ñòð. 6 |

2.3 156

S-equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 158

2.5 The moduli space of semi-stable bundles . . . . . . . . . . . . 159

2.6 The spectral cover construction . . . . . . . . . . . . . . . . . 161

2.7 Symplectic bundles . . . . . . . . . . . . . . . . . . . . . . . . 162

Flat SU(n)-connections . . . . . . . . . . . . . . . . . . . . .

2.8 163

Flat G-bundles and holomorphic GC-bundles . . . . . . . . .

2.9 166

The coarse moduli space for semi-stable holomorphic GC -

2.10

bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

2.11 Exercises: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

3 The Parabolic Construction 173

3.1 The parabolic construction for vector bundles . . . . . . . . . 173

3.2 Automorphism group of a vector bundle over an elliptic curve 175

Parabolics in GC . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 180

3.4 The distinguished maximal parabolic . . . . . . . . . . . . . . 181

3.5 The unipotent subgroup . . . . . . . . . . . . . . . . . . . . . 183

3.6 Unipotent cohomology . . . . . . . . . . . . . . . . . . . . . . 185

3.7 Exercises: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

4 Bundles over Families of Elliptic Curves 188

4.1 Families of elliptic curves . . . . . . . . . . . . . . . . . . . . 188

4.2 Globalization of the spectral covering construction . . . . . . 190

4.3 Globalization of the parabolic construction . . . . . . . . . . 192

2

Holomorphic bundles over elliptic manifolds

4.4 The parabolic construction of vector bundles regular and semi-

stable with trivial determinant on each ber . . . . . . . . . . 197

4.5 Exercises: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

5 The Global Parabolic Construction for Holomorphic Princi-

pal Bundles 199

5.1 The parabolic construction in families . . . . . . . . . . . . . 199

5.2 Evaluation of the cohomology group . . . . . . . . . . . . . . 200

5.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 201

References 203

139

Holomorphic bundles over elliptic manifolds

Introduction:

In this series of lectures we shall examine holomorphic bundles over com-

pact elliptically bered manifolds. We shall examine constructions of such

bundles as well as (duality) relations between such bundles and other geo-

metric objects, namely K3-surfaces and del Pezzo surfaces.

We shall be dealing throughout with holomorphic principal bundles with

structure group GC where G is a compact, simple (usually simply connected)

Lie group and GC is the associated complex simple algebraic group. Of

course, in the special case G = SU(n) and hence GC = SLn (C), we are con-

sidering holomorphic vector bundles with trivial determinant. In the other

cases of classical groups, G = SO(n) or G = Sympl(2n) we are consider-

ing holomorphic vector bundles with trivial determinant equipped with a

non-degenerate symmetric, or skew symmetric pairing. In addition to these

classical cases there are the nite number of exceptional groups. Amazingly

enough, motivated by questions in physics, much interest centers around the

group E8 and its subgroups. For these applications it does not suce to

consider only the classical groups. Thus, while often rst doing the case of

SU(n) or more generally of the classical groups, we shall extend our discus-

sions to the general semi-simple group. Also, we shall spend a good deal of

time considering elliptically bered manifolds of the simplest type { namely,

elliptic curves.

The basic references for the material covered in these lectures are:

1. M. Atiyah, Vectors bundles over an elliptic curve, Proc. London Math.

Soc. 7 (1967) 414-452.

2. R. Friedman, J. Morgan, E. Witten, Vector bundles and F-theory,

Commun. Math. Phys. 187 (1997) 679-743.

, Principal G-bundles over elliptic curves, Math. Research Letters

3.

5 (1998) 97-118.

, Vector bundles over elliptic brations, J. Alg. Geom. 8 (1999)

4.

279-401.

1 Lie Groups and Holomorphic Principal GC-bundles

In this lecture we review the classication of compact simple groups or equiv-

alently of complex linear simple groups. Then we turn to a review of the

140

Holomorphic bundles over elliptic manifolds

basics of holomorphic principal bundles over complex manifolds. We n-

ish the section with a discussion of isomorphism classes of G-bundles (G

compact) over the circle.

1.1 Generalities on roots, the Weyl group, etc.

A good general reference for Root Systems, Weyl groups, etc. is [3]. Let G

be a compact group and T a maximal torus for G. Of course, T is unique up

to conjugation in G. The rank of G is by denition the dimension of T. We

denote by W the Weyl group of T in G, i.e., the quotient of the subgroup of G

conjugating T to itself modulo the normal subgroup of elements commuting

with T (which is T itself). This is a nite group. We denote by g the Lie

algebra of G and by t g the Lie algebra of T. The group G acts on its Lie

algebra g by the adjoint representation. The exponential mapping exp: t ! T

is a covering projection with kernel t, where is the fundamental group

of T. The adjoint action of W on t covers the conjugation action of W on

T.

The complexication gC decomposes into the direct summand of sub-

spaces invariant under the conjugation action of the maximal torus T. The

subspace on which the action is trivial is the complexication of the Lie al-

gebra t of the maximal torus. All other subspaces are one dimensional and

are called the root spaces. The non-trivial character by which the torus acts

on a root space is called a root of G (with respect to T) for this subspace.

By denition the roots of G are non-zero elements of the character group

(dual group) of T. This character group is a free abelian group of dimension

equal to the rank of G. In the case when G is semi-simple, the roots of G

span a subgroup of nite index inside entire character group. When G is

not semi-simple the center of G is positive dimensional, and the roots span a

sublattice of the group of characters of codimension equal to the dimension

of the center of G.

Equivalently, the roots can be viewed as elements of the dual space t of

the Lie algebra of T, taking integral values on . Since all the root spaces

are one-dimensional, we see that the dimension of G as a group is equal to

the rank of G plus the number of roots of G.

The collection of all roots forms an algebraic object inside t called a

root system. By denition a root system on a vector space V is a nite

subset V of roots such that for each root a 2 = there is a dual coroot

=a_ 2 V such that the `re

ection' ra : V ! V dened by ra(b) = b hb; a_ ia

141

Holomorphic bundles over elliptic manifolds

normalizes the set . It is easy to see that if such a_ exists then it is unique

and furthermore that the fra ga2 generate a nite group. The element ra is

called the re

ection in the wall perpendicular to a. The wall, denoted Wa ,

xed by ra is simply the kernel of the linear map a_ on V . The group W

generated by these re

ections is the Weyl group. Dually we can consider the

wall Wa V determined by a = 0. There is the dual re

ection ra : V ! V

given by the formula ra (v) = v ha; via_ . These re

ections generate the

adjoint action of W on V . The set V generates a lattice in V called

the root lattice and denoted root . Dually the coroots span a lattice in V

called the coroot lattice and denoted . Back to the case of a root system of

a Lie group G with maximal torus T, since any root of G must take integral

values on 1 (T ) we see that the lattice 1 (T ) t is contained in the integral

dual of the root lattice. Furthermore, one can show that the coroot lattice

is contained in 1 (T ).

The walls fWa g divide V into regions called Weyl chambers. Each cham-

ber has a set of walls. We say that a set of roots is a set of simple roots if (i)

the walls associated with the roots are exactly the walls of some chamber C,

and (ii) the roots are non-negative on this chamber. It turns out that a set of

simple roots is in fact an integral basis for the root lattice. Also, every root

is either a non-negative or a non-positive linear combination of the simple

roots, and roots are called positive or negative roots depending on the sign

of the coecients when they are expressed as a linear combination of the

simple roots. (Of course, these notions are relative to the choice of simple

roots.)

Since the Weyl group action on V is nite, there is a Weyl-invariant

inner product on V . This allows us to identify V and V in a Weyl-invariant

fashion and consider the roots and coroots as lying in the same space. When

we do this the relative lengths of the simple roots and the angles between

their walls are recorded in a Dynkin diagram which completely classies the

root system and also the Lie algebra up to isomorphism. If the group is

simple, then this inner product is unique up to a positive scalar factor. In

general, we can use this inner product to identify t and t . When we do we

have _ = h;i .

2

There is one set of simple roots for each Weyl chamber. It is a simple

geometric exercise to show that the group generated by the re

ections in the

walls acts simply transitively on the set of Weyl chambers, so that all sets

of simple roots are conjugate under the Weyl group, and the stabilizer in

the Weyl group of a set of simple roots is trivial. Thus, the quotient t=W is

142

Holomorphic bundles over elliptic manifolds

identied with any Weyl chamber.

A Lie algebra is said to be simple if it is not one-dimensional and has no

non-trivial normal subalgebras. A Lie algebra that is a direct sum of simple

algebras is said to be semi-simple. A Lie algebra is semi-simple if and only

if it has no non-trivial normal abelian subalgebras. A compact Lie group is

said to be simple, resp., semi-simple, if and only if its Lie algebra is simple,

resp., semi-simple. Notice that a simple Lie group is not necessarily simple

as a group. It can have a non-trivial (nite) center, when then produces

nite normal subgroups of G. These are the only normal subgroups of G if

G is simple as a Lie group.

