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SCHOOL ON

ALGEBRAIC GEOMETRY

26 July - 13 August 1999

Editor

Lothar Gottsche

The Abdus Salam ICTP

Trieste, Italy

SCHOOL ON ALGEBRAIC GEOMETRY { First edition

Copyright c 2000 by The Abdus Salam International Centre for Theoretical Physics

The Abdus Salam ICTP has the irrevocable and inde nite authorization to reproduce and dissem-

inate these Lecture Notes, in printed and/or computer readable form, from each author.

ISBN 92-95003-00-4

Printed in Trieste by The Abdus Salam ICTP Publications & Printing Section

PREFACE

One of the main missions of the Abdus Salam International Cen-

tre for Theoretical Physics in Trieste, Italy, founded in 1964 by Abdus

Salam, is to foster the growth of advanced studies and research in de-

veloping countries. To this aim, the Centre organizes a large number of

schools and workshops in a great variety of physical and mathematical

disciplines.

Since unpublished material presented at the meetings might prove

of great interest also to scientists who did not take part in the schools

the Centre has decided to make it available through a new publica-

tion titled ICTP Lecture Note Series. It is hoped that this formally

structured pedagogical material in advanced topics will be helpful to

young students and researchers, in particular to those working under

less favourable conditions.

The Centre is grateful to all lecturers and editors who kindly autho-

rize the ICTP to publish their notes as a contribution to the series.

Since the initiative is new, comments and suggestions are most wel-

come and greatly appreciated. Information can be obtained from the

Publications Section or by e-mail to \pub; off@ictp.trieste.it". The

series is published in house and also made available on-line via the ICTP

web site: \http://www.ictp.trieste.it".

M.A. Virasoro

Director

v

Contents

Christoph Sorger

Lectures on Moduli of Principal G-Bundles Over Algebraic Curves . . . . . . 1

Geir Ellingsrud and Lothar Gottsche

Hilbert Schemes of Points and Heisenberg Algebras . . . . . . . . . . . . . . . . . . . . 59

Lothar Gottsche

Donaldson Invariants in Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 101

John W. Morgan

Holomorphic Bundles Over Elliptic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 135

C.S. Seshadri

Degenerations of the Moduli Spaces of Vector Bundles on Curves . . . . . 205

Eduard Looijenga

A Minicourse on Moduli of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Richard Hain

Moduli of Riemann Surfaces, Transcendental Aspects . . . . . . . . . . . . . . . . . 293

Makoto Matsumoto

Arithmetic Fundamental Groups and Moduli of Curves . . . . . . . . . . . . . . . 355

vii

Introduction

This is the rst volume of a new series of lecture notes of the Abdus Salam

International Centre for Theoretical Physics. These new lecture notes are

put onto the web pages of the ICTP to allow people from all over the world

to access them freely. In addition a limited number of hard copies is printed

to be distributed to scientists and institutions which otherwise possibly do

not have access to the web pages.

This rst volume contains the lecture notes of the School on Algebraic

Geometry which took place at the Abdus Salam International Centre for

Theoretical Physics from 26 July to 13 August 1999 under the direction

of Lothar Gottsche (ICTP), Joseph Le Potier (Universite Paris 7), Eduard

Looijenga (University of Utrecht), M.S. Narasimhan (ICTP).

The school consisted of 2 weeks of lecture courses and one week of con-

ference. This volume contains the notes of most of the lecture courses in the

rst two weeks. The topic of the school was moduli spaces. More specif-

ically the lectures were devided into three subtopics: principal bundles on

Riemann surfaces, moduli spaces of vector bundles and sheaves on projective

varieties, and moduli spaces of curves.

The school was nancially supported by ICTP and by a grant of the

European commission. I take this opportunity to thank the other organizers.

We are grateful to all the lecturers and speakers at the conference for their

contribution to the success of the school.

Lothar Gottsche

April, 2000

Lectures on moduli of principal

G-bundles over algebraic curves

Christoph Sorger

Mathmatiques, Universit de Nantes,

e e

BP 92208, 2, Rue de la Houssinire, F-44322 Nantes Cedex 03, France

e

Lecture given at the

School on Algebraic Geometry

Trieste, 26 July { 13 August 1999

LNS001001

christoph.sorger@math.univ-nantes.fr

Contents

1. Introduction 5

2. Generalities on principal G-bundles 6

3. Algebraic stacks 8

4. Topological classication 20

5. Uniformization 22

6. The determinant and the pfaan line bundles 26

7. Ane Lie algebras and groups 33

8. The innite Grassmannian 39

9. The ind-group of loops coming from the open curve 43

10. The line bundles on the moduli stack of G-bundles 45

References 56

Moduli of G-bundles 5

1. Introduction

These notes are supposed to be an introduction to the moduli of G-bundles

on curves. Therefore I will lay stress on ideas in order to make these notes

more readable. My presentation of the subject is strongly in

uenced by

the work of several mathematicians as Beauville, Laszlo, Faltings, Beilinson,

Drinfeld, Kumar, Narasimhan and others.

In the last years the moduli spaces of G-bundles over algebraic curves have

attracted some attention from various subjects like from conformal eld the-

ory or Beilinson and Drinfeld0s geometric Langlands program [5]. In both

subjects it turned out that the \stacky" point of view is more convenient

and as the basic motivation of these notes is to introduce to the latter sub-

ject our moduli spaces will be moduli stacks (and not coarse moduli spaces).

As people may feel uncomfortable with stacks I have included a small in-

troduction to them. Actually there is a forthcoming book of Laumon and

Moret-Bailly based on their preprint [15] and my introduction merely does

the step -1, i.e. explains why we are forced to use them here and recalls the

basic results I need later.

So here is the plan of the lectures: after some generalities on G-bundles,

I will classify them topologically. Actually the proof is more interesting

than the result as it will give a

avor of the basic theorem on G-bundles

which describes the moduli stack as a double quotient of loop-groups. This

\uniformization theorem", which goes back to A. Weil as a bijection on sets,

will be proved in the section following the topological classication.

Then I will introduce two line bundles on the moduli stack: the deter-

minant and the pfaan bundle. The rst one can be used to describe the

canonical bundle on the moduli stack and the second to dene a square-

root of it. Unless G is simply connected the square root depends on the

choice of a theta-characteristic. This square root plays an important role in

the geometric Langlands program. Actually, in order to get global dieren-

tial operators on the moduli stack one has to consider twisted dierential

operators with values in these square-roots.

The rest of the lectures will be dedicated to describe the various objects

involved in the uniformization theorem as loop groups or the innite Grass-

mannian in some more detail.

6

2. Generalities on principal G-bundles

In this section I dene principal G-bundles and recall the necessary back-

ground I need later. Principal G-bundles were introduced in their generality

by Serre in Chevalley0s seminar in 1958 [19] based on Weil0 s \espaces brs

e

algbriques" (see remark 2.1.2).

e

2.1. Basic denitions. Let Z be a scheme over an algebraically closed eld

k, G be an ane algebraic group over k.

2.1.1. Denition. By a G-bration over Z, we understand a scheme E on

which G acts from the right and a G-invariant morphism : E ! Z. A

morphism between G-brations : E ! Z and 0 : E 0 ! Z is a morphism

of schemes ' : Z ! Z 0 such that = 0 '.

A G-bration is trivial if it is isomorphic to pr1 : Z G ! Z, where G

acts on Z G by (z; g):

= (z; g

).

