nite di-

M Y

mensional variety Y under a morphism . Work of Mumfordon the Chow groupof

p Y CH X

surfaces with g suggests that should be replaced by , which is neither ®nite

dimensional nor a variety [192]. This `non-geometric' behaviour of makes it more dif®- s

cult to extend it over a suitable compacti®cation of M . For many purposes the following

is suf®cient.

M M r c c

Corollary 10.3.9 Ð There exists a morphism from a projective

variety , which is birational over M and such that the pull-back of any two-form

on M extends to .

w E

Proof. This is a consequence of 4.B.5. Indeed, if E , where is the family

s s

w j j M X

M M

on , then , since the pull-back of a

s s

X M E 2 quasi-universal family on M to is equivalent to constructed in 4.B.5.

10.4 Non-Degeneracy of Two-Forms

H X K

In the previous section we constructed for each global section X a two-form

M M r c c E

on the stable part of the moduli space . Moreover, if is a closed

s s

Ext E E M M Ext E E T M

point in the good part , i.e. if , then ,

and with respect to this identi®cation is given by the map

E E Ext E E Ext E E Ext

H X O H X K k

X X

(cf. 10.2.1 and 10.3.3.) Thus the question whether is non-degenerate in good points

E M

is answered by the following `local' proposition:

10.4 Non-Degeneracy of Two-Forms 227

Proposition 10.4.1 Ð The form is non-degenerate if and only if multiplication by in-

Ext E E Ext E E K X

duces an isomorphism . Similarly, the restric-

X X

Ext E E

tion of to the subspace is non-degenerate if and only if the homomorphism

Ext E E Ext E E K

is an isomorphism.

X X

Proof. In order to prove the proposition, we need to relate the de®nition of (involving

cup product and trace) to Serre duality. Let E be a ®nite locally free resolution. Note

j i i j

H omE E H omE E

that the isomorphism can be obtained by the pairing

i j j i j j

O H omE E H omE E H omE E

H om E E

More generally, if A , then

A A A O

A H om A O

is a perfect pairing and leads to an isomorphism X . Hence for any

O K X

section X there is a commutative diagram

ev al

A K A H om A K A K

X X X

x x

A A A O X

Passing to cohomology we get

i i j ij

Ext E E K Ext E E E xt A K H A H X K

X X X

x x

j ij

i ij

Ext E E Ext E E Ext E E H X O

X X

j Observe that for i , is the map from the lower left corner of the diagram to the

upper right corner. n

Serre duality in its general form says that for a smooth variety X of dimension and a

bounded complex A of coherent sheaves the pairing

ni i n

E xt A K H X A H X K k

X X

is perfect (cf. [96]). If we apply this to the diagram above with i in the case of a

surface , we get: is a non-degenerate if and only if is an isomorphism, thus proving

the ®rst part of the proposition.

id f A U U X

For the second observe, that for any local section , , and E

tr f tr f A O kertr E

one certainly has . Thus the splitting X is orthogonal

A A A O

with respect to the bilinear map X . This implies that the splitting

i i

Ext E E H X O Ext E E

X is orthogonal with respect to . It is also re- X

spected by Serre duality. Hence one concludes as before. 2

228 10 Symplectic Structures

O X K X

X Corollary 10.4.2 Ð Let be a surface with X , i.e. is either abelian or K3.

Then the pairing

Ext E E Ext E E k

X X

is a non-degenerate alternating form, and the same holds for the restriction of E to the

Ext E E 2

linear subspace . X

Combining this corollary with the fact that under the same hypotheses the smoothness

Ext E E

obstruction group vanishes for any stable sheaf, we get Mukai's celebrated X result on the existence of a holomorphic symplectic structure on the moduli space of sheaves

on K3 and abelian surfaces:

O X K M r c c

X Theorem 10.4.3 Ð If is a smooth projective surface with X , then

admits a non-degenerate symplectic structure. 2

K O

X If X one does not expect to be non-degenerate at every point of the mod-

uli space. The best one can hope for is a generic non-degeneracy. Of course, a necessary

F M M

condition is that the moduli space is of even dimension. Suppose is a

good point. According to Proposition 10.4.1, F is non-degenerate at if and only if

E E K E E Ext Ext X

is an isomorphism.

X X

O K K j D

X X D

Using the exact sequence X , where is the divi-

H X K

sor de®ned by X , one sees that a suf®cient condition is the vanishing of

HomE E K j M r Q c

X . If one restricts to the moduli space of sheaves with ®xed

HomE E K j

determinant it suf®ces to show X in order to have non-degeneracy

E of at . The following is a crucial result in the theory. It is only known for the rank two case but hopefully true in general. Recently, Brussee pointed out that the assertion is a consequence

of the relation between Seiberg-Witten invariants and Donaldson polynomials and the fact

that the only Seiberg-Witten class of a minimal surface of general type is X .

X D Z jK j

Theorem 10.4.4 Ð Let be a surface of general type and let X be a

O c Q mo d c

reduced connected canonical divisor. If X , then for the

M Q c

symplectic structure on is generically non-degenerate.

Q mo d O c dim M

Proof. Note that the assumption X is equivalent to

mo d . Otherwise the symplectic structure could never be non-degenerate.

The proof of the theorem consists of two parts. First, one establishes the existence of a

F D HomF F K j

rank two vector bundle on such that X . Next, one uses this to

M show that the restriction of the generic bundle E shares the same property. Once the

existence of F is known, the proof goes through in the higher rank case as well. The ®rst

step is highly non-trivial even when D is smooth. The proof is omitted. 10.4 Non-Degeneracy of Two-Forms 229

Let us sketch the second part of the proof. Here one makes use of some results in de-

formation theory. By the same method as in the proof of Theorem 9.3.3 one shows that

c E M r Q c HomE E K

for the generic sheaf satis®es , and

Ext E E K E X

hence, X . The in®nitesimal deformations of on with ®xed

Q Ext E E E j Ext E E

D D D

determinant are given by and of by . For a locally

E Ext E E Ext E j E j

D D

free the cokernel of the natural map is contained

Ext E E K E j

in X . Thus all in®nitesimal deformations of can be lifted to de- X

formations of E on . The same procedure works for the deformations of higher order.

Consequently, there exists a deformation E of which restricts to a generic bundle on

D F E j

. The assumption on the bundle implies that the generic bundle D has vanishing

HomE j E j K 2

D X D .