A simply connected compact semi-simple group is a product of simple

groups. In general a compact semi-simple group is nitely covered by a

product of simple groups.

There are a nite number of simple Lie groups with a given Lie algebra

g. All are obtained in the following fashion. There is exactly one simply

connected group G with g as Lie algebra. For this group the fundamental

group of the maximal torus is identied with the coroot lattice of the

group. This group has a nite center CG which is in fact identied with the

dual to the root lattice modulo the coroot lattice. Any other group with the

same Lie algebra is of the form G=C where C CG is a subgroup. Thus, the

fundamental group of the maximal torus of G=C is a lattice in t containing

the coroot lattice and contained in the dual to the root lattice. Its quotient

by the coroot lattice is C.

Each compact simple group G embeds as the maximal compact subgroup

of a simple complex linear group GC . The Lie algebra of GC is the complex-

ication of the Lie algebra of G and there are maximal complex tori of GC

containing maximal tori of G as maximal compact subgroups. For example,

the complexication of SU(n) is SLn (C) whereas the complexication of

SO(n) is the complex special orthogonal group SO(n; C). We can recover

the compact group from the semi-simple complex group by taking a maximal

compact subgroup (all such are conjugate in the complex group and hence

are isomorphic). Any such maximal compact subgroup is called the compact

form of the group. The fundamental group of a complex semi-simple group

is the same as the fundamental group of its compact form.

1.2 Classication of simple groups

Let us look at the classication of such objects.

143

Holomorphic bundles over elliptic manifolds

1.2.1 Groups of An-type

The rst series of groups is the series SU(n+1), n 1. These are the groups

of An -type. The maximal torus of SU(n+1) is usually taken to be the group

of all diagonal matrices with entries in S 1 and the product of the entries being

Q

one. We identify this in the obvious way with f(1 ; : : : ; n+1 )j n+1 i = 1g.

i=1

The rank of SU(n + 1) is n. The Lie algebra su(n) is the space of matrices

of trace zero, and the root space gij ; 1 i; j n; i 6= j consists of matrices

with non-trivial entry only in the ij position. The root associated to this

root space is denoted ij and is given by ij (1 ; : : : ; n+1 ) = i j 1 . Writing

P

things additively, we identify t with f(z1 ; : : : ; zn+1 )j i zi = 0g and then

ij = ei ej where ei is the linear map which is projection onto the ith

coordinate. The usual choice of simple roots are 12 ; 23 ; : : : ; nn+1 . With

this choice the positive roots are the ij where i < j. When i < j we

have ij = i(i+1) + + (j 1)j . The Weyl group is the symmetric group

on n + 1 letters. This group acts in the obvious way on Rn+1 and leaves

invariant the subspace we have identied with t. This is the Weyl group

action on t. In particular, the restriction of the standard inner product on

Rn+1 to t is Weyl invariant. Also, it is easy to see that all the roots are

conjugate under the Weyl action. Notice that each simple root has length 2

and meets the previous simple root and the succeeding simple root (in the

obvious ordering) in 1, and is perpendicular to all other simple roots. All

this information is recorded in the Dynkin diagram for An . Since each root

has length two, under the induced identication of t with t every root is

identied with its coroot _ . Thus, in this case, and as we shall see, in all

other simply laced cases, one can identify the roots and coroots and hence

their lattices.

1 1 1 1 1

The center C(SU(n + 1)) is the cyclic group of order n + 1 consisting of

diagonal matrices with diagonal entry an n+1 root of unity. Thus for each

cyclic subgroup of Z=(n + 1)Z there is a form of SU(n) with fundamental

group this cyclic subgroup. The full quotient SU(n+1)=CSU(n+1) is often

called PU(n). The intermediate quotients are not given names.

144

Holomorphic bundles over elliptic manifolds

1.2.2 Groups of Bn-type

For any n 3, the group Spin(2n + 1) is a simple group of type Bn . The

standard maximal torus of this group is the subgroup of matrices that project

into SO(2n+1) to block diagonal matrices SO(2)SO(2) SO(2)f1g.

The Lie algebra of this torus is naturally a product of n copies of R, one

for each SO(2), and we let ei be the projection of t onto the ith -factor. The

Lie algebra is all skew symmetric matrices. For any 1 i; j n let gij ; 1

i < j n; i 6= j be the subspace of skew symmetric matrices with non-zero

entries only in places (2i 1; 2j 1); (2i 1; 2j); (2i; 2j 1); (2i; 2j) and the

symmetric lower diagonal positions. This is a four-dimensional subspace of

the Lie algebra of SPin(2n + 1). Thus, there are four roots associated with

this space, they are ei ej . There are also subspaces gi ; 1 i n, where

gi has non-zero entries only in positions (2i 1; 2n + 1) and (2i; 2n + 1) as

well as the symmetric lower diagonal positions. The two roots associated to

gi are ei . Thus, Spin(2n + 1) is of rank n. We can identify the Lie algebra

of its maximal torus with Rn in such a way that the roots are ei ej for

i 6= j and ei , where ei is the projection onto the ith coordinate. The dual

coroots to these roots are ei ej and 2ei , so that the coroot lattice

for Spin(2n + 1) is the even integral lattice in Rn , whereas the fundamental

group of the maximal torus of SO(2n + 1) is the full integral lattice Zn . Of

course, the quotient =Zn = Z=2Z is the fundamental group of SO(2n + 1).

The Weyl group of Spin(2n + 1) is the group generated by re

ections in the

simple roots. It is easy to see that these re

ections generate the group of all

permutations of the coordinates and also all sign changes of the coordinates.

Thus, abstractly the group is (f1gn n . Clearly, the standard metric on

Rn is a Weyl invariant metric. Notice that something new happens here {

not all the roots have the same length. In our normalization ei ej has

length squared 2, whereas ei has length squared 1. In particular, not all

the roots are conjugate under the action of the Weyl group. In this case

there are two orbits { one orbit for each length. This fact is expressed by

saying that the group is not simply laced.

Since the center of Spin(2n + 1) is the cyclic group of order 2, there are

only two groups with this Lie algebra spin(2n + 1) and SO(2n + 1).

The Dynkin diagram of type Bn is

1 2 2 2 2 2

>

145

Holomorphic bundles over elliptic manifolds

1.3 Groups of Cn-type

Let J be the 2n 2n matrix of block 2 2 diagonal entries

0 1:

10

The symplectic group consists of all matrices A such that Atr JA = J. These

are the linear transformations that leave invariant the standard skew sym-

metric pairing on R2n , the one given by J. The Lie algebra of this group

consists of all matrices A such that

Atr = JAJ 1 = JAJ:

For any n 2 the symplectic group Sympl(2n) is a group of type Cn so

that the complex symplectic group is a complex semi-simple group. There is

a complication in that the real symplectic group is non-compact; it is rather

what is called the R-split form of the group. Its maximal algebraic torus is a

product of n copies of R and is given by the group of diagonal matrices with

diagonal entries (1 ; 1 ; : : : ; n ; n 1 ). For example, Sympl(2) is identied

with SL2 (R). By general theory there is a compact form for the complex

symplectic group SymplC(2n). It is given as the group of quaternion linear

transformations of Hn , so that as one would expect, the compact form of

SymplC(2) is SU(2). The maximal torus of this group is again a product

of circles so that t is again identied with Rn . The roots are ei ej for

1 i < j n and 2ei . Thus, once again the group is non-simply laced.

Its Weyl group is the same as the Weyl group of Bn . The coroots dual to

the roots are ei ej and ei so that the coroot lattice is the integral

lattice Zn . The dual to the root lattice consists of all fx1 ; : : : ; xn g such that

xi 2 (1=2)Z for all i and xi xj (mod Z) for all i; j. This lattice contains

=

the coroot lattice with index two, so that the center of the simply connected

form of this group is Z=2Z and there is one non-simply connected form of

these groups.

The Dynkin diagram of type Cn is

1 1 1 1 1

<

It turns out that Spin(5) is isomorphic to Sympl(4) which is why we

start the B-series at n = 3. The group Symp(2) is isomorphic to SU(2)

which is why we begin the C-series at n = 2.

146

Holomorphic bundles over elliptic manifolds

1.3.1 Groups of Dn-type

For any n 4 the group Spin(2n) is a group of type Dn . The usual maximal

torus for Spin(2n) is the subgroup that projects onto SO(2) SO(2)

SO(2n). Thus, t is identied with R2n with the factors being tangent to

the factors in this decomposition. The roots of Spin(2n) are ei ej for

1 i < j n. This group is simply laced and in the given Weyl invariant

p

inner product all roots have length 2. Thus, we can identify the roots with

their dual coroots in this case. The coroot lattice is then the even integral

lattice. The fundamental group of the maximal torus for SO(2n) is the

integral lattice which contains the coroot lattice with index two re

ecting

the fact that the fundamental group of SO(2n) is z=2Z. The Weyl group

consists of all permutations of the coordinates and all even sign changes of

the coordinates. This is a simply laced group.