A principal G-bundle in the Zariski, resp. tale, resp. fppf, resp. fpqc sense

e

is a G-bration which is locally trivial in the Zariski, resp. tale, resp. fppf,

e

resp. fpqc topology. This means that for any z 2 Z there is a neighborhood

U of z such that EjU is trivial, resp. that there is an tale, resp.

at of

e

'

nite presentation, resp.

at quasi-compact covering U 0 ! U such that

' (EjU ) = U 0 U EjU is trivial.

2.1.2. Remark. In the above denition, local triviality in the Zariski sense

is the strongest whereas in the fpqc sense is the weakest condition. If G is

smooth, then a principal bundle in the fpqc sense is even a principal bundle

in the tale sense ([9], x6). In the following we will always suppose G to be

e

smooth and we will simply call G-bundle a principal G-bundle in the talee

sense. If G = GLr or if Z is a smooth curve (see Springer0s result in [22],

1.9), such a bundle is even locally trivial in the Zariski sense, but it was

Serre0 s observation that in general it is not. He dened those groups for

which local triviality in the Zariski sense implies always local triviality in

the tale sense to be special. Then, for semi-simple G, Grothendieck (same

e

seminar, some exposs later) classied the special groups: these are exactly

e

the direct products of SLr 0 s and Sp2r 0 s.

Remark that if the G-bundle E admits a section, then E is trivial. Dene

the following pointed (by the trivial bundle) set :

He1t(Z; G) = fG-bundles over Zg=isomorphism:

Moduli of G-bundles 7

2.2. Associated bundles. If F is a quasi-projective k-scheme on which G

acts on the left and E is a G-bundle, we can form E(F) = E G F the

associated bundle with ber F. It is the quotient of E F under the action

of G dened by g:(e; f) = (e:g; g 1 f). The quasi-projectivity1 of F is needed

in order to assure that this quotient actually exists as a scheme.

There are two important cases of this construction.

2.2.1. The associated vector bundle. Let F be a vector space of dimension

n. Suppose G = GL(F). Then G acts on F from the left and we can

form for a G-bundle E the associated bundle V = E(F). This is actually a

vector bundle of rank n. Conversely, for any vector bundle V of rank n the

n

associated frame bundle E (i.e. IsomOZ (OZ ; V )) is a GLn-bundle.

2.2.2. Extension of structure group. Let : G ! H be a morphism of al-

gebraic groups. Then G acts on H via , we can form the extension of the

structure group of a G-bundle E, that is the H-bundle E(H). Thus, we have

dened a map of pointed sets

He1t (Z; G) ! He1t (Z; H)

Conversely, if F is an H-bundle, a reduction of structure group is a G-

bundle E together with an isomorphism of G-bundles : E(H) ! F.

2.2.3. Lemma. Suppose : G ,! H is a closed immersion. If F is an

H-bundle, denote F(H=G) simply by F=G. There is a natural one to one

correspondence between sections : Z ! F=G and reductions of the structure

group of F to G.

Proof. View F ! F=G as a G-bundle and consider for : Z ! F=G the

pullback diagram

F F

G

G G

G

F=G

Z

which denes the requested reduction of the structure group.

1 In fact it is enough to suppose that F satises the property that any nite subset of

F lies in an ane open subset of F.

8

2.3. G-bundles on a curve. Let X be a smooth and connected curve. By

the above quoted theorem of Springer ([22], 1.9), all G-bundles over X are

locally trivial in the Zariski topology, so the reader might ask why I insisted

on the tale topology in the above denition. The reason is that in order to

e

study G-bundles on X, we will study families of G-bundles parameterized

by some k-scheme S. By denition, these are G-bundles on XS = X S,

and here is where we need the tale topology.

e

A warning: it is not a good idea to dene families point-wise. Let0 s look

at the example of Or . Then we may view (considering Or GLr and

using Lemma 2.2.3) an Or -bundle as a vector bundle E together with an

isomorphism : E ! E such that = (I denote here and later the

transposed map of by ). The point is as follows. If E is a vector bundle

over XS together with an isomorphism : E ! E such that for all closed

point s 2 S the induced pair (Es ; s ) is an Or -bundle, this does not imply

in general that (E; ) itself is an Or -bundle.

3. Algebraic stacks

3.1. Motivation. Given a moduli problem such as classifying vector bun-

dles over a curve, there are essentially two approaches to its solution: coarse

moduli spaces and algebraic stacks. The former, introduced by Mumford,

are schemes and are constructed, after restricting to a certain class of ob-

jects such as semi-stable bundles in the above example, as quotients of some

parameter scheme by a reductive group using geometric invariant theory.

However they do not - in general - carry a universal family and may have ar-

ticial singularities coming from the quotient process in their construction.

So in order to construct objects on the coarse moduli space, one consid-

ers generally rst the parameter space (which carries a universal family) and

then tries to descend the constructed object to the moduli space which might

be tricky or impossible.

In our case of the geometric Langlands program a special line bundle on

the moduli space (i.e. a certain square root of the dualising sheaf) will play

a particular role. However, it can be shown, that even if there is a functo-

rial construction of this line bundle (hence a line bundle on the parameter

scheme), it does not - for general G - descend to the coarse moduli space of

semi-stable G-bundles.

It turns out, for this and other reasons, that in order to study the questions

related to the geometric Langlands program, one has to consider the latter,

i.e. the \stacky" solution to the moduli problem. So in my lectures I will

Moduli of G-bundles 9

concentrate on the moduli stack of principal G-bundles and as there are not

many references for stacks for the moment, I will recall in this section the

ideas and properties of stacks I need in order to properly state and prove

the basic results for the program.

3.1.1. The moduli problem. The basic moduli problem for G-bundles on a

projective, connected, and smooth curve X=k is to try to represent the func-

tor which associates to a scheme S=k the set of isomorphism classes of fam-

ilies of G-bundles parameterized by S:

MG;X : (Sch=k)op ! Set

E

S 7! # =s

G

SX

Now, as G-bundles admit in general non trivial automorphisms (the auto-

morphism group of a G-bundle contains the center of G), we can0 t expect

to be able to solve the above problem, i.e. nd a scheme M that represents

the above functor. Loosely speaking, if it would exist we should be able,

given any morphism ' from any scheme S to M, to recover uniquely a fam-

ily E parameterized by S such that the map dened by s 7! [Es ] denes

the morphism '. As this should in particular apply to the closed points

Spec(k) 2 M, the above translates that not only for every G-bundle one is

able to choose an element in its isomorphism class with the property that

this choice behaves well under families, but also that there is only one such

choice with this property. This clearly is an obstruction which makes the

existence of M unlikely and can be turned into a rigorous argument.

However, I will not do this here, but rather discuss the rst non trivial

case of G = k , i.e. the case of rank 1 vector bundles. Then a possible

candidate for M is the jacobian J(X). We know that J(X) parameterizes

isomorphism classes of line bundles on X and that there is a Poincar bundle

e

P on J(X) X. Hence we get, for every j 2 J(X), a canonical element in

the isomorphism class it represents, namely Pj , and this choice is compatible

with families (pullback P to the family). The point in this example is that

this choice is not unique as P

pr1 (A) is also a Poincar bundle for A 2

e

Pic(J(X)).

Actually what we can do here is to consider a slightly dierent functor, by

xing a point x 2 X and looking at pairs (L; ) of line bundles together with

an isomorphism : Lx ! k. Such a choice determines uniquely a Poincar e

bundle P and J(X) (together with P) actually represents the functor dened

10

by such pairs. The above process of adding structure to the functor in order

to force the automorphism group of the considered objects to be trivial is

sometimes called to \rigidify" the functor.