Comments: Ð Mukai was the ®rst to construct algebraically a symplectic structure on the moduli space of sim-

ple sheaves on K3 and abelian surfaces [186]. Theorem 10.4.3 is due to him. Later Tyurin [247] gen-

p eralized his construction for surfaces with g . He also considered Poisson structures. Trace and

pairing were treated by Artamkin in connection with the deformation theory of sheaves [5], though

the sign in 10.1.3 is missing in [5]. Ð For a very detailed treatment of the Atiyah class of (complexes of) sheaves we refer to the article

of Angeniol and Lejeune-Jalabert [2]. E Ð 10.3.2 was proved by O'Grady [206] for smooth S and locally free . Also compare [33]. A reference for the Albanese mapping is [252].

Ð The existence of the bundle F in the proof of 10.4.4 is due to Oxbury [213] and O'Grady [206]

if D is smooth and to J. Li [149] in general. 230

11 Birational properties

Moduli spaces of bundles with ®xed determinant on algebraic curves are unirational and very often even rational. For moduli spaces of sheaves on algebraic surfaces the situation differs drastically and, from the point of view of birational geometry, discloses highly interesting features. Once again, the geometry of the surface and of the moduli spaces of sheaves on the surface are intimately related. For example, moduli spaces associated to rational surfaces are expected to be rational and, similarly, moduli spaces associated to minimal surfaces of general type should be of general type. We encountered phenomena of this sort already at various places (cf. Chapter 6). There are essentially two techniques to obtain information about the birational geometry of moduli spaces. First, one aims for an explicit parametrization of an open subset of the moduli space by means of Serre correspondence, elementary transformation, etc. Second, one may approach the question via the positivity (negativity) of the canonical bundle of the moduli space. The ®rst step was made in Section 8.3. The best result in this direction is due to Li saying that on a minimal surface of general type with a reduced canonical divisor the moduli spaces of rank two sheaves are of general type. This and similar results concerning the Kodaira dimension are presented in Section 11.1. The use of Serre correspondence for a birational description is illustrated by means of two examples in Section 11.3. Both examples treat moduli spaces on K3 surfaces, where this technique can be applied most success- fully. In Section 11.2 we survey more results concerning the birational geometry of moduli spaces. For precise statements and proofs we refer to the original articles.

11.1 Kodaira Dimension of Moduli Spaces

For the convenience of the reader we brie¯y recall some of the main concepts in birational

geometry. As a general reference we recommend Ueno's book [252].

n X

Let X be an integral variety of dimension over an algebraically closed ®eld. is ra-

n m

P X X

tional if it is birational to P . If there exists a dominant rational map , then is m

called unirational. Note that by replacing P by a general linear subspace we can assume

m .

kodX De®nition 11.1.1 Ð Let X be a smooth complete variety. Its Kodaira dimension

is de®ned by:

h X O mK m kodX

If X for all , then .

11.1 Kodaira Dimension of Moduli Spaces 231

h X O mK kodX

If X or , but not always zero, then .

h X O mK m kodX

If X , then .

kodX n X

It turns out that the Kodaira dimension satis®es . If is not smooth

kodX

or not complete but birational to a smooth complete variety X , then we de®ne

X X

kod . This de®nition does not depend on due to the fact that the Kodaira dimension n

is a birational invariant. Last but not least, an integral variety X of dimension is of general

X n

type if kod .

M c Pic X

Let us begin with the rank one case. Obviously, H is either empty

or an abelian variety. In particular, in the latter case the Kodaira dimension is zero. The mor-

M r c c Pic X

phism H de®ned by the determinant is locally trivial in the Âetale

topology (cf. the proof of 4.5.4). Thus one is inclined to study the geometry of the ®bre

M r Q c Q Pic M Q c Hilb X X

H H over separately. Since , the fol-

lowing result computes the Kodaira dimension in the rank one case.

kodHilb X n kodX

Theorem 11.1.2 Ð If n then .

n n

Hilb X S S X X

Proof. We ®rst introduce some notations: Let M , ,

X X S M S

X , and let and be the natural morphisms. The tensor

p O K X

product X is a line bundle on with a natural linearization for the action of

S X

the symmetric group n . The isotropy subgroupsof all points in act trivially. Therefore, S

the line bundle descends to a line bundle on . Moreover,

n m n S

S H X O mK H S H X p O mK

X X

O K

We use the following facts: i) M , which follows from a local calculation in

O x x x x S X x x O

S n i M

points with , and ii) , which is an easy

i ii

M S H M O mK H M

consequence of the normality of and . Then M

m n

kodM n kodX 2 H S S H X O mK

X . This yields .

Hilb X Corollary 11.1.3 Ð If X is a surface of general type, then is of general type as

well. 2

Let us now come to the higher rank case. Here we ®rst mention a consequence of The-

q X h X O

orem 5.1.6 in Chapter 5. Note that a surface with X can never be

unirational. For such surfaces we have:

q X

Theorem 11.1.4 Ð If X is an irregular surface, i.e. , then the moduli spaces

M r Q c c

H are not unirational for .

Proof. This follows easily from the observation that for the Albanese map de®nes

M r Q c AlbX AlbX

a surjective morphism H (cf. 5.1.6). Since is a torus, any

AlbX 2 morphism P must be constant.

232 11 Birational properties H

Theorem 11.1.5 Ð Let X be a minimal surface of general type. Fix an ample divisor and

Q PicX D jK j

a line bundle . Assume: i) there exists a reduced canonical divisor X

O c Q c M Q c

and ii) X . Then for the moduli space is a

kodM Q c dimM Q c

H normal irreducible variety of general type, i.e. H .

Proof. We ®rst prove the theorem under the additional assumption that X contains no

-curves. This is equivalent to X being ample. We will indicate the necessary modi-

®cations for the general case at the end of the proof. By the results of Chapter 9 (Theorem

M r Q c

9.3.3 and Theorem 9.4.3) we already know that H is normal and irreducible for c

suf®ciently large . Thus it remains to verify the assertion on the Kodaira dimension. The-

orem 4.C.7 shows that for any two polarizations H and the corresponding moduli spaces

r Q c M r Q c M c

H and are birational for . Therefore, it suf®ces to prove

Q c H K M M

the theorem in the case X . To simplify notations we write

M E M Ext E E

and denote by the open subset of stable sheaves with vanishing . M

Note that is smooth (4.5.4).