The Dynkin diagram of type Dn is

1

Â¡Â¡

Â¡Â¡

1 2 2 2 2 Â¡Â¡Â¡

Â¡

gg

gg

gg

g

1

1.3.2 The exceptional groups

A good reference for the exceptional Lie groups is [1]. In addition to the

classical groups there are ve exceptional simply connected simple groups.

Their names are E6 ; E7 ; E8 ; G2 and F4 . The subscript is the rank of the

group. There are natural inclusions D5 E6 E7 E8 . The fundamental

group of Er is a cyclic group of order 9 r. Both G2 and F4 are simply

connected.

Here are their Dynkin diagrams:

147

Holomorphic bundles over elliptic manifolds

1 2 3 2 1

2

E6

1 2 3 4 3 2

2

E7

2 3 4 5 6 4 2

3

E8

2 4 3 2

<

F4

3 2

<

G2

We shall not say too much about these groups now, but let me give

the lattices E6 ; E7 ; E8 . These are viewed as the fundamental group of the

maximal torus of the simply connected form of the group. We give these

lattices with an inner product. This is the Weyl invariant inner product. In

all cases these groups are simply laced and the coroots are the elements in the

lattice of square two. We describe all these lattices at once { for any r 8

P

consider the indenite integral quadratic form q(x; a1 ; : : : ; ar ) = r a2 x2

i=1 i

148

Holomorphic bundles over elliptic manifolds

on Zr+1 . We let k = (3; 1; 1; : : : ; 1). Then q(k) = r 9 < 0. For 6 r 8,

the lattice Er is the orthogonal subspace in Zr+1 of k. It is of rank r and has

the induced quadratic form which is easily seen to be even, positive denite,

and of discriminant 9 r. For lower values of r it turns out that the lattice

dened this way is the lattice of a classical group: E5 = D5 , E4 = A4 ,

E3 = A2 A1 .

1.4 Principal holomorphic GC-bundles

Let GC be a complex linear algebraic group and let X be a complex mani-

fold. A holomorphic principal GC-bundle over X is determined by an open

covering fUi g of X and transition functions gij : Ui \ Uj ! GC . The tran-

sition functions are required to be holomorphic and to satisfy the cocycle

conditions: gji = gij 1 and gjk (z) gij (z) = gik (z) for all z 2 Ui \ Uj \ Uk . As

usual, we can use the gij as gluing data to glue Ui GC to Uj GC along

uij GC, by the rule (z; g) 2 Ui GC maps to (z; gij (z) g) in Uj GC pro-

vided that z 2 Ui \ Uj . The cocycle condition tells us that the triple gluings

are compatible so that we have dened an equivalence relation and the result

of gluing E is a Hausdor space. The projection mappings Ui GC ! Ui

then t together to dene a continuous map p: E ! X. The fact that the gij

are holomorphic implies that the natural complex structures on the Ui GC

are compatible and hence dene a complex structure on E for which p is

a holomorphic submersion with each ber isomorphic to GC. The complex

manifold E is called the total space of the principal bundle and p is called

the projection. There is a natural (right) free, holomorphic GC-action on E

such that p is the quotient projection of this action. Two open coverings and

gluing functions dene the isomorphic principal GC -bundles if the resulting

total spaces are biholomorphic by a GC-equivariant mapping commuting

with the projections to X.

If is a holomorphic principal GC -bundle over X and : GC ! Aut(V )

is a complex linear representation, then there is an associated holomorphic

vector bundle (). If E is the total space of then the total space of ()

is GC V where GC acts on V via the representation .

Example: Let GC be C . (Notice that this is not a simple group.) Then

a holomorphic principal C bundle determines a holomorphic line bundle un-

der the natural representation given by complex multiplication C C ! C.

The holomorphic line bundles associated to and 0 are isomorphic as holo-

morphic line bundles if and only if and 0 are isomorphic as holomorphic

149

Holomorphic bundles over elliptic manifolds

principal C -bundles. Notice that the total space of the C -bundle can

be identied with the complement of the zero section in the corresponding

holomorphic line bundle.

Along the same lines, let GC be SLn (C). Then using the dening com-

plex n-dimensional representation, a principal SLn (C)-bundle over X de-

termines a holomorphic n-dimensional vector bundle V over X. But this

bundle has the property that its determinant line bundle ^nV is trivialized

as a holomorphic line bundle. Of course, given a holomorphic n-dimensional

vector bundle V over X with a trivialization of its determinant line bundle

we can dene the associated bundle of special linear frames in V . The ber

over x 2 X consists of all bases ff1 ; : : : ; fn g for Vx such that f1 ^ ^ fn

is identied with 1 2 C under the given trivialization of ^n(Vx ). The lo-

cal trivialization of the vector bundle, produces a local trivialization of this

bundle of frames. The holomorphic structure on the total space of V de-

termines a holomorphic structure on the bundle of frames. The obvious

SLn (C)-action on the bundle of frames then makes it a holomorphic prin-

cipal SLn (C)-bundle. This sets up a bijection between isomorphism classes

of holomorphic principal SLn (C)-bundles over X and holomorphic vector

bundles over X with trivialized determinant line bundle.

In the same way we can identify a holomorphic principal SO(n; C)-

bundle over X with a holomorphic rank n vector bundle V over X with

holomorphically trivialized determinant and with a holomorphic symmetric

form V

V ! C which is non-degenerate on each ber. A holomorphic

principal Symp(2n)-bundle over X is identied with a holomorphic rank 2n

vector bundle V over X with holomorphically trivialized determinant and

with a holomorphically varying skew-symmetric bilinear form on the bers

which is non-degenerate on each ber.

Up to questions of nite covering groups, this exhausts the list of classical

simple groups: SLn (C), SO(n; C), and Sympl(2n; C).

Associated to any complex group GC there is the adjoint representation

of GC on its Lie algebra gC. Thus, associated to any holomorphic principal

GC-bundle is a vector bundle denoted ad. Its rank is the dimension of GC

as a group. In the case of E8 this is the smallest dimensional representation;

it is of course of the same dimension as the group 248. All other simple groups

have smaller representations: G2 has a seven dimensional representation

even though it has dimension 14; F4 has a 28-dimensional representation;

E6 has a 27 dimensional representation (which we discuss later), and E7 has

a 54-dimensional representation. Still, it is not clear that the best way to

150

Holomorphic bundles over elliptic manifolds

study principal bundles over these groups is to look at the vector bundles

associated to these representations. For example, it is not obvious what

extra structure a 248-dimensional vector bundle carries if it comes from a

principal E8 -bundle, nor what extra information it takes to determine the

structure of that bundle.

1.5 Principal G-bundles over S 1

Let us begin with a simple problem. Fix a compact, simply connected, sim-

ple group G. Let be a principal G-bundle over S 1 and let A be a

at

G-connection on . (The reason for the passage from GC to G and the in-

troduction of a

at connection will be explained in the next lecture.) The

holonomy of A around the base circle is an element of G, determined up to

conjugacy, which completely determines the isomorphism class of (; A) and

sets up an isomorphism between the space of conjugacy classes of elements

in G and the space of isomorphism classes of principal G-bundles with

at

connections over S 1 . Thus, we have reduced our problem to that of under-

standing the space of conjugacy classes of elements in a compact group. To

some extent this is a classical and well-understood problem, as we show in

the next section.

1.5.1 The ane Weyl group and the alcove structure

This leads us to the question of what the space of conjugacy classes of ele-

ments in G looks like. To answer this question we introduce the ane Weyl

group and the alcove structure on the Lie algebra t of the maximal torus T

of G. For a root and k 2 Z we denote by W;k the codimension-one ane-

linear subspace of t determined by the equation f = kg. By denition, the

ane Weyl group, Wa , is the group of ane isometries of t generated by

re

ections in all walls of the form W;k . It is easy to see that there is an

exact sequence of groups

0 ! ! Wa ! W ! 0:

(Recall that t is the coroot lattice.) Here, the map Wa ! W is the

dierential or linearization of the ane map. (Recall that is the coroot

lattice, i.e., 1 (T ) t.) This sequence is split by including W as the group

generated by the re

ections in the walls W;0 , but the action of W on

is non-trivial: it is the obvious action. Thus, Wa is isomorphic to the

semi-direct product W with the natural action of W on .