Let us return to our original moduli problem. As I explained above, the

main problem is the existence of non trivial automorphisms and there is

nothing much we can do about this, without adding additional structures

which may be complicated in the general case and denitively changes the

moduli problem. Grothendieck0 s idea to avoid the diculties posed by the

existence of these non trivial automorphisms is simple : keep them. However,

as we will see, carrying out this idea is technically quite involved.

So how to keep the automorphisms? If we do not want to mod out the

automorphisms what we can do is to replace the set of automorphisms classes

of G-bundles over S X by the category which has as objects such bundles

and as morphisms the isomorphisms between them.

By denition, the categories we obtain have the property that all arrows

are invertible; categories with this property are called groupoids. In the

following the category of groupoids will be denoted by Gpd. It will be con-

venient to write groupoids and categories in the form fobjectsg + farrowsg:

Applying the above idea to our moduli problem gives a \functor"

MG;X : Sch=kop !Gpd

S 7 !fG-bundles on X Sg +

fisomorphisms of G-bundles on X Sg

Actually this is not really a functor as before, but a broader object, called

a \lax functor": if f : S 0 ! S is a morphism of k-schemes the pullback

denes a functor f : MG;X (S) ! MG;X (S 0 ). If g : g3(0)Tj/T2-334TT316091T5eIeTf6803285 0 148

(the)Tj

167

(S) ! cs2 0 T1 k Tf

65 TD33 TD TDD(

de8509 14 T43.92

Moduli of G-bundles 11

maps between sets), and natural transformations between morphisms (this

is new).

3.1.2. The quotient problem. Suppose that the linear group H acts on the

scheme Z. Suppose moreover that the action is free. Then Z=H exists as a

scheme and the quotient morphism : Z ! Z=H is actually an H-bundle.

f

What are the points of Z=H? If S ! Z=H, we get a cartesian diagram

Z0 Z

G

H H

fG

Z=H

S

So f denes an H-bundle Z 0 H S and an H-equivariant morphism . If

!

the action of H is not free, the quotient Z=H does not, in general, exist as

a scheme, however what we can do is to consider the following lax functor

[Z=H] : Sch=kop !Gpd

S 7 !f(Z 0 ; ) =Z 0 H S is a H-bundle and : Z 0 ! Z

!

is a H-equivariant morphismg +

fisomorphisms of pairsg

This denition makes sense for any action of H on Z and the \quotient

map" Z ! [Z=H] (we will see in a moment what this means) behaves like

an H-bundle.

3.1.3. The idea then is to dene a stack exactly as such lax functors, after

imposing some natural topological conditions on them. Of course this may

seem to be somehow cheating but Grothendieck showed us that one can

actually do geometry with a certain class of such stacks which he called

algebraic.

After the above motivation, the plan for the rest of the section is:

Grothendieck Topologies

k-spaces and k-stacks

Descent

Algebraic stacks

12

3.2. Grothendieck Topologies. Sometimes in algebraic geometry we need

to use topologies which are ner than the Zariski topology, especially when

interested in an analogue of the inverse function theorem. Over C , there

is the classical topology, although using it leads to worries about the alge-

braicity of analytically dened constructions. Otherwise one has to use a

Grothendieck topology such as the tale topology.

e

A Grothendieck topology is a topology on a category. The category might

be similar to the category Zar(Z) of Zariski open sets of a k-scheme Z, or it

might be an ambient category like Sch=k or A=k. Grothendieck topologies

are most intuitively described using covering families, which describe a basis

or a pretopology for the topology.

3.2.1. Covering families. In this approach a Grothendieck topology (or pre-

topology) on a category C with ber products is a function T which assigns

to each object U of C a collection T(U) consisting of families fUi 'i Ugi2I

!

of morphisms with target U such that

if U 0 !'U is an isomorphism, then fU 0 ! Ug is in T(U);

if fUi ! Ugi2I is in T(U), and if U 0 ! U is any morphism, then the

i

family fUi U U 0 ! U 0 gi2I is in T(U 0 );

if fUi 'i Ugi2I is in T(U), and if for each i 2 I one has a family

!

fVij ! Uigj2Ii in T(Ui), then fVij ! Ui ! Ugi2I;j2Ij is in T(U).

The families in T(U) are called covering families for U in the T-topology.

A site is a category with a Grothendieck topology.

3.2.2. Small sites. Let0 s look at some examples:

(i) If Z is a k-scheme consider the category of Zariski open subsets of Z.

S

A family fUi Ugi2I is a covering family for U if i2I Ui = U. The

resulting site is the small Zariski site or Zariski topology on Z written

ZZar .

(ii) If Z is a k-scheme, let Et=Z be the category whose objects are talee

maps U ! Z and whose morphisms are tale maps U 0 ! U compatible

e

with the projections to Z. A family fUi ! Ugi2I is a covering family

if the union of the images of the Ui is U (such a family is called a

surjective family). This is the small tale site or tale topology on Z

e

e

written Zt .

e

(iii) Replacing \tale" by \smooth" gives a topology on Smooth=Z called

e

the smooth topology. The small smooth site on a scheme is Zsm .

Using \

at of nite presentation" gives the fppf topology and a small

Moduli of G-bundles 13

site Zfppf. The letters \fppf" stand for \dlement plat de prsentation

e e

nie." There are also letters \fpqc" standing for \dlement plat et

e

quasi-compact." Intuitively, each of these successive topologies is ner

than the previous one because there are more open sets.

3.2.3. Big sites. One can also dene a topology on all schemes at once. The

category Sch=k of all k-schemes may be given the Zariski, tale, smooth,

e

fppf, and fpqc topologies. In these topologies the covering families of a

scheme U are surjective families fUi 'i Ugi2I of, respectively, inclusions of

!

open subschemes, tale maps, smooth maps,

at maps of nite presentation,

e

and

at quasi-compact maps. Each successive topology has more covering

families than the previous one and so is ner.

One can do the same thing to the category A=k of ane k-schemes.

3.2.4. Sheaves. A presheaf of sets on a category C with a Grothendieck

topology (of covering families) is a functor F : C op ! Set. A presheaf

is separated if for all objects U in C, all f; g 2 F(U), and all covering

families fUi 'i Ugi2I of U in the topology, the condition fjUi = gjUi for

!

all i implies f = g. A presheaf is a sheaf if it is separated and in addition,

whenever one has a covering family fUi 'i Ugi2I in the topology and a

!

system ffi 2 F(Ui )gi2I such that for all i; j, one has F(p1 )(fi ) = F(p2 )(fj )

i;j i;j

in F(Ui U Uj ), then there exists an f 2 F(U) such that fjUi = fi for all

i. A compact way to say the above is to say that F(U) is the kernel of the

following double arrow

F(p1 ) Y

Y i;j

! F(Ui U Uj )

F(Ui) 2

F(pi;j ) i;j

i2I

3.3. k-spaces and k-stacks. By a k-space (resp. k-group) we understand

a sheaf of sets (rep. groups) over the big site (A=k)fppf . A lax functor

X from A=kop to Gpd associates to any U 2 ob(A=k) a groupoid X(U)

and to every arrow f : U 0 ! U in A=k a functor f : X(U) ! X(U 0 )

together with isomorphisms of functors g f ' (f g) for every arrow

g : U 00 ! U 0 in A=k. These isomorphisms should satisfy the following

compatibility relation: for h : U 000 ! U 00 the following diagram commutes

h g f G h (f g)

o o

(g h) f

G (f g h)

14

If x 2 ob(X(U)) and f : U 0 ! U it is convenient to denote f x 2 ob(X(U 0 )

by xjU 0. A lax functor will be called a k-stack if it satises the following two

topological properties:

(i) for every U 2 ob(A=k) and all x; y 2 ob(X(U)) the presheaf

Isom(x; y) : A=U !Set

(U 0 ! U) 7 ! HomX(U 0 ) (xjU 0; yjU 0)

is a sheaf (with respect to the fppf topology on A=U).