In order to show that M is of general type we have to control the space of global sec-

H M O mK M M W

tions for some desingularization . Let i denote the ir- M

reducible components of codimension one of the exceptional divisor of . Then we claim

O nK n L u PicM O a W L

that i , where is positive and

(for the notation see Section 8.1). Indeed, by Theorem 8.3.3 there exists a positive integer n

L j O nK co dimM n M M n M

such that M . Moreover, , since is contained

Ext E E

in the subset of sheaves which are not good, i.e. either not -stable or ,

and that this subset has at least codimension two is a consequence of Theorem 9.3.2. This is

L O nK

enough to conclude that and only differ by components of the exceptional

divisor.

d m m

c m c m h M L M L d

By Corollary 8.2.16 we have h , where

dimM c c m a

, the constant is positive, and comprises all terms of lower degree. If i

n mn

O nK H M O mnK i L H M L

for all , then and hence .

M M

Hence M is of general type. The rest of the proof deals with the case that at least one of the a

coef®cients i is negative. Here we apply a result of Chapter 10 (see 10.3.9 and also 4.B.5),

M M

where we constructed a desingularization such that M admits a regular two-

H M j H X K

form with . Here is the section

s M

de®ning D and is the open dense subset of stable sheaves.

From now on let r . Then Theorem 10.4.4 applies and shows that , and hence

H M O K

, is generically non-degenerate. Let be the Pfaf®an of . Then is

a non-vanishing section. We claim that vanishes on all components i : By construction,

M M E

the desingularization has the following properties: There exists a family

M X s r t M E E

over of rank such that for all the sheaf t is isomorphic to for some

E E t s

semistable sheaf with . Moreover, E . In order to show that

vanishes on a component i it suf®ces to show that degenerates at the generic point of W

i . As this is a local problem we may use Luna's Etale Slice Theorem (see Theorem 4.2.12)

E r m to assume that s , i.e. is a family of semistable sheaves of rank . Fix an integer

11.1 Kodaira Dimension of Moduli Spaces 233

R M PGLP m

as in the construction of the moduli space and let be the principal -

p E q O m

bundle associated to . With the notations of Chapter 4 and 9 there exists

R R QuotV O m P M

a classifying morphism , where R is a good

PGLP m M W

quotient and a principal -bundle over . Let i be an irreducible component

W W i of codimension one of the exceptional divisor of and let i . So we have the

following diagram.

R W R

W M M

Moreover, is an isomorphism over a dense open subset. The compatibility of the two-

forms constructed in Chapter 10 gives E , where is the univer-

R X W M n M R W R i

sal quotient sheaf over .If , then i , where is the

co dimR

closed subset of -unstable sheaves. By Theorems 9.1.1 and 9.1.2 we have

W R c W M

i i

for . Hence has positive ®bre dimension. If , then

W R W M i

i has positive ®bre dimension and so has . Hence, in both cases

degenerates on the component i . But then the same is true for the two-form

F W

on i .

Having proved that vanishes along i we may consider as a section of the sheaf

O K W a maxfa g

i . Let . Then the multiplication with de®nes an in- M

jection

O mnK a W L

i i

O mn aK a aW

i i

O mn aK

h M O mn aK c mn c mn kodM

Hence and therefore

M d dim M M

kod , i.e. is of general type.

X K

We now come to the case that contains -curves. Then X is no longer ample, but

f X Y X Y K

still big and nef. If is the morphism from to its canonical model , then X

H Y H X

is the pull-back of an ample divisor Y on . Let be an arbitrary polarization on and

M M r Q c M

consider the moduli space H . As before, is normal and irreducible for

. Moreover, copying the arguments of the proof of Theorem 4.C.7 and using that

K c F M

X is in the positive cone we ®nd that for the set of sheaves which are

not K -stable is at least of codimension two. X

The Bogomolov Restriction Theorem 7.3.5 (cf. Remark 7.3.7) applied to a smooth curve

C jnK j n M M C

X ( ) yields a rational map which is regular on the open sub-

X n C

set of K -stable sheaves with singularities in . The complement of this open set has at X

least codimension two. As before, to conclude the proof it suf®ces to verify that is gener-

F G H omE F

ically injective. Let E and be two locally free sheaves and let . The re-

H X G H C Gj

striction homomorphism C is surjective if and only if its Serre dual

234 11 Birational properties

H C G j K H X G K C

C X

C is injective. If avoids all -curves, which

jnK j

the generic curve in X does, then there is a commutative diagram

H X G j K H X G K

C C X

H C f G j K H Y f G H

C C Y

R f G R f G

For the second vertical isomorphism use that and that is zero-

H Y R f G H Y

dimensional and hence . The kernel of

H C f G j K H Y f G H

C C Y

H Y f G n H n H

Y Y

is a quotient of which clearly vanishes for , since

E F M

is ample. Therefore we may assume that for all locally free the restriction

F j HomE F HomE j F j E j

C C C C

C is surjective. In particular, if , then there

E F j C

exists a homomorphism with C . Thus is generically injective and

detE n M M detF since , it is in fact bijective. Thus, for the map C is

injective on the locally free part. 2

Remark 11.1.6 Ð The proof has been presented in a way indicating that the moduli spaces

M r Q c

H for a minimal surface of general type are expected to be of general type with-

out the assumptions i), ii) and r . In fact, if the singularities of the moduli spaces are a canonical, i.e. all coef®cients i are nonnegative, then the proof goes through: For all three assumptions were only used to ensure the existence of a generically non-degenerate two- form which would not be needed in this case.

Along the same line of arguments, only much simpler, one also proves

X H Q PicX c

Theorem 11.1.7 Ð Let be a surface, a polarization, , and .

X M r Q c i) If is a Del Pezzo surface, then H is a smooth irreducible variety of Ko-

daira dimension .

O K X M r Q c O

X H ii) If X , i.e. is abelian or K3, then is a normal irreducible

variety of Kodaira dimension zero.

X kodX kodM r Q c

iii) If is minimal and , then H and equality holds

E M r Q c Ext E E

if all H are stable and .

H K O nK L n

Proof. For i) and the case X we use for some (cf. The-

M O mnK H m

orem 8.3.3) to conclude M for all . For a polarization different

O K O nK n

from X we again use 4.C.7 ii) and iii) follow from for some which

k H M O mnK H M O kodM

immediately yields and hence .If

M M O K O

X , then the distinguished desingularization constructed in Appendix

H M O K

4.B admits a generically non-degenerate two-form. Hence . Under the

M M O nK O

additional assumptions in iii) one has and thus M which also gives

H M O nK kodM 2

M . Hence in both cases.