151

Holomorphic bundles over elliptic manifolds

The set of walls W;k is a locally nite set and divides t into (an innite

number of) regions called alcoves. If G is simple (or even semi-simple),

then the alcoves are compact. In the case when G is simple, the alcoves

are simplices. Clearly, each alcove is contained in some Weyl chamber. An

alcove containing the origin in fact contains a neighborhood of the origin

in the Weyl chamber that contains it. Its walls are the walls of the Weyl

chamber containing it together with one more. This extra wall is given by an

equation of the form = 1 for some root determined by the Weyl chamber

˜

(or equivalently by the set of simple roots f1 ; : : : ; r g determined by the

Weyl chamber). It turns out that is a positive linear combination of the i

˜

and it has the largest coecients. In particular, it is the unique nonnegative

linear combination of the simple roots with the property that its sum with

any simple root is not a root. This root is called the highest root of G.

˜

The numbers displayed on the Dynkin diagrams above are the coecients

of the simple roots in their unique linear combination which is the highest

root. If is a set of simple roots for G, then the associated set [ is ˜

denoted by and is called the extended set of simple roots.

e

As in the case of the Weyl group, it is a nice geometric argument to show

that Wa acts simply transitively on the set of alcoves and that the quotient

t=Wa is identied with any alcove.

Lemma 1.5.1 Let G be a compact, simply connected semi-simple group.

Then the space of conjugacy classes of elements in G is identied with an

alcove A in t. The identication associates to t 2 A the conjugacy class of

exp(t) 2 T.

Proof. Every g 2 G is conjugate to a point t 2 T. Two points of T

are conjugate in G if and only if they are in the same orbit of the Weyl

group action on T. Thus, the space of conjugacy classes of elements in G

is identied with T=W. Since G is simply connected, T = t= and hence

T=W = t=Wa which we have just seen is identied with an alcove.

Example: If G PSU(n + 1), then the alcove is the subset (t1 ; : : : ; tn+1) 2

=

Rn+1 satisfying j tj = 0 and tj 0 and tj tj+1 for all 1 j n + 1.

Every element in SU(n + 1) is conjugate to a diagonal matrix with diagonal

entries (1 ; : : : ; n+1 ), the j being elements of S 1 . We can do a further

conjugation under j = exp(2itj for 0 tj 1. The determinant condition

Q P

is j j = 1, which translates into j tj = 0. By a further Weyl conjugation

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we can arrange that the tj are in increasing order. This makes explicit the

isomorphism between the simplex in Rn+1 and the space of conjugacy classes

in SU(n+1 ).

1.6 Flat G-bundles over T 2

Let G be a compact, simply connected group.

Lemma 1.6.1 Let x; y 2 G be commuting elements. Show that there is a

torus T G containing both x and y. If (x; y) and (x0 ; y0 ) are pairs of

elements in a torus T G and if there is g 2 G such that g(x; y)g 1 =

(x0 ; y0 ), then there is an element n normalizing T and conjugating (x; y) to

(x0 ; y0 ).

The proof is an exercise.

Corollary 1.6.2 Let G be a compact simply connected group. Then the

space of isomorphism classes of principal G-bundles with

at connections

over T 2 is identied with the space (T T)=W, where T is a maximal torus

of G and W is the Weyl group acting on T T by simultaneous conjugation.

Notice that this implies that the space of such bundles is connected. Of

course, this is not too surprising since for G simply connected, all G-bundles

over T 2 are topologically trivial.

1.7 Exercises

1. Let G be a compact connected Lie group. Show that the exponential

mapping from the Lie algebra g to G is onto. Use this to show that every

element of G is contained in a maximal torus.

2. Show all maximal tori of G are conjugate.

3. Suppose that G is compact. Show that the centralizer of a maximal torus

in G is the torus itself.

4. Let G be a compact simply connected group. Show that the center of G

is the intersection of the kernels of all the roots. Show that if G is simply

connected and semi-simple then the center of G is identied with the quotient

(root ) = where t is the coroot lattice, where root t is the root

lattice, and (root ) is the algebraically dual lattice in t. Use this to show

that the center of a semi-simple group is nite.

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Holomorphic bundles over elliptic manifolds

5. Show that if G is a compact Lie group whose Lie algebra is semi-simple,

then any normal subgroup of G is a nite central subgroup.

6. Count the number of roots for groups of An , Bn , Cn , and Dn type and

determine the dimension of each of these groups.

6. For any r; 3 r 8, let Zr+1 be the free abelian group of rank r + 1 with

basis h; e1 ; : : : ; er and with non-degenerate quadratic form Q with Q(h) =

1, Q(ei ) P 1; 1 i r, and the basis being mutually orthogonal. Let

=

r e . Then k? is a lattice of rank r with a positive denite

k = 3h i=1 i

pairing of determinant 9 r. Show that a basis for k? is e1 e2 ; : : : er 1

er ; H e1 e2 e3 and that these vectors are all of square 2. Count the

number of vectors in this lattice of square two. Show that for each vector of

square 2, re

ection in the vector is an integral isomorphism of the lattice k?

and its form. Show that for r = 3 the lattice is the coroot lattice of A2 A1 ,

for r = 4, the lattice is the coroot lattice of A4 , for r = 5 the lattice is the

coroot lattice for D5 . In all cases the coroots are exactly the vectors of square

2. It follows of course that the group generated by re

ections in the vectors

of square two is the Weyl group. It turns out that for r = 6; 7; 8 the lattice

is the coroot lattice of Er . The statements about the roots, coroots and the

Weyl group remain true for these cases as well. Assuming this, compute the

dimensions of the Lie groups E6 ; E7 ; E8 .

8. Let E be an elliptic curve. Give a 1-cocycle which represents the generator

of H 1 (E; OE )). Give a 1-cocycle that represents the principal C -bundle

O(q p0).

9. Show that if V is a semi-stable vector bundle of degree zero over an elliptic

curve E with a non-degenerate skew-form, then det(V ) is trivial. Show that

this is not necessarily true it V supports a non-degenerated quadratic form

instead.

10. Dene the compact form of SymplC (2n) in terms of quaternion linear

mappings of a quaternionic vector space.

11. Show that the conjugacy class of the holonomy representation determines

an isomorphism between the space of isomorphism classes of principal G-

bundles over S 1 with

at connections and the space of conjugacy classes of

elements in G.

12. Show that the space of conjugacy classes of elements in G is identied

with the alcove of the ane Weyl group action on t.

13. Show that if x; y are commuting elements in a compact simply connected

group G then there is a torus in G containing both x and y. [Hint: Show

that the centralizer of x, ZG (x) is connected.] Show that this fails to be true

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Holomorphic bundles over elliptic manifolds

if G is compact but not simply connected (e.g. G = SO(3)). Show that if

X and X 0 are ordered subsets of T which are conjugate in G, then they are

conjugate by an element of the normalizer of T.

14. Show that if G is a compact semi-simple group, then G-bundles over

T 2 are classied up to topological equivalence by a single characteristic class

w2 2 H 2 (T 2 ; 1 (G)) = 1 (G). Show that if a G-bundle admits a

at connec-

tion with holonomy (x; y) around the generating circles, then its character-

istic class in 1 (G) G is computed as follows. One chooses lifts x; y in the

˜ ˜

universal covering group G of G for x; y. Then [x; y] 2 G lies in the kernel of

˜

the projection to G, i.e., lies in 1 (G) G. This is the characteristic class

of the bundle.

15. Show that the space of isomorphism classes of SO(3)-bundles with

at

connection over T 2 has two components. Show that one of these components

has dimension 2 and the other is a single point.

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Holomorphic bundles over elliptic manifolds

2 Semi-Stable GC-Bundles over Elliptic Curves

In this lecture we introduce the classical notions of stability, semi-stability,

and S-equivalence for vector bundles. We then consider in some detail semi-

stable vector bundles over an elliptic curve and the moduli space of their

S-equivalence classes. We extend these results to the one situation where

it has an easy and direct analogue { namely complex symplectic bundles.

Then we switch and consider

at SU(n)-bundles and state the Narasimhan-

Seshadri result relating these bundles to semi-stable holomorphic bundles.

Then we generalize this result to compare

at G-bundles and semi-stable

holomorphic GC -bundles. Lastly, we discuss Looijenga's theorem which, in

the case that G is simple and simply connected, describes quite explicitly

these moduli spaces in terms of the coroot integers for the group G.

2.1 Stability

Let C be a compact complex curve. The slope of a holomorphic vector

bundle V ! C is (V ) = deg(V )=rank(V ) where deg(V ) is the degree of the

determinant line bundle of V . A vector bundle V ! C is said to be stable if

for every proper subbundle W V we have (W) < (V ). The bundle V is

said to be semi-stable if (W) (V ) for every proper subbundle W V .

A subbundle W which violates these inequalities is called destabilizing or

de-semistabilizing.