(ii) Every descent datum is eective.

Recall that a descent datum for X for a covering family fUi 'i Ugi2I is

!

a system of the form (xi ; ji )i;j2I with the following properties: each xi is

an object of X(Ui ), and each ji : xijUji ! xjjUji is an arrow in X(Uji ).

Moreover, we have the co-cycle condition

kijUkji = kjjUkji jijUkji

where Uji = Uj U Ui and Ukji = Uk U Uj U Ui , for all i; j; k.

A descent datum is eective if there exists an object x 2 X(U) and in-

vertible arrows i : xj Ui ! xi in X(Ui ) for each i such that

jjUji = ji ijUji

for all i; j 2 I:

Any k-space X may be seen as a k-stack, by considering a set as a groupoid

(with the identity as the only morphism). Conversely, any k-stack X such

that X(R) is a discrete groupoid (i.e. has only the identity as automor-

phisms) for all ane k-schemes U, is a k-space.

3.3.1. Example. (The quotient stack) Let us consider again the quotient

problem of (3.1.2), in the more general setup of a k-group acting on a

k-space Z, which we will actually need in the sequel. The quotient stack

[Z= ] is dened as follows. Let U 2 ob(A=k). The objects of [Z= ](U)

are pairs (Z 0 ; ) where Z 0 is a -bundle over U and : Z 0 ! Z is -

equivariant, the arrows are dened in the obvious way and so are the functors

[Z= ](U) ! [Z= ](U 0 ).

3.4. Morphisms. A 1-morphism F : X ! Y will associate, for every U 2

f

ob(A=k), a functor F(U) : X(U) ! Y(U) and for every arrow U 0 ! U an

Moduli of G-bundles 15

isomorphism of functors (f) : fX F(U 0 ) ! F(U) fY

F(U) G

X(U) Y(U)

dr

Â¨

Â¨Â¨Â¨ Â¨Â¨ Â¨ Â¨ (f) fY

fX

"

X(U 0 ) Y(U 0 )

CG

F(U 0 )

satisfying the obvious compatibility conditions: (i) if f = 1U is an identity,

then (1U ) = 1F(U) is an identity and (ii) if f and g are composable, then

F(gf) is the composite of the squares (f) and (g) further composed with

the composition of pullback isomorphisms g f ' (f g) for X and Y (I

will not draw the diagram here).

A 2-morphism between 1-morphisms : F ! G associates for every

U 2 ob(A=k), an isomorphism of functors (U) : F(U) ! G(U):

F(U)

A

X(U) Y(U)

(U) S

G(U)

There is an obvious compatibility condition which I leave to the reader.

3.4.1. Remark. The above denitions of 1- and 2-morphisms make sense for

any lax functor. The compatibility conditions, which will be automatically

satised in our examples, may seem complicated, however can0 t be avoided

with this approach. The point is that typically in nature the pullback objects

f

f x for every x 2 ob(X)(U) and U 0 ! U are well dened up to isomorphism,

but that the actual object f x is arbitrary in its isomorphism class. Let0 s

have a closer look at our example MG;X . In this case taking the pullback

(f id) E of a G-bundle E on X U to X U 0 corresponds to take a

tensor product. This is well dened up to canonical isomorphism (it is the

solution of a universal problem) and we are so used to choose an element in

its isomorphism class that we generally (and safely) forget about this choice.

However, when comparing the functors g f and (f g) this choice comes

up inherently and we get only something very near to \equality" namely

a canonical isomorphism of functors. So once we see that our functors are

only lax (as opposed to strict) in general we see that we have to choose these

isomorphisms of functors in the denitions and then all sorts of compatibility

conditions pop up naturally.

16

There is another, less intuitive but more intrinsic approach to lax functors

using k-groupoids. This is an essentially equivalent formalism which avoids

the choice of a pullback object, hence reduces the compatibility conditions.

As this is the point of view of [15], I will describe brie

y the relation between

the two (which may also help to facilitate the reading of the rst chapter

of [15]). I start with a lax functor X : (A=k)op ! Gpd to which I will

associate a category X together with a functor : X ! (A=k) (actually I

should denote X by X as well, but here I want to distinguish the two). The

objects of X are

a

ob X(U)

U2ob(A=k)

the morphisms going from x 2 ob X(U) to y 2 ob X(V ) are pairs (; f) with

f : U ! V an arrow in (A=k) and an arrow in X(U) from x to f y. A

convenient way to encode these pairs is as follows2

G f

x fy y

G

With these notations, the composite of two arrows

G f G g

x fy y gz z

G G

is dened to be

G f G G gf G

(gf) z

x fy fgz z

The functor is dened to send an object of X(U) to U and an arrow (; f)

to f. Looking at : X ! (A=k) we see that the categories X(U) are the

ber categories XU with objects the objects x of X such that (x) = U and

arrows the arrows f of X such that (f) = 1jU .

The functor : X ! (A=k) satises the following two properties (exer-

cise: prove this)

f

(i) for every arrow U 0 ! U in (A=k) and every object x in XU , there is

u

an arrow y ! x in X such that (u) = f

(ii) for every diagram z ! x u y in X with image U 00 ! U g U 0 in

f

v

h

(A=k) there is for every arrow U 00 ! U 0 such that f = gh a unique

w

arrow z ! y such that u = vw and (w) = h.

2 I learned this from Charles Walter0 s lectures on stacks in Trento some years ago.

Moduli of G-bundles 17

A functor : X ! (A=k) satisfying (i) and (ii) is called a k-groupoid

in [15]. So a lax functor denes a k-groupoid. On the other hand, given a

k-groupoid we can dene a lax functor as follows. To every U 2 ob(A=k)

f

we associate the ber category XU . If U 0 ! U is an arrow in (A=k) and

u

x 2 ob XU then by (i) we know that there is y ! x in X. From (ii) it follows

u

that y ! x is unique up to isomorphism. Now we choose { once and for all

u u

{ for every f and x such an arrow y ! x which we denote by f x ! x.

u

Moreover, for every arrow x0 ! x in X, we denote by f (u) the unique arrow

which make the following diagram commutative

f x0 x0

G

f (u) u

f x x

G

g f

We get a functor f : XU ! XU 0, and also for U 00 ! U 0 ! U an isomorphism

of functors g f ! (f g) satisfying the conditions of a lax functor.

For k-groupoids most of the basic denitions such as 1- and 2-morphisms

are more elegant: a 1-morphism is a functor F : X ! Y strictly compatible

with the projection to (A=k); the 2-morphisms are the isomorphisms of

1-morphisms.

3.5. Descent. The word \descent" is just another name for gluing appro-

priate for situations in which the \open sets" are morphisms (as in the tale

e

topology) rather inclusions of subsets (as in the Zariski topology). The basic

descent theorem says that morphisms of schemes can be \glued" together in

the

at topology if they agree on the \intersections". The same applies to

at families of quasi-coherent sheaves. Having the notion of a sheaf and a

stack to our disposition, faithfully

at descent can be stated as follows:

Theorem. Faithfully

at descent ([SGA 1], VIII 5.1, 1.1 and 1.2):

(i) (Faithfully

at descent for morphisms) For any k-scheme Z the functor

of points

Hom(A=k) ( ; Z) : (A=k)op ! Set is a k-space.