11.2 More Results 235

H r Q c

Remark 11.1.8 Ð There are explicit numerical conditions on such that for a

E M r Q c

minimal surface of Kodaira dimension zero all H are stable (cf. 4.6.8 ).

K r Ext E E If the order of X does not divide the rank then for any stable sheaf. Thus the conditions in iii) are frequently met.

11.2 More Results

Following the Enriques classi®cation of algebraic surfaces we survey known results related

to the birational structure of moduli spaces.

X P

11.2.1 Kodaira dimension . Let ®rst be the projective plane . One certainly ex-

pects moduli spaces of sheaves on P to be rational. In general, it is not hard to prove that

they are unirational. Already in the seventies moduli spaces of stable rank-two bundles on

N c

P were intensively studied. Barth [21] announced that the moduli spaces of sta-

c c c N c

ble rank-two bundles with are irreducible and rational. Note that

is non-empty if and only if c . The analogous problem for odd ®rst Chern number, i.e.

c N c N c

, was discussed by Hulek [107]: is rational and irreducible. Here

is non-empty if and only if c . Unfortunately, there was a gap in Barth's approach to the

rationality. Hulek remarked in [107] that for this could easily be ®lled. Ellingsrud

n N c

and Strùmme [59, 60] proved the rationality of N and with dif-

N n ferent techniques. They also proved the rationality of an Âetale P -bundle over .

Maruyama discussed the problem further [169]. In an Appendix to his paper Noruki proved

the rationality of N . Partial results are known for the higher rank moduli spaces: Le

r r r

Bruyn [140] proved that N is rational for . The rationality problem for the

r c

rank two case was eventually solved by Katsylo [119]. He proved that N is rational

cdr c if g or with the exception of ®nitely many cases.

More generally, let X be a rational surface. Ballico [9] showed that there exists a polariza-

H N r c c

tion such that H is smooth, irreducible and unirational. Combining this with

M r c c

9.4.2 one ®nds that the moduli spaces H are unirational for any polarization if

. X

To complete the case of negative Kodaira dimension, let X be a ruled surface, i.e. is the

PE C E C g

projectivization of a rank two vector bundle on a curve of genus . As

det M r c c

in the general situation there always exist two canonical morphisms H

Pic C AlbC Pic C Pic M r c c AlbX X

and H (cf. Chap-

M r c c

ter 10). Loosely speaking, one expects the moduli space H together with these

C Pic C

two morphisms to be birational to a projective bundle over Pic or at least to

C Pic C g have unirational ®bres over Pic . For rational ruled surfaces, i.e. , this

is certainly true ([9]). The rank two case has been studied in detail by many people. Hoppe

O L E r c

and Spindler [105] considered the case , , and such that the intersection

c f f N c c

H with the ®bre class is odd. They showed that indeed is birational to

236 11 Birational properties

Pic C Pic C P . Brosius [37], [38] gave a thorough classi®cation of all rank two

bundles on ruled surfaces. He distinguishes between bundles of type U and E according to

c f

whether is odd or even. Bundles of type U are constructed via Serre correspondence

L E M I E O Z C as extension Z and corresponds to . Bun-

dles of type E can be described by means of elementary transformations along ®bres of .

E In this case corresponds to the divisor of the ®bres where the elementary transformation is performed. Brosius' results allow to generalize the birational description of Hoppe and Spindler. Friedman and Qin combined these results with a detailed investigation of the

chamber structure of a ruled surface [217, 218]. One of the ideas is the following: The con-

c f tribution coming from crossing a wall can be explicitely described. In the case , when for a polarization near the ®bre class the moduli space is always empty (cf. 5.3.4), this

is enough to deduce the birational structure of the moduli space with respect to an arbitrary

polarization. For rational ruled surfaces and r this was further pursued in [154] and

[85]. For related results see also Brinzanescu's article [34]. The explicit example N

PO O n P

on the Hirzebruch surfaces X was treated by Buchdahl in [39]. Vector bundles of rank on ruled surfaces have also been studied by Gieseker and Li [82]. They use elementary transformations along the ®bres to bound the dimension of the `bad' locus of the moduli space. 11.2.2 Kodaira dimension 0. According to the classi®cation theory of surfaces there are four types of minimal surfaces of Kodaira dimension zero: K3, abelian, Enriques, and hyperelliptic surfaces. The Kodaira dimension of the moduli spaces is by Theorem 11.1.7 known for K3 and abelian surfaces and under additional assumptions also for Enriques and hyperelliptic surfaces. According to a result of Qin [215], with the exception of three special cases,

the birational type of the moduli space of -stable rank two bundles does not depend on the polarization. Some aspects of moduli spaces on K3 surfaces were studied in Chapter 5, 6, and 10. The upshot is that sometimes the moduli space is birational to the Hilbert scheme and that it is in general expected to be a deformation of a variety birational to some Hilbert scheme. The birational correspondence to the Hilbert scheme is achieved either by using Serre correspondence or, if the surface is elliptic, by elementary transformations. In the latter case Friedman's result for general elliptic surfaces apply [67, 68]. There are also birational descriptions of some moduli spaces of simple bundles on K3 surfaces available [224, 246]. Two examples of moduli spaces of sheaves on K3 surfaces will be discussed in the next section. Moduli spaces on abelian surfaces behave in many respects similar to moduli spaces on K3 surfaces. In particular, they are sometimes birational to Hilbert schemes or to products of them. For examples see Umemura's paper [254]. The Hilbert scheme itself ®bres via the group operation over the surface. Beauville showed that the ®bres are irreducible symplectic [25]. The same phenomenon should be expected for the higher rank case. The Fourier-Mukai transformation, which can also be used to study birational properties of the moduli space on abelian surfaces, was introduced by Mukai [185, 188]. It was further studied in [62], [159], [23].

11.2 More Results 237

X X The universal cover of an Enriques surface is a K3 surface. The pull-back of

sheaves de®nes a two-to-one map from the moduli space on X to a Lagrangian of maximal

dimension in the moduli space on X . This and some explicit birational description of moduli spaces using linear systems on the Enriques surface can be found in Kim's thesis [122]. Hyperelliptic surfaces are special elliptic surfaces and, therefore, Friedman's results ap-

ply. Special attention to the hyperelliptic structure has been paid in the work of Takemoto

and Umemura [255], who studied projectively ¯at rank two bundles, i.e. bundles with

c .