Note: One usually requires (W) < (V ), resp., (V ) for every vector

bundle over C which admits a vector bundle mapping W ! V which is in-

jective on the generic ber. The image of such a mapping will not necessarily

be a subbundle of V , but rather is a subsheaf of its sheaf of local holomorphic

sections. Nevertheless, there is a subbundle W 0 V containing the image of

W under the given mapping with the property that W 0 modulo the image

of W is supported at a nite set of points (a sky-scraper sheaf). In this case

deg(W 0 ) = deg(W ) + `(W 0 =W) where `(W 0 =W) is the total length of the

sky-scraper sheaf. It follows that (W 0 ) (W) so that if W destabilizes or

de-semistabilizes V then so does W 0 . Thus, in the case of curves it suces

to work exclusively with subbundles. Let me re-iterate that this is special

to the case of curves.

The main reason for introducing stability is that the space of all bundles

can be studied in terms on stable bundles (the so-called Harder-Narasimhan

ltration) and that the space of isomorphism classes of stable bundles forms

a reasonable space.

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Holomorphic bundles over elliptic manifolds

Over high dimensional manifolds X n we need a couple of modications.

First of all we need a Khler class ! so that we can dene the degree of a

a

R

bundle V , deg(V ), to be X !n 1 ^c1 (V ). (Of course, the degree, and hence

the slope depends on the choice of Khler class.) Then as indicated above

a

we must also consider all torsion-free subsheaves W of V . Each of these has

a rst Chern class and hence a degree, again depending on ! and computed

by the same formula as above. With these modications one denes slope

stability and slope semi-stability exactly as before. There are more rened

notions, for example Gieseker stability, which are often needed over higher

dimensional bases, especially if one hopes to obtain compact moduli spaces.

2.2 Line bundles of degree zero over an elliptic curve

We x a smooth elliptic curve E. By this we mean that E is a compact

complex (smooth) curve of genus 1, and we have xed a point p0 2 E. This

determines an abelian group law on E for which p0 is the identity element.

Consider a line bundle L ! E of degree zero. According to Riemann-

Roch,

rank H 1 (E; L) = rank H 0 (E; L)

which tells us nothing about whether L has a holomorphic section. However,

if we consider L

O(p0 ), then RR tells us that this bundle has at least one

holomorphic section. Let : O ! L

O(p0 ) be such a section. Of course,

since the degree of L

O(p0 ) is 1, vanishes once, say at a point q 2 E. This

means that factors to give a holomorphic mapping 0 : O(q) ! L

O(p0 )

which is generically an isomorphism. In general a map between line bundles

which is generically one-to-one has torsion cokernel and the total length of

the cokernel is the dierence of the degrees of the two bundles. In our case,

the domain and range both have degree one, so that the cokernel is trivial,

i.e., 0 is an isomorphism. This proves that L is isomorphic to O(q)

O( p0),

which is also written as O(q p0 ). It is easy to see that associating to L the

point q sets up an isomorphism between the space of line bundles of degree

zero on E, Pic0 (E), and E itself.

2.3 Semi-stable SLnC-bundles over E

Let V be a semi-stable vector bundle of rank n and degree zero. Semi-

stability implies that any subbundle of V has non-positive degree. Let us

rst show that there is a line bundle of degree zero mapping into V . We

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Holomorphic bundles over elliptic manifolds

proceed in the same manner as with line bundles. The bundle V

O(p0 ) is

also easily seen to be semi-stable and of slope 1. By RR V

O(p0 ) has n

holomorphic sections. Let H 0 be the space of these sections. Then evaluation

determines a vector bundle mapping from the trivial bundle H 0

OE to

V

OE (p0 ). If no non-trivial section of V

OE (p0 ) vanished at any point,

then this map would be an isomorphism of vector bundles, which is absurd

since the bundles have dierent degree. We conclude that there is a non-zero

section of V

OE (p0 ) vanishing at some point. Consider the cokernel of this

section. Generically it is a vector bundle of rank n 1, but it has non-trivial

torsion, say T, corresponding to the zeros of the section. Then there is a

line bundle LT tting into an exact sequence

0 ! O ! LT ! T ! 0

and an extension of to a map 0 : LT ! V

O(p0 ) whose quotient is a

vector bundle W of rank one less than that of V . Of course, the degree

of LT is the total length of T. By the fact that the slope of V

OE (p0 )

is one and is semi-stability, we know the total length of T is at most one.

But on the other hand we know that T is nontrivial, so that it has length

exactly one. It follows that it is of the form O=O(q) for some point q 2 E.

Thus, we have a map O(q) ! V with torsion-free cokernel and hence a map

O(q p0) ! V whose quotient is a semi-stable bundle of degree n 1.

Continuing inductively we see that V is written as a successive extension

of n line bundles of degree zero. We can associate to V , the n points that are

identied with these line bundles. Let us now think about the extensions.

For any line bundle of degree zero, RR tells us that since H 1 (E; L) = 0 unless

L is trivial. It follows immediately that if L and L0 are non-isomorphic line

bundles of degree zero then any extension

0 ! L ! V ! L0 ! 0

is trivial. An easy inductive argument shows that if W1 written as a suc-

cessive extension of line bundles Li of degree zero and W2 as a successive

extension of line bundles Mj of degree zero, and no Li is isomorphic to any

Mj , then any extension

0 ! W1 ! V ! W2 ! 0

is trivial. Consequently, we can split V into pieces q2E Vq where Vq is a

successive extension of line bundles isomorphic to O(q p0 ). Lastly, there

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Holomorphic bundles over elliptic manifolds

is, up to equivalence, exactly one non-trivial extension

0 ! O(q p0 ) ! V ! O(q p0 ) ! 0:

An easy argument shows that if V is such a nontrivial extension, then it has

a unique nontrivial extension by O(q p0 ) and so forth.

Notice that since every vector bundle of degree zero over an elliptic curve

E has a subline bundle of degree zero, the only stable degree zero vector

bundles over E are line bundles. Thus, the best we can hope for in higher

rank and degree zero is that the bundle be semi-stable. As we shall eventually

see, there are stable bundles of positive degree.

This allows us to establish the following theorem rst proved by Atiyah:

Theorem 2.3.1 Any semi-stable vector bundle of degree zero over E is iso-

morphic to a direct sum of bundles of the form O(q p0 )

Ir where the Ir

are dened inductively as follows; I1 = O and Ir is the unique non-trivial

extension of Ir 1 by O.

Clearly, the determinant of q2E O(q p0)

Ir(q) is

q2E O(q p0 )

r(q) ,

or under our identication of line bundles of degree zero with points of E,

P

the determinant of V is identied with q2E r(q)q, where the sum is taken

in the group law of the elliptic curve. Thus, V has a trivial determinant if

P

and only if q r(q)q = 0 in E.

2.4 S-equivalence

We just classied semi-stable vector bundles of degree zero on an elliptic

curve in the sense that we enumerated the isomorphism classes. But we have

not produced a moduli space (even a coarse one) of all such bundles. The

trouble, as always in these problems, is that the natural space of isomorphism

classes is not separated, i.e., not a Hausdor space. The reason is exemplied

by the fact that there is a bundle over E H 1 (E; O) whose restriction to

Efag is the bundle which is the extension of O by O given by the extension

class a. This bundle is isomorphic to I2 for all fag 6= 0 and is isomorphic to

O O if fag = 0. Thus, we see that any Hausdor quotient of the space of

isomorphism classes of bundles I2 and O O must be identied.

This phenomenon is an example of S-equivalence. We say that a semi-

stable bundle V is S-equivalent to a semi-stable bundle V 0 if there is a family

of V ! E C, where C is a connected smooth curve, so that for generic

c 2 C the bundle VjC fcg is isomorphic to V and so that there is c0 2 C

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Holomorphic bundles over elliptic manifolds

for which VjE fc0 g is isomorphic to V 0 . This relation is not an equivalence

relation (it is not symmetric), so we take S-equivalence to be the equivalence

relation generated by this relation.

Then we take as the moduli space of semi-stable vector bundles of degree

zero the space of S-equivalence classes. Since Ik is S-equivalent to k O, it

follows immediately from Atiyah's theorem that:

Theorem 2.4.1 The set of S-equivalence classes of semi-stable bundles of

rank n is identied with the set of unordered n-tuples of points (e1 ; : : : ; en )

E. The subset of those with trivial determinant is the subset of unordered

P

n-tuples (e1 ; : : : ; en ) for which b ei = 0 in the group law of E.

i=1

2.5 The moduli space of semi-stable bundles

Everything we have done so far is at the level of points { that is to say we

are describing all isomorphism classes or S-equivalence classes of bundles.

Now we wish to see that the symmetric product of E with itself n-times is

actually a coarse moduli space for the S-equivalence classes of semi-stable

bundles of rank n and degree zero. Since we have already established a

one-to-one correspondence between the points of this symmetric product

and the set of S-equivalence classes of bundles, the question is whether this

identication varies holomorphically with parameters. By this we mean that

any time we have a holomorphic family V ! E X of semi-stable bundles

rank n bundles on E (that is to say V is a rank n vector bundle and its

restriction to each slice E fxg is a semi-stable bundle), there should be

a unique holomorphic mapping X ! (E E)=Sn which associates to

each x 2 X the point of the symmetric product which characterizes the S-

equivalence class of VjEfxg. Of course, this does dene a function from X

to the symmetric product; the only issue is whether it is always holomorphic.