(ii) (Faithfully

at descent for

at families of quasi-coherent sheaves) For

any scheme Z, the lax functor (A=k)op ! Gpd dened by

S 7! fquasi-coherent OZk S -modules

at overSg + fisomorphismsg

is a k-stack.

18

Other descent results can be derived from these two. For instance, faith-

fully

at descent for principal G-bundles follows from (ii), i.e. the lax functor

MG;X of (3.1.1) is a k-stack.

3.6. Algebraic stacks. I now come to the denition of an algebraic stack,

then I will show in the next section that our k-stack MG;X is actually alge-

braic.

3.6.1. The ber of a morphism of stacks. Fiber products exist in the category

of k-stacks. I will not dene them here, but rather explain what is the ber

of a morphisms of stacks, as this is all I need here. Let F : X ! Y be a

morphisms of stacks, let U 2 ob(A=k) and consider a morphism : U ! Y,

that is an object of Y(U). The ber X is the following stack over U:

X : A=U !Gpd

(U 0 ! U) 7 !f(; ) = 2 ob(X)(U 0 ); : F() ! jU 0g +

f f

f(; ) ! (0 ; 0 ) = ! 0 s.t. F(f) = 0 g

3.6.2. Representable morphisms. The morphism F is representable if X is

representable as a scheme for all U 2 ob(A=k) and all 2 ob Y(U), i.e.

\the bers are schemes". All properties P of morphisms of schemes which

are stable under base change and of local nature for the fppf topology make

sense for representable morphisms of stacks. Indeed, one denes F to have P

if for every U 2 ob(A=k) and every 2 ob(Y ()) the canonical morphism

of schemes X ! U has P. Examples of such properties are

at, smooth,

surjective, tale, etc. ; the reader may nd a quite complete list in [15].

e

3.6.3. Denition. A k-stack X is algebraic if

(i) the diagonal morphism X ! X X is representable, separated and

quasi-compact p

(ii) there is a k-scheme P and a smooth, surjective morphism P ! X.

Actually the representability of the diagonal is equivalent to the following:

for all U 2 ob(A=k) and all 2 ob Y(U) the morphism of stacks U ! X is

representable. Hence (i) implies that p is representable (and so smoothness

and surjectivity of p make sense)

Suppose F : X ! Y is a representable morphism of algebraic k-stacks and

that Y is algebraic. Then X is algebraic also.

3.6.4. Proposition. Suppose Z is a k-scheme and H is a linear algebraic

group over k acting on Z. Then the quotient k-stack [Z=H] is algebraic:

Moduli of G-bundles 19

Proof. This follows from the denitions : a presentation is given by the

morphism p : Z ! [Z=H] dened by the trivial H-bundle on Z.

3.6.5. Proposition. The k-stack MGLr ;X of 3.1.1 is algebraic.

Proof. ([15],4.14.2.1)

3.6.6. Corollary. The k-stack MG;X of 3.1.1 is algebraic.

Proof. Choose an embedding G GLr . Using Lemma 2.2.3 we may (and

will) view a G-bundle E over a k-scheme Z as a GLr -bundle V together

with a section 2 H 0 (Z; V=G). Consider the morphism of k-stacks

' : MG;X ! MGLr ;X

dened by extension of the structure group. The corollary follows from the

above proposition and the following remark:

3.6.7. The above morphism is representable. Let U be a k-scheme and :

U ! MGLr ;X be a morphism, that is a GLr -bundle F over XU = X k U.

For any arrow U 0 ! U in A=k the GLr -bundle F denes a GLr -bundle

over XU 0 which we denote by FU 0 .

We have to show that the ber MG;X (), as dened in (3.6.2), is repre-

sentable as a scheme over U. As a U-stack, MG;X () associates to every

arrow U 0 ! U the groupoid dened on the level of objects by pairs (E; )

where E is a G-bundle over XU 0 and : E(GLr ) ! FU 0 is an isomorphism

of GLr -bundles. On the level of morphisms we have the isomorphisms of

such pairs, dened as follows: the pair (E1 ; 1 ) is isomorphic to the pair

(E2 ; 2 ) if there is an isomorphism : E1 ! E2 such that 2 (GLr ) = 1 .

Such an isomorphism is, if it exists, unique for, since G acts faithfully on

GLr , (GLr ) = 2 1 1 uniquely determines . Therefore, the ber is

a U-space. Moreover, the set of pairs (E; ) is canonically bijective to the

set HomXU 0 (XU 0; (F=G)U 0). An easy verication shows that this bijection

is functorial, i.e. denes an isomorphism between the U-space of the above

pairs and the functor which associates to U 0 ! U the above set of sections.

So we are reduced to show that the latter functor is representable. But

this follows from Grothendiecks theory of Hilbert schemes ([10], pp. 19{20),

once we know that F=G ! XU is quasi-projective. In order to see this last

statement we use Chevalley0 s theorem on semi-invariants: there is a repre-

sentation V of GLr with a line ` such that G is the stabilizer (in GLr ) of `.

We get an embedding GLr =G P(V ), hence the required embedding

F=G P(F (V )):

20

(Actually the line bundle on F=G which corresponds to the above embedding

is nothing else than the line dened by extension of the structure group of

the G-bundle F ! F=G via 1 where is the character dened by the

action of G on `.)

3.6.8. Proposition. Suppose G is reductive. The algebraic stack MG;X is

smooth of dimension dim(G)(g 1).

This follows from deformation theory. I will be rather sketchy here as

rendering precise the arguments below is quite long. Let E be a G-bundle.

Consider the action of G on g given by the adjoint representation and then

the vector bundle E(g). The obstruction to smoothness of MG;X lives in

H 2(X; E(g)) which vanishes since X is of dimension 1. The innitesimal

deformations of E are parameterized by H 1 (X; E(g)) with global automor-

phisms parameterized by H 0 (X; E(g)). Over schemes in order to calculate

the dimension we would calculate the rank of its tangent bundle. We can do

this here also but on stacks one has to be careful about how one understands

the \tangent bundle". We see this readily here: for example for G = GLr the

tangent space H 1 (X; End(E; E)) is not of constant dimension over the con-

nected components but only over the open substack of simple vector bundles.

Of course dim H 1 (X; End(E; E)) jumps exactly when dim H 0 (X; End(E; E))

jumps, so again one has to take care of global automorphisms. However, we

may consider the tangent complex on MG;X . In our case this complex is

Rpr1(E(g)) where E is the universal G-bundle over MG;X X, which may

be represented by a perfect complex of length one (see section 6.1.1 for this).

By denition, the dimension of the stack MG;X at the point E is the rank of

the cotangent complex at E, which is (E(g)). If G is reductive there is an

isomorphism g ! g of G-modules. Therefore we know that deg(E(g)) = 0

and then Riemann-Roch gives dim MG;X = dim(G)(g 1). If g(X) = 0, then

its dimension is dim(G), which may be surprising, but which is, in view of

the above, the only reasonable result we may ask for (the standard example

of a stack with negative dimension is BG = [=H] which is of dimension

dim H).