11.2.3 Kodaira dimension 1. Surfaces in this range all are elliptic. Sheaves of rank two

have been studied by Friedman [67, 68]. As for ruled surface, sheaves of even and odd ®-

c bre degree are treated differently. [67] deals with bundles with . In particular, they have even intersection with the ®bre class. Friedman gives an upper bound (depending on

the geometry of the surface) for the Kodaira dimension of the moduli space of rank two

c c bundles with and which are stable with respect to a suitable polarization (cf. Chapter 5). Moreover, the moduli space is birationally ®bred by abelian varieties. If the elliptic surface has no multiple ®bres then the base space is rational. Bundles with odd ®- bre degree are studied in [68] via elementary transformations. Under certain assumptions on the elliptic surface, e.g. if there are at most two multiple ®bres, the moduli space is shown to be birational to the Hilbert scheme of an elliptic surface naturally attached to the original one. Hence the Kodaira dimension is known in these cases by 11.1.2. A result in the spirit of 11.1.5 and 11.1.7 is missing. It would be interesting to see which Kodaira dimensions moduli spaces on elliptic surfaces can attain. Do they ®ll the gap between varieties with Kodaira dimension zero and those of general type? For the case that rank and ®bre degree are coprime O'Grady [210] suggests that the canonical model of the moduli space should be an appropriate symmetric product of the base curve of the elliptic ®bration. In particular, he expects that the Kodaira dimension of the moduli space should be half its dimension.

11.2.4 Surfaces of general type. The only known result is Li's Theorem 11.1.5.We do not even know a single example where one can show that the moduli space is of general type

without refering to the general theorem. Note that there is a big difference between surfaces

p p g of general type with g and those with . The Chow group of surfaces of the ®rst type is huge [192], but according to a conjecture of Bloch the Chow group is trivial in the latter case. Is this re¯ected by the birational geometry of the moduli space?

Recently, O'Grady [210] slightly generalized Li's result 11.1.5. He showed that the higher

M r Q c c

rank moduli spaces H are also of general type for if one in addition as-

sumes that the minimal surface of general type admits a smooth irreducible canonical curve

C h K j deg Qj mo dr

C C such that X . 238 11 Birational properties

11.3 Examples

We wish to demonstrate how Serre correspondence can be used to give a birational description of moduli spaces. Both examples deal with sheaves on a K3 surface. The techniques can certainly be applied to other surfaces as well, but they almost never work as nicely as

in the following two examples.

H X

Let X be a K3 surface and let be an ample divisor on . Consider the moduli space

M Q c Q

H of semistable sheaves of rank two with determinant and second Chern

c O mH H

number . Since twisting with does not change stability with respect to , the map

E E mH M Q c M QmH c m H

de®nes an isomorphism H

Q mc QH

. Thus we may assume that is ample from the very beginning. We wish to

M Q c show that in many instances the birational structure of the moduli space H can

be compared with the one of the Hilbert scheme of the same dimension. In order to state the

n n n c Q

theorem we need to introduce the following quantities: k

n n n c Q

and l .

Q n M Q k n

Theorem 11.3.1 Ð If is ample and , then the moduli space H is

birational to Hilb .

M Q k n M Q k n Q

Proof. By Theorem 4.C.7 the two moduli spaces H and

n O H Q

are birational for . Thus we may assume . Theorems 9.4.3 and 9.4.2 say

n M Q k n E

that for the moduli space Q is irreducible and the generic sheaf

M Q k n N M Q k n Q

is -stable and locally free. Let Q be the open dense

Hilb X

subset of all -stable locally free sheaves. We will construct a rational map

dim Hilb X

N which is generically injective. Since both varieties are smooth and

Hilb X Q dimN l n k n c

, this is enough to conclude that and

M Q k n

Q are birational.

By the Hirzebruch-Riemann-Roch formula

h X O n H H l n

Z Hilb X H X I n H

Hence for the generic we have Z . Using the

exact sequence

H X O n H H X O H X I n H

X Z Z

h X I n H Z Z

this implies Z for generic . In other words, for generic there

is a unique non-trivial extension

O F I n H

X Z

11.3 Examples 239

O n H Z

Moreover, such an F is locally free, for satis®es the Cayley-Bacharach

H L F

property (5.1.1). If F is not -stable, then there exists a line bundle with

c LH L O F

. Since such a line bundle cannot be contained in , there exists a curve

jL n H j Z Z

C containing . We show that this cannot happen for generic . Since

dim jL n H j

X is regular, it suf®ces to show that can be bounded from above by

l n C jL n H j h O C h g C C

. Let . Then C

c L c L

C n H c LH

. Together with the exact

sequence

O O C O C

X X C

this proves

h O C c L

c L c LH H c LH n H

By Hodge Index Theorem and , for

H l n L O n H h O C

. Hence X .

Z Hilb X

Thus, for the generic there exists a unique extension

O F I n H

Z Z

F Z F Z

and Z is -stable and locally free. Hence, associating the subscheme to the sheaf

Hilb X N h X F 2

de®nes a rational map , which is injective, since Z .

Our second example is very much in the spirit of the ®rst one. We use Serre correspondence to prove that certain moduli spaces on a K3 surface of special type are birational to the Hilbert scheme. Specializing to elliptic K3 surfaces enables us to handle a more exhaustive

list of moduli spaces; in particular those of higher rank sheaves.

X P X

Let be an elliptic K3 surface with a section . We assume that

X Z O Z O f f

Pic , where is the ®bre class. In particular, all ®bres are irre-

v v v v v f

ducible. Let be a Mukai vector such that and consider sheaves

E r rkE v c E v ch E r v

with , , and . (For the de®nition of the Mukai vector see Section 6.1.) If we consider stability with respect to a suitable polarization,

then a sheaf is -semistable if and only if the restriction to the generic ®bre is semistable

f f

5.3.2. Since , any semistable sheaf on the ®bre is stable. Therefore, with

X c f

respect to a suitable polarization semistable sheaves on with are -stable

(5.3.2, 5.3.6). Moreover, since the stability on the ®bre is unchanged when the sheaves are

f E

twisted with O , a sheaf is -stable with respect to a suitable polarization if and only

E f O f ch E f ch E

if is -stable. Thus, by twisting with and using , we

r f r can reduce to the case that the Mukai vector is of the form . The following

theorem is Proposition 6.2.6 in Section 6.2, which was stated there without proof.

r f r H mf

Theorem 11.3.2 Ð Let v and let be a suitable polarization

v M v Hilb X n

with respect to . Then H is irreducible and birational to , where

r r .