To establish this we need a direct algebraic construction which goes from

a vector bundle to an unordered n-tuple of points in E. Let V be a semi-

stable vector bundle of degree 0 and rank n. Then H 0 (E; V

O(p0 )) is

n-dimensional. We have the evaluation map from sections to the bundle

which we can view as a map from the trivial bundle H 0 (E; V

O(p0 ))

OE

to V

O(p0 ). Taking the determinants we get a map of line bundles

^nH 0 (E; V

O)

OE ! detV

O(np0):

This map is non-trivial. The domain is a trivial line bundle and the range has

degree n. Thus, the cokernel of the map is a torsion sheaf of total degree n.

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Holomorphic bundles over elliptic manifolds

Taking the support of this torsion module, counted with multiplicity gives

the unordered n-tuple in E. As one can see directly, for any sum of line

bundles of degree zero, this map exactly picks out the unordered n points in

E associated with the n-line bundles. More generally, one can easily check

that the sections of Ir

O(q) all vanish to rst order at p0 so that such a

factor produces a zero of order r at q in the above determinant map.

Theorem 2.5.1 The coarse moduli space of S-equivalence classes of semi-

stable vector bundles of rank n and degree zero over an elliptic curve E is

identied with (E E)=Sn . The coarse moduli space of those with trivial

determinant is identied with the subspace of unordered n-tuples which sum

to zero in the group law of E.

Proof. To each semi-stable vector bundle of degree zero we associate the

unordered n-tuple of points in E which corresponds to the set of line bundles

of degree zero on E which are the successive quotients of V . By Atiyah's

classication we see that this is a well-dened function. Clearly, S-equivalent

semi-stable bundles are mapped to the same point and in light of Atiyah's

classication, two bundles which map to the same point are S-equivalent.

It remains to see that if V ! E X is an algebraic family of semi-stable

rank n bundles of degree zero over E, parametrized by X, then the resulting

map X ! (E E)=Sn is holomorphic. To see this let : E X ! X be

the projection and consider the cohomology of along the bers of V

O(p0 ).

Let p: E X ! E be the projection to the other component. Since we have

already seen that for each x 2 X, the cohomology H 0 (E; V j(Efxg

O(p0 ))

has rank n, it follows that the cohomology along the bers R0 (V

p O(p0 ))

is a vector bundle of rank n over X. We take its nth exterior power and pull

back to a line bundle L on E X, trivial on each E fxg. As before, the

evaluation mapping induces a map from ev: L ! ^n V

p O(np0 ), which

ber-by-ber in X is the map we considered above. In particular, the zero

locus of ev is a subvariety of E X whose projection to X is an n-sheeted

ramied covering. Its intersection (counted with multiplicity) with each E

fxg gives the unordered n-points in E associated with the bundle VjEfxg.

This proves that the map X ! (E E)=Sn is algebraic.

(This argument works in either the classical analytic topology or in the

Zariski topology.)

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Holomorphic bundles over elliptic manifolds

The space we just obtained of S-equivalence classes of semi-stable vector

bundles of rank n with trivial determinant has another, extremely useful

description.

Theorem 2.5.2 The coarse moduli space of S-equivalence classes of semi-

stable rank n bundles with trivial determinant on an elliptic curve E is identi-

ed with the projective space associated with the vector space H 0 (E; O(np0 )).

Proof. Given n points e1 ; : : : ; en in E whose sum is zero there is a mero-

morphic function on E vanishing at these n points (with the correct mul-

tiplicities) with a pole only at p0 . Of course, the order of this pole is at

most n and is exactly equal to the number of i; 1 i n, for which ei

is distinct from p0 . (Our evaluation mapping constructed such a function.)

But a non-zero meromorphic function on E is determined up multiple by its

zeros and poles.

Corollary 2.5.3 The coarse moduli space of semi-stable vector bundles on

E of rank n and trivial determinant is isomorphic to a projective space Pn 1 .

In particular, as we vary the complex structure on E, the complex structure

on this moduli space is unchanged.

2.6 The spectral cover construction

Let us construct a family of semi-stable bundles on E parametrized by

P(H 1 (E; OE (p0 )), which to simplify notation we denote by Pn . There is a

covering T ! Pn dened as follows: a point of T consists of a pair ([f] 2

jOE (np0)j; e) where e 2 E is a point of the support of the zero locus of f.

In other words, letting Sn 1 Sn be the stabilizer of the rst point, T =

(E {z E) =Sn 1 . Clearly, T ! Pn is an n-sheeted ramied covering.

| }

n times

There is of course a natural mapping g: T ! E which associates to (f; e) to

point e. We let L be pullback of the Poincar bundles OEE ( Efp0g) to

e

T E under gId. Then the pushforward Vn = (g Id) (L) of L over T E

to Pn E is a rank n vector bundle. A generic point of Pn has n distinct

preimages in T and at such points Vn restricts to be a sum of n distinct line

bundles of degree zero { the sum of bundles associated with the n-points in

the preimage. If one asks what happens at points where g ramies, it turns

out that the pushforward bundle develops factors of the form O(q p0 )

Ir

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Holomorphic bundles over elliptic manifolds

when r points come together. Thus, in fact, this construction produces

a family of regular semi-stable bundles on E. Let q: T E ! T be the

projection to the rst factor. Notice that for any line bundle M on T, we

have that (gId) (L

qM) is also a rank n vector bundle on Pn E which is

isomorphic ber-by-ber to the bundle (g Id) L. Of course, globally these

bundles can be quite dierent, and for example have dierent characteristic

classes over Pn E.

This construction is universal in the following sense, which we shall not

prove (see Theorem 2.8 in Vector Bundles over elliptic brations).

Lemma 2.6.1 Suppose that S is an analytic space and U ! S E is a rank

n vector bundle which is regular and semi-stable with trivial determinant on

each ber fsgE. Let 'U : S ! Pn be the map that associates to each s 2 S

˜ ˜˜

the S-equivalence class of UjfsgE . Let S = S Pn T and let ': S ! S and

˜

be the natural maps. Then there is a line bundle M over S such that U

is isomorphic to

(' Id) (( Id) L

p M);

1

˜ ˜

where p1 : S E ! S is the projection.

2.7 Symplectic bundles

Let us make an analysis of principal SymplC (2n)-bundles which follows the

same lines as the analysis for SLn . We can view a holomorphic principal

Sympl(2n)-bundle over E as a holomorphic vector bundle V with a nonde-

generate skew symmetric (holomorphic) pairing V

V ! C. Equivalently,

we can view the pairing as an isomorphism ': V ! V which is skew-adjoint

in the sense that ' = '. We say that such a bundle is semi-stable if

the underlying vector bundle is. If we consider rst a sum of line bundles

of degree zero, 2n Li this bundle will support a skew symmetric pairing

i=1

if and only if we can number the line bundles so that L2i 1 is isomorphic

to L . In this case we take the pairing to be an orthogonal sum of rank

2i

two pairings, the individual rank two pairings pairing L2i 1 and L2i via the

duality isomorphism. This means that if a semi-stable rank 2n vector bun-

dle with trivial determinant supports a symplectic form then the associated

points (e1 ; : : : ; e2n ) in E are invariant (up to permutation) under the map

e 7! e. It turns out that we can identify the coarse moduli space of rank

2n semi-stable symplectic bundles over E with the subset of n unordered

points in E=fe eg. Of course, E=fe eg is the projective line P 1 .

= =

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Holomorphic bundles over elliptic manifolds

Then the space of n unordered points in P 1 is the n-fold symmetric product

of the projective line, which is well-known to be projective n-space P n . In

terms of linear systems, n unordered points on P 1 is the projective space of

H 0(P 1 ; O(n)).

The double covering map from E to P 1 is given by the Weierstrass p-

function, and the set of 2n points in E invariant under e 7! e is the zeros

of a polynomial of degree at most n in p. This polynomial has a pole only

at p0 and that pole has order at most 2n.

Once again one can make a spectral covering construction. We dene

TSymp as the space of pairs (f 2 jOE (2np0 )f( x) = f(x); e) where e is

in the support of the zero locus of f. Then over T E there is a two plane

bundle with a non-degenerate skew symmetric pairing. The pushforward of

this bundle to the projective space of even functions on E with pole less

than 2np0 times E is then a family of symplectic bundles over E.

It turns out that these are the only two families of simply connected

groups for which a direct construction like this, relating the moduli space to

the projective space of a linear series can be made. For the other groups the

coarse moduli space is not a projective space, but rather a space of a type

which is a slight generalization called a weighted projective space. For the

groups SO(n), which of course are not simply connected, it is possible to

make a similar construction producing a projective space, but it is somewhat

delicate and I shall not give it here.

