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ICTP Lecture Notes

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Since unpublished material presented at the meetings might prove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
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

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
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

Mathmatiques, Universit de Nantes,
e e
BP 92208, 2, Rue de la Houssinire, 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 classi cation 20
5. Uniformization 22
6. The determinant and the pfaan line bundles 26
7. Ane Lie algebras and groups 33
8. The in nite 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 classi cation.
Then I will introduce two line bundles on the moduli stack: the deter-
minant and the pfaan bundle. The rst one can be used to describe the
canonical bundle on the moduli stack and the second to de ne 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 di eren-
tial operators on the moduli stack one has to consider twisted di erential
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 in nite Grass-
mannian in some more detail.
6

2. Generalities on principal G-bundles
In this section I de ne 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 brs
e
algbriques" (see remark 2.1.2).
e
2.1. Basic de nitions. Let Z be a scheme over an algebraically closed eld
k, G be an ane algebraic group over k.
2.1.1. De nition. 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 de nition, 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 de ned 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 exposs later) classi ed 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. De ne
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 de ned 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
de ned 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 de nes the requested reduction of the structure group.
1 In fact it is enough to suppose that F satis es the property that any nite subset of
F lies in an ane 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 de nition. 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 de nition, 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 de ne 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-
ti cial 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

SX
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 de ned by s 7! [Es ] de nes
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 di erent 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 de ned
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
which may be complicated in the general case and de nitively changes the
moduli problem. Grothendieck0 s idea to avoid the diculties 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 de nition, 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
de nes 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(
de8509 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 de nes 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 de nition 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 de ne 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 de ned 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 Zt .
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 \ dlement plat de prsentation
e e
nie." There are also letters \fpqc" standing for \ dlement 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 de ne 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 ane 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 satis es 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 e ective.
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 e ective 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 ane 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 de ned 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 de ned 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 de nitions of 1- and 2-morphisms make sense for
any lax functor. The compatibility conditions, which will be automatically
satis ed 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 de ned 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 de ned up to canonical isomorphism (it is the
solution of a universal problem) and we are so used to choose an element in
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 de nitions 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 de ned to be
G  f  G    G gf G
(gf) z
x fy fgz z
The functor  is de ned 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) satis es 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 de nes a k-groupoid. On the other hand, given a
k-groupoid we can de ne 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 de nitions 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 de ned by
S 7! fquasi-coherent OZk 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 de nition 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 de ne 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 de nes 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. De nition. 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 de nitions : a presentation is given by the
morphism p : Z ! [Z=H] de ned 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
de ned 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 de nes a GLr -bundle
over XU 0 which we denote by FU 0 .
We have to show that the ber MG;X (), as de ned 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 de ned 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, de ned 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 veri cation shows that this bijection
is functorial, i.e. de nes 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 de ned by extension of the structure group of
the G-bundle F ! F=G via  1 where  is the character de ned 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 in nitesimal
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 de nition, 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. De ne 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 de nition, 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 de ned 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 ane algebraic
group over k. We choose a closed point x0 2 X and consider X  = X fx0 g.
Remark that X  is ane (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 identi es 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 de ne 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 shea cation 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 ane 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 gU 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)jXfug (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 di er 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 ane k-scheme and q be an element of QG (U).
By de nition 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)) di er 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 ) di er 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 de ne a map MG;X ! LX GnQG as follows. Let U be an ane 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 de nes 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 de nes 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; ) de nes 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 RpF 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 de ned by
max max
^
0
( ^ K 1) 1
det(K  ) = K
The determinant of our family F of vector bundles parameterized by S is
de ned by4
DF = det(RpF) 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 de nition 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
qE where E is a vector bundle on X. We obtain the line bundle
DF
qE , 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
qu
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
qu = 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
qOp 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 de ned
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 de ned 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 pfaan 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  ! RpF ! 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 di erential acquires a 1 sign.
30

The problem here is that this isomorphism is only de ned 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 ane. 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 di erential 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 pfaan 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 de ne a canonical square root of DF ,
called the pfaan 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 pfaan 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 pfaan 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 de ne the determinant resp. pfaan 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-
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