240 11 Birational properties

X n r r

Proof. Let us begin with a dimension check: dim Hilb and

dim M v v v r r r r H

H . Next, since is suitable,

E M v M v E M v

H H

any H is -stable. Hence is smooth. Moreover, if , then

m f m m m

E is -stable with respect to for all . Thus we may assume .

i n ii f

The assertion is proved by induction over the rank. De®ne v

i i r M v Hilb X

for . Note that H . We will de®ne an open dense subset

i i

U Hilb X U M v

and injective dominant morphisms H . Since all varieties n

are smooth of dimension , this suf®ces to prove the theorem.

n n

U Hilb X Z Hilb X H X I

Let be the open subset of all with Z

U n f

. We show that is non-empty: By the Hirzebruch-Riemann-Roch formula

O n f n h O n f h O

. Serre duality gives

H X O n f n f . The vanishing of the ®rst cohomology can be

computed as follows: By the exact sequence

H F O j H X O H X O n f

j F

F F

where are distinct generic ®bers, one has

h X O h X O n f h X O h F O j

j F j

The exact sequence

H X O H O H X O H X O

X O n f h X O h X O

and Serre duality imply h .

O n f O n f n

Therefore, h . Thus for the generic

Z Hilb X H X I n f U

the cohomology Z vanishes, i.e. .

U Hilb X

Let be the inclusion and assume we have already constructed an in-

i i

U M v

jective morphism H satisfying

i i

Z U E Z

(Ai ) If and , then

i i i

E f h E f h E f

h .

i h E

(Bi ) If , then .

i U

Note that (Ai ) holds true for be de®nition of . The Hirzebruch-Riemann-Roch for-

i i

E f h E f

mula gives and by (Ai ) one knows . Hence there

exists a unique non-trivial extension

i i

O E E f

11.3 Examples 241

i i i i

v E v c E f c E f H

Then . Since and is suitable with respect

i i

E to v , the sheaf (which is in fact locally free, but we do not need this) is -stable if

and only if the restriction of the extension to a generic ®bre F is non-split. Since the exten-

i i

H X E f Ext E f O

sion space is one-dimensional, this is the case

i i

Ext E f O Ext E j O F

if and only if the restriction homomorphism F is

i i

H F E j H X E f

non-trivial or, dualizing, if and only if F is non-trivial.

H X E f

The kernel of the latter map is a quotient of which vanishes by (Ai ). The

H F E j i i i

space F is non-trivial. Indeed, for this follows from (B ) and for

H F I nf j H F O E

F F

from Z . Hence is -stable and we de-

i i i

U M v Z E

®ne a morphism H by . The map is injective, because

i i i

h E h O h E f E

by (Ai ). This also shows that satis®es

i i

E E

(Bi ) for . To make the induction work we have to verify (A ) for . The van-

i i

H X E f H X E f

ishing of follows immediately from (Ai ) and

i i

X E f E H is implied by the stability of with respect to a suitable polariza-

tion.

i i

V M v E

i H i

If we denote by the open subset of sheaves satisfying (Ai ) and (B ),

i i

V M v H

then the arguments show that there exists a morphism i commuting

i i

with and .

M v

To conclude the proof we have to show that H is irreducible or, equivalently, that

i i i

U M v i U V i i

H is dominant for all . By construction for all . Let us

E M v h E

®rst assume that the generic H has exactly one global section, i.e. .

i i

i E

Then we prove the dominance of by induction over . Assume is dominant. Let

M v h E

H with , then there exists an exact sequence

O E E

E j F

Since F is stable for the generic ®bre , can only vanish along divisors contained in

®bres. Since all ®bres are irreducible, could only vanish along complete ®bers, which is

E E

excluded by h . Hence, does not vanish along any divisor. This implies that is

i i

E E f M v

torsion free and, as one easily checks, also -stable. Thus H . We

E V E h E h E h E

claim that i . Indeed, the stability of and imply .

h E if h E if

Moreover, using the stability of E on the generic ®bre, one gets

i h E f

for . Next, the Hirzebruch-Riemann-Roch formula yields and

E f h E f

h . Using the long exact cohomology sequence we obtain

i i

h E h O E f h E f h O f h E

, h , and

V E E h E f h O f

. Thus, indeed i . Hence, is in the image of

i i i

V M v U V

H i

i . Since by the induction hypothesis is dense in , this is

i i

U M v

enough to conclude that also is dense in .

E M v

We still have to verify that the generic H has exactly one global section.

E M v h E E h E j

Consider H with . Since is torsion free of rank

F P h E h E

( a generic ®bre), it is in fact a line bundle on . Using , we

E O

conclude . Thus there is an exact sequence

242 11 Birational properties

O f E E

E E

with h . Moreover, since the restriction of to the generic ®bre is a stable bun-

dle of degree one, an explicit calculation shows that the same is true for E . The generic

deformation of E also has -dimensional space of global sections if and only if the

f E E

inclusion O deforms in all direction with . We claim that this implies that the

E E Ext O f E

natural map Ext is trivial.

E E

Indeed, consider the relative Quot-scheme Q that parametrizes quotients . Then

Q M v E

H is dominant, and hence for generic the tangent map is surjective. Using the

notations of Proposition 2.2.7 this implies that the obstruction map

T S T M v Ext E E Ext K F Ext O f E

s H

vanishes. That the obstruction map is the natural one follows from the arguments in Section

Ext E E

2.A. We show that this leads to a contradiction whenever . Note that

Ext O f E Ext O f E O f E

Ext factorizes through the injection .

E E Ext O f E Thus, it suf®ces to consider the homomorphism Ext which

sits in the exact sequence

Ext E E Ext O f E Ext E E Ext E E Ext O f E

Ext E E Ext E E Ext E E k k

In this sequence , and as well, because of

E E E E

Hom and the fact that any deformation of the quotient would produce a

f E E E deformation of O and thus more global sections of (Here we use that is

stable and, therefore, does not admit any non-scalar automorphisms). Furthermore, since

O f E Ext E E Ext O f E Ext , the homomorphism is surjective.

In the exact sequence

Ext O f O f Ext O f E Ext O f E Ext O f O f

the ®rst term vanishes and the last term is isomorphic to k . This yields the lower bound

O f E h E f E

ext . Using the stability of , one checks that

h E f h E f

and, hence,

h E f E f E f

O f E Ext E E Ext O f E

Thus ext and, therefore, the map does

2 not vanish for .