2.8 Flat SUn-connections

Let us switch gears now to state a general result linking principal holomor-

phic vector bundles to SU(n)-bundles equipped with a

at connection. This

is a variant of the famous Narasimhan-Seshadri theorem.

Suppose that W ! E is an SU(n)-bundle equipped with a connection A.

Consider the complex vector bundle associated to the dening n-dimensional

representation: WC = W SU(n) Cn. Let dA :

0 (E; WC ) !

1 (E; WC ) be

the covariant derivative determined by the connection A. We can take the

(0; 1)-part of the covariant derivative @ A :

0 (E; WC ) !

0;1 (E; WC ). Since

the base is a curve, for dimension reasons the square of this operator van-

ishes, and hence it denes a holomorphic structure on WC. It is a standard

argument to see that this bundle is semi-stable, and in fact is a sum of stable

bundles, i.e. of line bundles of degree zero. The Narasimhan-Seshadri result

is a converse to this computation.

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Theorem 2.8.1 Let V be a semi-stable holomorphic vector bundle of rank n

with trivial determinant over an elliptic curve E. Then there is an SU(n)-

bundle W ! E and a

at SU(n)-connection on W such that the induced

holomorphic structure on the bundle WC is S-equivalent to V . This bundle

and

at connection are unique up to isomorphism. Thus, the space of iso-

morphism classes of

at SU(n)-bundles is identied with the moduli space

of S-equivalence classes of semi-stable holomorphic vector bundles of rank n

with trivial determinant.

Remarks: (1) As we have already pointed out the holomorphic bundles pro-

duced by this construction are sums of line bundles of degree zero. Generi-

cally, of course, these are the unique representatives in their S-equivalence

class, but when two or more of the line bundles coincide there are other

isomorphism classes in the same S-equivalence class.

(2) This theorem is usually stated for curves of genus at least 2. Over such

curves there are stable (as opposed to properly semi-stable) vector bundles.

In this context, the notion of S-equivalence is not necessary (or more pre-

cisely it coincides with isomorphism). The result is that associating to a

at

SU(n)-bundle its associated holomorphic vector bundle determines an iso-

morphism between the space of the space of conjugacy classes of irreducible

representations of 1 (C) ! SU(n) and the space of stable holomorphic vec-

tor bundles of rank n and trivial determinant. In our case, when the base is

a curve of genus one, there are now irreducible representations (for n > 1)

re

ecting the fact that there are no stable vector bundles with trivial de-

terminant. In our case we have an equivalence between the S-equivalence

classes of properly semi-stable bundles.

(3) A

at connection on an SU(n)-bundle over E is given by homomorphisms

1(E) ! SU(n) up to conjugation. Of course, choosing a basis for H1(E), or

equivalently choosing a pair of one-cycles on E which intersect transversely in

a single point, with +1 interesection at that point, identies 1 (E) with a free

abelian group on two generators. A homomorphism of this group into SU(n)

is then simply a pair of commuting elements in SU(n). We consider two

pairs as equivalent if they are conjugate by a single element of SU(n). As is

well-known, two commuting elements in SU(n) are simultaneously conjugate

into the maximal torus T (the diagonal matrices) of SU(n). Thus, we can

assume that our elements lie in this maximal torus. The only conjugation

remaining is simultaneous Weyl conjugation. Thus, the moduli space of

homomorphisms of 1 (E) ! SU(n) up to conjugation is identied with

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Holomorphic bundles over elliptic manifolds

(T T)=W. (Notice that in this description we have ignored the complex

structure.) This generalizes the picture we established in the last lecture for

at SU(n)-bundles over S 1 . Still we have one more step to complete this

picture. Before we identied T=W with the alcove for the ane Weyl group

action on t. We have not yet identied (T T)=W.

Let us give another description of (T T)=W in a way that will keep

track of the complex structure. We write E = C=1 (E) and T = t= where

t Rn is the subspace of points whose coordinates sum to zero, and where

the coroot lattice is the intersection of this subspace with the integral

lattice. A homomorphism 1 (E) ! T dualizes under Pontrjagin duality to a

homomorphism Hom(T; S 1 ) ! Hom(1 (E); S 1 ). The rst group is identied

with the dual to the lattice and the second is identied with the dual

curve E to E. Since we have chosen a point p0 on E we have identied E

and E holomorphically. Thus, the Pontrjagin dual of the map 1 (E) ! T

is a map ! E, or equivalently an element of

E. We have the exact

sequence

0 ! ! Zn ! Z ! 0

where the last map is the sum of the coordinates. Tensoring with E yields

0 !

E ! n E ! E

where again the last map is the sum of the coordinates. Thus, we see that

E is identied with the subset (e1 ; : : : ; en ) 2 n E of points which sum to

zero. Following through the action of the Weyl group, which is the symmetric

group on n letters acting in the obvious way on Zn , we see that the

Weyl action on Hom(1 (E); T) becomes the permutation action on the space

on n points summing to zero. Thus, we see a direct isomorphism

Hom(1 (E); T)=W ! (

E)=W

which realizes the Narasimhan-Seshadri theorem. Notice that the complex

structure on E induces a complex structure on

E which is clearly in-

variant under the Weyl action. Thus, this structure descends to a complex

structure on (

E)=W, and hence determines a holomorphic structure on

the moduli space. This structure agrees with the usual functorial one of

the coarse moduli space. Notice that since, as we have already seen by dif-

ferent methods, the quotient is a projective space, in the end, the complex

structure is independent of the complex structure on E.

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Holomorphic bundles over elliptic manifolds

2.9 Flat G-bundles and holomorphic GC-bundles

The above result for vector bundles generalizes to an arbitrary complex

semi-simple group. Let GC be a complex semi-simple group with compact

form G. Suppose that W ! E is a principal G-bundle equipped with a

at G-connection A. We take an open covering of E by contractible open

sets fUi g. Then for each i, there is a trivialization of WjUi in which A

is the product connection. This induces a trivialization of W G GCjUi .

The overlap functions on gij : Ui \ Uj ! G in the given trivializations for W

are locally constant and hence holomorphic functions when viewed as maps

of Ui \ Uj into G G GC . Thus, we have produced a holomorphic bundle

structure. (A better argument shows that any G-connection produces a

holomorphic bundle structure on the associated GC-bundle.)

Denition 2.9.1 A holomorphic GC-bundle over E is semi-stable if its ad-

joint bundle is semi-stable. We say that two semi-stable GC-bundles V1 ; V2

on E are S-equivalent if there is a connected holomorphic family of semi-

stable GC-bundles on E containing bundles isomorphic to each of V1 and

V2.

Here is the general version of the Narasimhan-Seshadri result [8] for holo-

morphic principal GC-bundles over an elliptic curve.

Theorem 2.9.2 Let E be an elliptic curve and let GC be a complex semi-

simple group (not necessarily simply connected). Let V ! E be a semi-stable

holomorphic GC -bundle. Then there is a

at G-bundle W ! E such that

the induced holomorphic GC-bundle structure on W G GC is S-equivalent

to V . This

at G-bundle is uniquely determined up to isomorphism.

Once again this result is usually stated for smooth curves of genus at least

two and establishes an isomorphism between the space of conjugacy classes

of irreducible representations of 1 (C) into G and the space of isomorphism

classes of stable GC -bundles. But over an elliptic curve there are no stable

GC-bundles (and no irreducible representations of 1 (E) into G) for any

semi-simple group G, and we must consider semi-stable bundles. As in

the case of vector bundles, we are then forced to work with the weaker

equivalence relation of S-equivalence instead of isomorphism.

We can examine

at G-bundles analogously to the way we did when G is

SU(n). First, it is a classical result [2] that in a simple connected Lie group

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Holomorphic bundles over elliptic manifolds

G any pair of commuting elements can be conjugated into the maximal torus

T, and any two pairs of elements in T are simultaneously conjugate if and

only if they are conjugate by a Weyl element. Thus:

Theorem 2.9.3 Let G be a compact simply connected semi-simple group

with maximal torus T, Weyl group W and coroot lattice t. Then, the

space of isomorphism classes of G-bundles with

at connections is identied

with (T T)=W. By Pontrjagen duality this space is identied with (

E)=W where W acts trivially on E and in the natural fashion on .

We have already unraveled all this for SU(n). Let us see what it says

in the case of Sympl(2n). In this case the coroot lattice is the integral

lattice in Rn and the Weyl group is the group (1)n o Sn with the 1's

acting as sign changes of the various coordinates and Sn permuting the

coordinates. Thus,

E is identied with nE and the action of the Weyl

group is by 1 in each factor and permutations of the factors. The quotient

nE=(1)n is nP1 and the symmetric group acts on this to produce a

quotient naturally identied with Pn . This recaptures what we saw directly

in terms of holomorphic bundles.