4. Topological classification

Here X is a compact connected oriented smooth real surface of genus g

and G a connected topological group. A topological G bundle E over X

is a topological space E on which G acts from the right together with a

G-invariant continuous map E ! X such that for every x 2 X there is an

Moduli of G-bundles 21

open neighborhood U of x such that EjU is trivial, i.e. isomorphic to U G

as a G-homogeneous space where G acts on U G by right multiplication.

4.1. Topological loop groups. Let x0 2 X and let D be a neighborhood

of x homeomorphic to a disc. Dene D = D x0 and X = X x0 .

Associated are the following three groups

LtopG = ff : D ! G=f is continuousg

LtopG = ff : D ! G=f is continuousg

+

LtopG = ff : X ! G=f is continuousg

X

By denition, we have the following inclusions:

Ltop G Ltop G Ltop G

+

X

Let Mtop be the set of isomorphism classes of topological G-bundles on X.

G;X

4.1.1. Proposition. There is a canonical bijection

! Mtop

top

LtopGnL G=Ltop G G;X

+

X

Proof. The basic observation is that if E is a topological G-bundle on X

then the restrictions of E to D and X are trivial. For the restriction to D

this is clear, since D is contractible; for the restriction to X we view X as a

CW-complex of dimension 2 and remark that, since G is connected, there is

no obstruction to the existence of a section of a G-bundle on X . It follows

that if we choose trivialization : EjD ! D G and : EjX ! X G

then the transition function

= jD1 is an element of Ltop G. On the other

hand, we may take trivial bundles on D and X and patch them together

by

in order to get a G-bundle E on X. Therefore there is a canonical

bijection

G

LtopG = f(E; ; ) =E ! X; : EjD ! D G; : EjX ! X Gg

Now, by construction, multiplying

2 Ltop G from the right by 2 Ltop G +

corresponds under this bijection to changing the trivialization by # ,

where # is the map D G ! D G dened by (z; g) 7! (z; g(z)) and

analogously multiplying from the left by 1 2 Ltop G corresponds to change

X top

the trivialization . It follows that dividing by L+ G forgets about the

trivialization and dividing by Ltop G forgets about the trivialization ,

X

hence the proposition.

4.1.2. Corollary. The set Mtop is in bijective correspondence with 1(G).

G;X

22

Proof. If

2 Ltop G, we denote by

: 1 (D ) ! 1 (G) the induced map.

Let be the positive generator of 1 (D ) and consider the map

f : Ltop G ! 1 (G)

7 !

()

Now f depends only on the double classes. In order to see this consider

for 2 Ltop G and 2 Ltop G the element 1

which we view as an

+ X

top G as follows: z 7! 1 (z)

(z)(z). Then remark that the

element of L

composite D ! D ! G is homotopically trivial since it extends to D.

For the composite D ! X ! G consider the induced map 1 (D ) !

1 (X ) ! 1(G) and remark (exercise) that the image of 1(D ) in 1(X )

has to sit inside the commutator subgroup. It follows that its image in 1 (G)

is trivial, since 1 (G) is abelian. Thus D ! X ! G is also homotopically

trivial. Therefore 1

is homotopic to

, hence f depends only on the

double classes. Then it is an easy exercise to see that the induced map on

the double quotient is indeed a bijection.

5. Uniformization

The uniformization theorem is the analogue of proposition 4.1.1 in the

algebraic setup. Let k be an algebraically closed eld, X be a smooth,

connected and complete algebraic curve over k and G be an ane algebraic

group over k. We choose a closed point x0 2 X and consider X = X fx0 g.

Remark that X is ane (map X to P1 using a rational function f with pole

of some order at x0 and regular elsewhere and remark that f 1(A 1 ) = X ).

What is the algebraic analogue of the \neighborhood of x0 homeomorphic

to a disc" of section 4? What we can do is to look at the local ring OX;x0 and

then consider its completion OX;x0 . Then Dx0 = Spec(OX;x0 ) will be conve-

b b

nient for if we choose a local coordinate z at x0 2 X then we may identify

OX;x0 with k[[z]], hence Dx0 with the \formal disc" D = Spec k[[z]] .

b

Moreover, Dx0 = D fx0 g is Spec(Kx0 ), where Kx0 is the eld of fractions

of OX;x0 . Using our local coordinate z we see that Kx0 identies to k((z)),

b

hence Dx0 to D = Spec k((z)) .

It will be convenient in the following to introduce the following notations:

if U = Spec(R) then we will denote DU = Spec R((z)) , DU = Spec R[[z]]

and XU = X U.

5.1. Algebraic loop groups. The algebraic analogue of the topological

loop group Ltop G is Homalg (D ; G), that is, the points of G with values in

Moduli of G-bundles 23

D , i.e. G k((z)) . This has to be made functorial so we will consider the

functor

LG : (A=k) !Grp

U = Spec(R) 7 !G R((z))

Actually that is a k-group (in the sense of 3.3). We dene the k-groups LX G

and L+ G as well by U 7! G O(XU ) and U 7! G R[[z]] respectively.

We denote QG the quotient k-space LG=L+G: this is the sheacation of

the presheaf

U = Spec(R) 7 ! G R((z)) G R[[z]] :

The k-group LX G acts on the k-space QG ; let [LX GnQG ] be the quotient

k-stack of 3.3.1.

5.1.1. Theorem. (Uniformization) Suppose G is semi-simple. Then there

is a canonical isomorphism of stacks

[L GnLG=L+ G] ! MG;X

X

Moreover, the LX G-bundle QG LX! MG;X is even locally trivial for the

G

tale topology if the characteristic of k does not divide the order of 1 (G(C )).

e

5.2. Key inputs. The theorem has two main inputs in its proof:

Trivializing G-bundles over XU (for this we need G semi-simple)

Gluing trivial G-bundles over XU and DU to a G-bundle over XU .

Both properties are highly non trivial in our functorial setup where U may

be any ane k-scheme, not necessarily noetherian. So I discuss them rst.

5.2.1. Trivializing G-bundles over the open curve. For general G it is not

correct that the restriction of a G-bundle to X is trivial. The basic examples

are of course line bundles. However, if we consider vector bundles with trivial

determinant (i.e. SLr -bundles) then this becomes true. The reason is that a

r

vector bundle E over X may be written as the direct sum OX det(EjX )

(translate to the analogue statement of nite module over a ring and use

that O(X ) is Dedekind as X is a smooth curve). Now if E is a vector

bundle with trivial determinant on XU we may ask whether, locally (for an

appropriate topology) on U, the restriction of E to XU is trivial. This is

indeed true (for the Zariski topology on U) and the argument proceeds by

induction on the rank r of E ([2], 3.5), the rank 1 case being trivial: consider

the divisor d = fx0 gU of XU and choose an integer n such that E(nd) has

24

no higher cohomology and is generated by its global sections. Then consider

a point u 2 U and a nowhere vanishing section s of E(nd)jXfug (count

dimensions in order to see its existence). Shrinking U, one may suppose

that this section is the restriction to E(nd) of a section which does not

vanish on XU . When restricting to XU we get an exact sequence

0 ! OXU ! EjXU ! F ! 0

where F is a vector bundle. But after shrinking U again we may assume

that F is trivial by induction and that the sequence splits, hence EjXU is

trivial.

The natural guess then is that the above trivialization property is true

for semi-simple G at least for the appropriate topology on U. This has been

proved by Drinfeld and Simpson.