Comments: Ð Theorem 11.1.2 was communicated to us by GÈottsche. 11.3 Examples 243

Ð Theorem 11.1.4 in the special case of irrational surfaces of negative Kodaira dimension, which

are all irregular, was proved by Ballico and Chiantini in [14].

Ð Theorem 11.1.5 is due to J. Li [149]. Using the special desingularization M we could simplify

some of the arguments. In the original version there is a numerical condition on the intersection of

c Q

with the -curves. Li explained to us how this can be avoided.

Ð 11.1.7 can be found in [112] and were certainly also known to Li.

Q O

Ð Theorem 11.3.1 was proved by Zuo [262] for X and generalized by Nakashima [197].

PicX Z

The case was also considered by O'Grady [205]. The result holds in fact without the

assumption n . Ð Theorem 11.3.2 is due to O'Grady [209]. Our presentation is slightly different, mostly because we were only interested in the birational description, whereas O'Grady aims for a description of the universal family as well. 244

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Index action of an algebraic group, 81 Ð expected of moduli space, 103 Albanese morphism, 225, 231 Ð of a sheaf, 3

Albanese variety, 126, 225 dimension estimate, 199±213 s

Albanses morphism, 229 Ð for M , 101

ample Ð for R , 104 Ð pseudo± divisor, 64 Ð for ¯ag schemes, 54 Ð vector bundle, 64 Ð for Quot-scheme, 44 Ð Ð tensor product of, 66 Ð general, 53 ample line bundle on moduli space, 184 discriminant, 71 ampleness criterion Donaldson-Uhlenbeck compacti®cation, 195 Ð for Cartier divisors, 64 Ð for vector bundles on curves, 65 elementary transformation, 129, 137 annihilator ideal sheaf, 3 Enriques classi®cation, 235 Atiyah class, 218, 229 equivariant morphism, 81 Auslander-Buchsbaum formula, 4 Euler characteristic Ð of a pair of sheaves, 141 Bertini Theorem, 8 Ð of a sheaf, 9 Bogomolov extension Ð Inequality, 72, 171±173 Ð small, 49 Ð Restriction Theorem, 174 Ð universal, 37 boundary, 201 exterior powers, 67 boundedness Ð Grothendieck Lemma, 29 family Ð Kleiman Criterion, 29 Ð bounded, 28 Ð of a family, 28 Ð ¯at, 32 Ð of semistable sheaves, 70 Ð quasi-universal, 105 Ð Ð on curves, 28 Ð universal, 105 Ð Ð existence of, 107 canonical class of moduli space, 195 ®ltration Cayley-Bacharach property, 123 Ð Harder-Narasimhan, 16, 26 chamber, see wall Ð Ð relative, 46 cohomology class Ð Ð under base ®eld extension, 17 Ð primitive, 151 Ð Jordan-HÈolder, 22, 26 connection, 61, 218 Ð torsion, 3 cup product, 216 ®nite coverings, 62 ¯ag-schemes, 48 deformation theory, 49 ¯atness criterion, 33 descent, 87 Flenner, Theorem of, 161 determinant bundle, 9, 37 form Ð of a family, 178 Ð one-, 223±226 Ð on the moduli space, 180 Ð two-, 215, 223±226, 232 determinantal variety, 121 Ð Ð non-degeneracy of, 226±229 differentials with logarithmic poles, 129 frame bundle, (projective), 83 dimension framed module, 111 262 Index functor Kleiman's Transversality Theorem, 121 Ð (universally) corepresentable, 38 Kobayashi-Hitchin Correspondence, 67 Ð pro-represented, 53, 101 Kodaira dimension, 230 Ð representable, 38 Ð of Hilbert scheme, 231 functor category, 38 Ð of moduli space, 232 Kodaira-Spencer map, 221 Geometric Invariant Theory, 85 Gieseker's construction, 109 Langton, Theorem of, 55, 189 Gieseker-stability, 13 Le Potier±Simpson Estimate, 68 GIT, see Geometric Invariant Theory limit point, 86, 96 Grassmannian, 38 linear determinant, 92 Ð tangent bundle of, 44 linearization Grauert-MÈulich Theorem, 57, 59, 61 Ð of a group action, 84 Grothendieck group, 36, 178 Ð of a sheaf, 83 Grothendieck Lemma, 29 local complete intersection, 44, 103

Grothendieck's Theorem 1.3.1 on vector bundles Ð general criterion, 53

on P , 14 Luna's Etale Slice Theorem, 86, 113 Grothendieck-Riemann-Roch formula, 195 group manifold Ð algebraic, 81 Ð hyperkÈahler, 150 Ð reductive, 85 Ð irreducible symplectic, 150 Mehta, Theorem of ± and Ramanathan, 164 Hermite-Einstein metric, 67 moduli functor, 80 Hilbert polynomial, 9 moduli space Ð reduced, 10 Ð canonical class of, 195 Hilbert scheme, 41, 92 Ð differential forms on, 225 Ð Kodaira dimension, 231 Ð ®ne, 105 Ð of K3 surface, 150, 156, 238 Ð local properties of, 101

Ð Ð Hodge structure, 155 Ð of -semistable sheaves, 190 Ð smoothness, 104 Ð of coherent sheaves, 80, 91

Hilbert-Mumford Criterion, 86, 96 Ð Ð Hodge structure, 148, 156

Hilbert-to-Chow morphism, 92 Ð Ð on P , 235 Hodge Index Theorem, 132, 172 Ð Ð on a curve, 100, 103, 187, 195, 204 Hodge structure Ð Ð on abelian surface, 228, 236 Ð of irreducible symplectic manifold, 155 Ð Ð on elliptic K3 surface, 135, 152, 239 Ð of moduli space, 148 Ð Ð on elliptic surface, 139, 237, 239 Ð of surface, 146 Ð Ð on Enriques surface, 237 Ð Ð on ®bred surface, 131 invariant morphism, 82 Ð Ð on hyperelliptic surface, 237 irreducibility Ð Ð on irregular surface, 231 Ð of moduli space, 203 Ð Ð on K3 surface, 133, 151, 156, 228, 236, 238 Ð of Quot-scheme, 157 Ð Ð on rational surface, 235 isotropic vector, 146 Ð Ð on ruled surface, 132, 235 isotropy subgroup, 82 Ð Ð on surface of general type, 228, 232, 237 Ð Ð two-dimensional, 144