Theorem (Drinfeld-Simpson). [7] Suppose G is semi-simple. Let E be

a G-bundle over XU . Then the restriction of E to XU is trivial, locally for

the fppf topology over U. If char(k) does not divide the order of 1 (G(C )),

then this is even true locally for the tale topology over U.

e

I will not enter into the proof, however I will invite the reader to have a

closer look at their note, as it uses some techniques which are quite useful

also in other contexts.

5.2.2. Gluing. Consider the following cartesian diagram

DU DU

G

XU XU

G

Given trivial G-bundles on XU and DU and an element

2 G R((z)) we

want to glue them to a G-bundle E on XU . The reader might say that this is

easy: just apply what we have learned about descent in section 3. However

some care has to be taken here: if U is not noetherian, then the morphism

DU ! XU is not

at! Nevertheless the gluing statement we need is true:

Theorem (Beauville-Laszlo). [3] Let

2 G R((z)). Then there exists

a G-bundle E on XU and trivializations : EjDU ! DU G, : EjXU !

XU G. Moreover the triple (E; ; ) is uniquely determined up to unique

isomorphism.

Moduli of G-bundles 25

Actually the above theorem is proved for vector bundles in [3] but the

generalization to G-bundles is immediate. Again, I will not enter into the

proof, but invite the reader to have a look at their note.

5.3. Proof of the uniformization theorem. Once the above two key

inputs are known, the proof of the uniformization theorem is essentially

formal.

We start considering the functor TG of triples:

TG : (A=k) !Set

G

U 7 !f(E; ; ) = E ! XU is a G-bundle with trivializations

: EjDU ! DU G; : EjXU ! XU G:g=

5.3.1. Proposition. The k-group LG represents the functor TG.

Proof. Let (E; ; ) be an element of TG (U). Pulling back the trivializations

and to DU provides two trivializations and of the pullback of E

over DU : these trivializations dier by an element

= 1 of G R((z))

(as usual U = Spec(R)). Conversely, if

2 G R((z)) , we get an element of

TG(U) by the Beauville-Laszlo theorem. These constructions are inverse to

each other by construction.

Now consider the functor of pairs PG :

PG : (A=k) !Set

G

U 7 !f(E; ) = E ! XU is a G-bundle with trivialization

: EjXU ! XU G:g=

5.3.2. Proposition. The k-space QG represents the functor PG .

Proof. Let U = Spec(R) be an ane k-scheme and q be an element of QG (U).

By denition of QG as a quotient k-space, there exists a faithfully

at ho-

momorphism U 0 ! U and an element

of G R0 ((z)) (U 0 = Spec(R0 )) such

that the image of q in QG (U 0 ) is the class of

. To

corresponds by 5.3.1

a triple (E 0 ; 0 ; 0 ) over XU 0 . Let U 00 = U 0 U U 0 , and let (E1 ; 100 ), (E2 ; 200 )

00 00

denote the pullbacks of (E 0 ; 0 ) by the two projections of XU 00 onto XU 0.

00 ((z)) dier by an element of G R00 [[z]] ,

Since the two images of

in G R

these pairs are isomorphic. So the isomorphism 200 100 1 over XU 00 extends to

00 00

an isomorphism u : E1 ! E2 over XU 00 , satisfying the usual co-cycle con-

dition (it is enough to check this over X , where it is obvious). Therefore

26

(E 0 ; 0 ) descends to a pair (E; ) on XR as in the above statement. Con-

versely, given a pair (E; ) as above over XU , we can nd a faithfully

at

homomorphism U 0 ! U and a trivialization 0 of the pullback of E over DU 0

(after base change, we may assume that the central ber of the restriction of

E to DU has a section then use smoothness to extend this section to DU ).

By 5.3.1 we get an element

0 of G R0 ((z)) such that the two images of

0

in G R00 ((z)) (with R00 = R0

R R0 ) dier by an element of G R00 [[z]] ; this

gives an element of QG (U). These constructions are inverse to each other

by construction.

5.3.3. End of the proof. The universal G-bundle over X QG (see 5.3.2),

gives rise to a map : QG ! MG;X . This map is LX G-invariant, hence

induces a morphism of stacks : LX GnQG ! MG;X . On the other hand

we can dene a map MG;X ! LX GnQG as follows. Let U be an ane k-

scheme, E a G-bundle over XU . For any arrow U 0 ! U, let T(U 0 ) be the set

of trivializations of EU 0 over XU 0 . This denes a U-space T on which the

group LX G acts. By Drinfeld-Simpson0s theorem, it is an LX G-bundle. To

any element of T(U 0 ) corresponds a pair (EU 0 ; ), hence by 5.3.2 an element

of QG (U 0 ). In this way we associate functorially to an object E of MG;X (U)

an LX G-equivariant map : T ! QG . This denes a morphism of stacks

MG;X ! LX GnQG which is the inverse of . The second assertion means

that for any scheme U over k (resp. over k such that char(k) does not divide

the order of 1 (G(C ))) and any morphism f : U ! MG;X , the pullback to U

of the bration is fppf (resp. tale) locally trivial, i.e. admits local sections

e

for the fppf (resp. tale) topology. Now f corresponds to a G-bundle E over

e

XU . Let u 2 U. Again by the Drinfeld-Simpson theorem, we can nd an fppf

(resp. tale) neighborhood U 0 of u in U and a trivialization of EjXU 0 . The

e

pair (E; ) denes a morphism g : U 0 ! QG (by 5.3.2) such that g = f,

that is a section over U 0 of the pullback of the bration .

6. The determinant and the pfaffian line bundles

Let X be a projective curve, smooth and connected over the algebraically

closed eld k.

6.1. The determinant bundle. Let F be a vector bundle over XS =

X k S, where S is a locally noetherian k-scheme. As usual we think of F

as a family of vector bundles parameterized by S.

Moduli of G-bundles 27

6.1.1. Representatives of the cohomology. In the following I will call a com-

plex K of coherent locally free OS -modules

0 ! K0 ! K1 ! 0

f

a representative of the cohomology of F if for every base change T ! S

gG

XT XS

p

u

f

T S

G

we have H i (f K ) = Ri u g F. In particular, if s 2 S is a closed point:

H i (Ks ) = H i(X; Fs )

Representatives of the cohomology of F are easy to construct in our setup.

Indeed, we may choose a resolution

0 ! P1 ! P0 ! F ! 0

of F by S-

at coherent OXS -modules such that p P0 = 0 (use Serre0 s theorem

A in its relative version to see its existence). Then we have p P1 = 0 and,

by base change for coherent cohomology, the complex

0 ! R1 p P1 ! R1 p P0 ! 0

is convenient. This result is generally quoted as choosing a perfect complex

of length one representing RpF in the derived category3 Dc (S)

6.1.2. The determinant bundle. The determinant of a complex K of locally

free coherent OS -modules 0 ! K 0 ! K 1 ! 0 if dened by

max max

^

0

( ^ K 1) 1

det(K ) = K

The determinant of our family F of vector bundles parameterized by S is

dened by4

DF = det(RpF) 1

3 All the derived category theory I need here and in the proof of 6.2.2 is in ([6],x1). The

category of complexes of OS -modules will be denoted by C(S); the category with the same

objects C(S) but morphisms homotopy classes of morphisms of C(S) will be denoted by

K(S). Finally D(S) is obtained by inverting the quasi-isomorphisms in K(S). A superscript

b (resp. subscript c) means that we consider the full sub-categories of bounded complexes

(resp. complexes with coherent cohomology).

4 The minus sign is chosen in order to get the \positive" determinant bundle.

28

In general, in order to calculate DF , we choose a representative K of the

cohomology of F and then calculate det(K ) 1 . This does not depend, up

to canonical isomorphism, on the choice of K (and this is the reason why

the above denition makes sense) [11].