K -groups, see Grothendieck group Ð Ð zero-dimensional, 143 Kleiman Criterion, 29 Ð of framed modules, 112 Index 263

Ð of polarized K3 surfaces, 152 quadratic form, 155, 156 Ð of simple sheaves, 118 Quot-scheme, 39 Ð tangent bundle of, 222 quotient Mukai vector, 142 Ð (universal) categorical, 82 multiplicity of a sheaf, 10 Ð (universal) good, 82 Mumford Criterion, see Hilbert-Mumford Crite- rion Ramanathan, Theorem of Mehta and, 164 Mumford-Castelnuovo regularity, 28 re¯exive hull of a sheaf, 6 Mumford-Takemoto-stability, 13 regular section, 7 regular sequence, 4, 9, 28, 68 Nakai Criterion, 64 regularity, see Mumford-Castelnuovo ± nef divisor, 64 resolution Newton polynomial, 219 Ð injective, 48 normality Ð locally free, 36

Ð of hyperplane section, 8 restriction of -semistable sheaves Ð of moduli space, 202 Ð Bogomolov's Theorems, 170±177 numerically effective divisor, see nef divisor Ð Flenner's Theorem, 160±164 Ð The Theorem of Mehta-Ramanathan, 164±170 obstruction, 43 restriction to hypersurface

Ð comparison of deformation ±, 51 Ð of -semistable sheaf, 58±62 Ð for deformation of a ¯ag of subsheaves, 51 Ð of pure sheaf, 8 Ð for deformation of a sheaf, 50 Ð of re¯exive sheaf, 8 Ð theory, 49 obstruction theory, 53 S-equivalence, 22, 80, 91 open property, 35, 45 saturation of a subsheaf, 4 openness semistability Ð of semistability, 45 Ð behaviour under ®nite coverings, 63 orthogonal, 178 Ð of exterior and symmetric powers, 67 Ð of tensor product, 61 period point, 155 Serre construction, see Serre correspondence Picard group Serre correspondence, 123, 136, 238 Ð equivariant, 87, 179, 182 Ð higher rank, 128

Ð of moduli space, 180, 182 Serre subcategory, 24 S

PlÈucker embedding, 42 Serre's condition k c , 4 point sheaf

Ð (semi)stable, 85 Ð m-regular, 27 Ð good, see good sheaf Ð degree, 13 Ð properly semistable, 85 Ð dual of, 5 Poisson structure, 229 Ð good, 202 polarization Ð maximal destabilizing sub-, 16 Ð change of, 114 Ð polystable, 23, 63, 67 Ð suitable, 131 Ð pure, 3, 45

principal G-bundle, 83, 91 Ð rank of, 10 pro-represented functor, see functor Ð re¯exive, 6 pseudo-ample divisor, 64 Ð regularity of, 28 purity of a sheaf, 3 Ð simple, 12, 45 Ð slope of, 14

264 Index n

Ð stable(semi), 11, 25 Ð on P , 19±21

P ÐÐ -, 14, 26 Ð on , 14 Ð Ð e-, 74, 207 Ð stable Ð Ð geometrically, 12, 23 Ð Ð existence, 125, 128, 130 Ð Ð properly, 80, 108

singular points of a sheaf, 9 wall, 114, 118 G slope, see sheaf weight, of a m -action, 96 smoothness Ð criterion, 102 Yoneda Lemma, 38 Ð generic ± of moduli space, 202 Ð of Hilbert scheme, 104 Zariski tangent space, 42 socle, 23 Ð of ¯ag scheme, 53 Ð extended, 23 Ð of Grassmannian, 44

Ð of a torsion sheaf, 157 Ð of moduli space, 101 splitting of vector bundles on P , 14 Ð of Quot-scheme, 44 stabilizer, see isotropy subgroup strati®cation Ð double-dual, 210 Ð ¯attening, 33 support of a sheaf, 3 surface, abelian etc., see moduli space of coherent sheaves on ... symmetric powers, 67 symmetric product, 91 symplectic structure, 215, see also irreducible symplectic manifold Ð closed, 215 Ð non-degenerate, 215, 228

tangent bundle n

Ð of P is stable, 21 Ð of Grassmann variety, 44 Ð of moduli space, 222 trace map, 63, 102, 113, 217 traceless Ð endomorphisms, 102, 202 Ð extensions, 102, 202 Ð Ð global bound for, 103 trivialization, universal, 85 variety Ð of general type, 231 Ð rational, 230 Ð unirational, 230 vector bundle Ð ample, see ample Ð globally generated, 121 265

Glossary of Notations

General notations

xc x

b round down of a real number .

xe x

d round up of a real number .

x maxfx g x

for a real number .

convention used in the de®nition of semistability, see p. 11.

Q T d

d vector space of polynomials of degree , p. 25.

Q T Q T Q T

d dd quotient vector space , p. 25.

m maximal ideal in a local ring.

r connection, p. 61.

V Hom V k k V

k dual of a -vector space . Schemes, varieties, morphisms

k ®eld, most of the time algebraically closed, in the second part of

the book in general of characteristic zero. k X in general scheme of ®nite type over , in the second part of the book a surface, which always means an irreducible smooth projective surface.

C mostly a smooth projective curve.

X X E E

Diagonal in a product , but see also: for a sheaf .

I Z X

Z ideal sheaf of a subscheme .

Z O Z

= Z , length of a zero-dimensional scheme . X

H often an ample or very ample divisor on .

X X H

deg degree of with respect to some ®xed ample divisor .

dim X X x dimO X x

x dimension of at = .

X X

kod Kodaira dimension of a variety , p. 230.

K X

X , canonical sheaf of a smooth variety.

X X

Pic Picard group of .

X G X

Pic equivariant Picard group of -linearized line bundles on , p. 87.

X X

Alb Albanese variety of , p. 126.

X X

CH Chow group of , p. 126.

K X X

Grothendieck group of coherent sheaves on .

X X

K Grothendieck group of locally free sheaves on .

K X K X K X X

, if is smooth, p. 178.

K c K

c , p. 180.

K K f h h g c

cH , p. 180.

X Pic X

Num , numerical equivalence, p. 172.

Num

K open cone in , p. 172.

X X

Coh category of coherent sheaves on , p. 3.

X X S symmetric product of , p. 91.

266 11 Glossary of Notations

X X Hilb Hilbert scheme of subschemes of of length , p. 41.