By construction, the ber of DF at s 2 S is given as follows:

max max

^

0 (X; F )) 1

^ H 1 (X; F )

DF (s) = ( H s s

We may also twist our family F by bundles coming from X, i.e. consider

F

qE where E is a vector bundle on X. We obtain the line bundle

DF

qE , and this line bundle actually depends only on the class of E in the

Grothendieck group K(X) of X (check this!). It follows that we get a group

morphism, Le Potier0 s determinant morphism [16]

F : K(X) ! Pic(S)

u 7 !DF

qu

If our bundle F comes from a SLr -bundle, i.e. has trivial determinant,

twisting F by an element u 2 K(X) then taking determinants just means

taking the r(u)-th power of DF :

6.1.3. Lemma. Suppose F is a vector bundle on XS such that Vmax F is

the pullback of some line bundle on X. Then

r(u)

DF

qu = DF in Pic(S)

where r(u) is the rank of u.

Proof. We may suppose that u is represented by a vector bundle L and even

{ after writing L as an extension { that L is a line bundle. But then it

is enough to check it for L = OX ( p), for p 2 X, where it follows, after

considering 0 ! OX ( p) ! OX ! Op ! 0, from the fact DF

qOp is trivial

under our hypothesis on F.

6.1.4. Theta-functions. Twisting is particularly useful in order to produce

sections of (powers of) the determinant bundle. Suppose S is integral and

that F is a vector bundle on XS with trivial determinant. Choose a vector

bundle E such that Fs

q E has trivial Euler characteristic for some s. If

0 ! K0 ! K1 ! 0

is a representative of the cohomology of F

q E, then we know that the

rank n of K 0 is equal to the rank of K 1 , hence

may be locally represented

r(E)

as a n n-matrix. We get a section E = det(

) of DF , well dened

Moduli of G-bundles 29

up to an invertible function on S: the theta-function associated to E. In

particular, its divisor E is well dened with support the points s 2 S such

that H 0 (Fs

E) 6= 0.

If we suppose moreover that Ft

q E has trivial cohomology for some

t 2 S then E 6= S, i.e. the section E is non trivial; if there is t0 2 S such

that H 0 (X; Et0

E) 6= 0 then E 6= ;.

6.2. The pfaan line bundle. Suppose char(k) 6= 2 in this subsection.

Let F be a vector bundle over XS = X S, together with a quadratic non

degenerate form with values in the canonical bundle !X . We will view as

an isomorphism F ! F _ such that = _ , where F _ = HomOXS (F; q !X ).

6.2.1. Lemma. If K is a representative of the cohomology of F, then

K [ 1] is a representative of the cohomology of F _ .

Here5 K [ 1] denotes the complex supported in degrees 0 and 1

0 ! K 1

! K 0 ! 0:

Proof. In the derived category Dc (S), we have

Rp(F _ ) ! Rp (RHomOXS (F; q !X )) (F is locally free)

! RHom(Rp(F ); OS )[ 1] (Grothendieck-Serre duality)

Now if K represents the cohomology of F we see that RHom(K ; OS )[ 1]

represents the cohomology of F _ . But this is nothing else than K [ 1] as

the K i are locally free.

6.2.2. Proposition. There exists, locally for the Zariski topology on S, a

representative of the cohomology K of F and a symmetric isomorphism:

: K ! K [ 1]

such that and induce the same map in cohomology.

Proof. Choose a representative K of the cohomology of F and remark that

e

induces an isomorphism in the derived category Dc (S)

b

e

e e

K ! RpF ! Rp (F _ ) ! K [ 1]

which is still symmetric (this follows from the symmetry of and standard

properties of Grothendieck-Serre duality).

K is supported in degrees 1

5 This is compatible with the usual signs: the dual of

and 0; when translated to the right by 1, the dierential acquires a 1 sign.

30

The problem here is that this isomorphism is only dened in the derived

category: the proposition actually claims that we can get a symmetric iso-

morphism of complexes and this we only get Zariski locally.

First we may suppose that S is ane. Then the category of coherent

sheaves on S has enough projectives and as the K i are locally free we see

e

that is an isomorphism in Kc (S). Let ' be a lift of to Cc (S). We get a

b b

e e

morphism of complexes

G

K0 K1

e e

'0 '1

G

K 1 K 0

e e

which needs neither to be symmetric nor an isomorphism (it is only a quasi-

isomorphism). First we symmetrize: i = ('i +' i )=2 for i = 0; 1. Remark

1

that is still a quasi-isomorphism, inducing in cohomology. Then we x

s 2 S. A standard argument shows that there is, in a neighborhood of s,

another length one complex K of free coherent OS -modules together with

a quasi-isomorphism u : K ! K , such that for the dierential d we have

e

djs = 0. Now

= u [ 1]u : K ! K [ 1]

is a symmetric quasi-isomorphism, inducing in cohomology, and js is an

isomorphism. Then, in a neighborhood of s, js will remain an isomorphism

which proves the proposition.

Let (K ; ) be as in the proposition and consider the following diagram

G1

K0 K

o 0

0 o

4

K 1 K 0

G

It follows that is skew-symmetric. Therefore the cohomology of F may

be represented, locally for the Zariski topology on S, by complexes of free

coherent OS -modules

0 ! K ! K ! 0

with skew. Such complexes will be called special in the following.

An immediate corollary is Riemann0s invariance mod 2 theorem:

Moduli of G-bundles 31

6.2.3. Corollary. 6 Let F be a vector bundle on XS equipped with a non

degenerate quadratic form with values in !X . Then the function

s 7! dim H 0 (X; Fs ) mod 2

is locally constant.

Proof. Locally there is a special representative K of the cohomology of F.

dim H 0 (X; Fs ) = rank K rank

Now use that the rank of is even as is skew.

6.3. The pfaan bundle. Let F be a vector bundle on XS equipped with a

non degenerate quadratic form with values in !X and cover S by Zariski open

subsets Ui such that F admits a special representative Ki of the cohomology

of F on Ui . Over Ui

max max

^ ^

K

Ki

DFjUi = i

which is a square. It turns out, because the K are special complexes, that

V

the max Ki glue together over S and dene a canonical square root of DF ,

called the pfaan bundle.

This gluing requires quite some work and is the content of ([14], x7). I

will not enter into the proof here: (loc.cit.) is self contained.

6.3.1. Theorem. Let F be a vector bundle over XS equipped with a non

degenerate quadratic form with values in !X . Then the determinant bundle

DF admits a canonical square root P(F;) . Moreover, if f : S 0 ! S is a mor-

phism of locally noetherian k-schemes then we have P(f F;f ) = f P(F;) .

6.4. The pfaan bundle on the moduli stack. Let r 3 and (F; )

be the universal SOr -bundle over MSOr ;X X. If we twist by a theta-

characteristic (i.e. a line bundle such that

= !X ), then F = F

q

comes with a non-degenerate form with values in !X . Then we may apply

6.3.1 in order to get the pfaan bundle P(F ;) which we denote simply by

P.

6 In fact the above arguments are valid for any smooth proper morphism Y ! S of

relative dimension 1. I only consider the situation of a product Y = X S here as this is

the one I need in order to dene the determinant resp. pfaan bundles.

32

6.5. The square-root of the dualizing sheaf. Suppose G is semi-simple

and consider its action on g given by the adjoint representation. It fol-

ñòð. 1 |