Theorem 2.9.3 does not hold for non-simply connected groups. The rea-

son is that we cannot simultaneously conjugate a pair of commuting ele-

ments in a non-simply connected group into a maximal torus. For exam-

ple, if G = SO(3) then rotations by radians in two perpendicular planes

commute but cannot be simultaneously conjugated into a maximal torus (a

circle) of SO(3). The reason is that if we lift these elements in any manner to

the double covering SU(2), then they generate a quaternion group of order

8, i.e., the commutation of the lifts in SU(2) is the non-trivial central ele-

ment of SU(2). If we could put both elements in a maximal torus of SO(3),

then they would lift to elements in a maximal torus of SU(2), and hence

there would be lifts which commuted. A similar phenomenon occurs in any

non-simply connected group G=C. It is an interesting problem to determine

the dimension of the space of representations of 1 (E) ! G=C which pro-

duce bundles of a given nontrivial topological type. There is a purely lattice

theoretic description of conjugacy classes of commuting elements in a non-

simply connected compact group, and hence using the Narasimhan-Seshdari

result, a lattice-theoretic description of the coarse moduli space of bundles

over a non-simply connected semi-simple complex group. This description

is quite interesting but much more complicated.

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Holomorphic bundles over elliptic manifolds

2.9.1 Looijenga's theorem

We have seen that in the cases of G = SU(n) and G = Sympl(2n) the space

of S-equivalence classes of semi-stable principal GC-bundles or equivalently

the space of conjugacy classes of representations of 1 (E) ! G is a projective

space. In fact, in each case we explicitly identied the moduli space with the

projective space of a linear system either on the elliptic curve or on the P 1

quotient of the elliptic curve by 1. We now give a theorem which determines

the nature of these moduli spaces for an arbitrary simply connected and

simple group G.

First, let us recall the notion of a weighted projective space. Suppose

that V is a k-dimensional complex vector space with a linear action of C .

Of course, this action can be diagonalized in an appropriate basis of V and

hence the action is completely determined up to isomorphism by k characters

on C. Any character of C is automatically of the form 7! r for some

integer r, and in this fashion the characters of C are identied with the

integers. The characters arising in the action on V are called the weights of

the action. If all the weights are non-zero and have the same sign, let us say

positive, then we say that the action is an action with positive weights. In

this case, there is a nice compact quotient space: V f0g=C which is called

a weighted projective space. For any set (g0 ; g1 ; : : : ; gk ) of positive integers,

the symbol P(g0 ; g1 ; : : : ; gk ) denotes the quotient of the action of C with

weights (g0 ; g1 ; : : : ; gk ). This quotient space is compact and of dimension k.

Indeed, is nitely covered in a ramied fashion by an ordinary projective

space Pk . A weighted projective space is not in general a smooth complex

variety, since the action of C has nite cyclic isotropy groups along certain

subspaces. Rather, the quotient space has cyclic orbifold-type singularities.

(The quotient space is locally isomorphic to the quotient of Ck by a nite

cyclic group.)

Theorem 2.9.4 [6,7] Let G be a compact, simple, simply connected group

and let E be an elliptic curve. Then the space of conjugacy classes of homor-

phisms 1 (E) ! G has a natural complex structure and with this structure

is isomorphic to a weighted projective space P(1; g1 ; : : : ; gr ) where g1 ; : : : ; gr

are the coecients that occur when the coroot dual to the highest root of G

is expressed as a linear combination of the coroots dual to the simple roots.

As we have seen, the space of conjugacy classes of homomorphisms

1(E) ! G is identied with (

E)=W . Since is abstractly a free abelian

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Holomorphic bundles over elliptic manifolds

group of rank r,

E is r E and hence has a natural complex structure

inherited from that of E. Clearly, the Weyl action is holomorphic, and thus

there is a possibly singular complex structure on the quotient space. This is

the one referred to in Looijenga's theorem.

2.10 The coarse moduli space for semi-stable holomorphic

GC-bundles

As we explained in the case of vector bundles, it is not enough simply to

nd the set of S-equivalence classes, one would like to identify a coarse

moduli space (one has to worry whether or not such a coarse moduli space

even exists). As a rst step in constructing the coarse moduli space for

semi-stable holomorphic GC-bundles, we claim that there is a holomorphic

family of GC-bundles over E parametrized by

E. Actually, this family

is a holomorphic family of TC -bundles over E. To construct this family

we choose an integral basis for , hence identifying it with Zr and hence

identifying

E with r E. This also identies the complexication TC of

the maximal torus T of G with a product r C . Then we let P ! E E

be the Poincar line bundle O( E fp0 g) (where is the divisor given

e

by the diagonal embedding of E). This is a family of line bundles of degree

zero over the second factor parametrized by the rst factor. Over (r E)E

we form r p P where pi : (r E) E ! E E is given by the product

i=1 i

of projection onto the ith -component in the rst factor and the identity in

the second factor. This sum of line bundles is then equivalent to a family

of principal r C -bundles, and though our identication, with a family of

TC-bundles. It is easy to trace through the identications and see that the

resulting family of TC -bundles is independent of the choice of basis for .

The Weyl group W acts, as we have already used several times, in the ob-

vious way on the parameter space

E. It also acts as outer automorphism

of the TC and hence changes one TC -principal bundle into a dierent one.

The family is equivariant under these actions: the TC -bundle parametrized

by

e is transformed by w 2 W acting on the set of TC -bundles to the

TC-bundle parametrized by w()

e. Thus, when we extend the structure

group from TC to GC the bundles parametrized by points in the same Weyl

orbit are isomorphic. Thus, our family of GC-bundles is equivariant under

the Weyl action. Consequently, if there is a coarse moduli space MGC for

S-equivalence classes of semi-stable holomorphic GC-bundles over E, then

we obtain a Weyl invariant holomorphic mapping

E ! MGC , and

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Holomorphic bundles over elliptic manifolds

hence holomorphic mapping (

E)=W ! MGC . Since the set of points

of (

E)=W are identied with the set of S-equivalence classes of such

bundles, this map would then be a bijection. In fact there is such a moduli

space (as can be established by rather general algebro-geometric arguments)

and it is (

E)=W. I will not establish this here, but I will assume it in

what follows. The precise statement is:

Theorem 2.10.1 The set of points of (

E)=W is naturally identied

with the set of S-equivalence classes of semi-stable GC-bundles over E. If

W ! E X is a holomorphic family of semi-stable GC-bundles over E

parametrized by X, then the function X ! (

E)=W induced by associating

to each x 2 X the point of (

E)=W identied with the S-equivalence class

of WjE fxg is a holomorphic mapping. This makes (

E)=W the coarse

moduli space for S-equivalence classes of semi-stable GC -bundles over E.

This completes the problem of understanding semi-stable holomorphic

GC-bundles over E. There is a coarse moduli space which is (

E)=W

with the identication of its points with bundles as given above. Further-

more, by Looijenga's theorem, this complex space is a weighted projective

space with positive weights given by the coecients of the simple coroots

in the linear combination which is the coroot dual to the highest root. The

only cases when this weighted projective space is in fact an honest projec-

tive space is when all the weights are 1 and this occurs only for the groups

SU(n) of A-type and the groups Sympl(2n) of C-type. In these two cases

we directly identied with projective space as being associated with an ap-

propriate linear series.

In the next lecture, we will give a construction which will describe the

other moduli spaces in terms of a C -action on an ane space. In the two

special cases given above this ane space will be a linear space and the action

will be the usual C -action hence producing a quotient projective space. In

the other cases, we will show that the ane action can be linearized to

produce a quotient which is a weighted projective space. This will provide

a proof of Looijenga's theorem, dierent from his original proof.

2.11 Exercises:

1. Show that if V; W are vector bundles over a smooth curve and that if

W ! V is a holomorphic map which is one-to-one on the generic ber, then

^

there is a subbundle W V which contains the image of W and so that

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Holomorphic bundles over elliptic manifolds

^ ^

W=W is a sky-scraper sheaf. Show that the degree of W is the degree of W

^

plus the total length of W=W.

2. Let E be a smooth projective curve over the complex numbers of genus

one. Show that the universal covering of E is analytically isomorphic to

C and that the fundamental group of E is identied with a lattice C.

Show that xing a point p0 2 E there is a holomorphic group law on E which

is abelian and for which p0 is the origin. Show this group law is unique given

p0. Using the Weierstrass p-function associated with this lattice show that

E can be embedded as a cubic curve in P2 with equation of the form

y2 = 4x3 + g2 x + g3

for appropriate constants g2 ; g3 .

3. Using the RR theorem show that if V is a semi-stable vector bundle of

positive degree over E, then H 1 (E; V ) = 0 and H 0 (E; V ) has rank equal

to the degree of V . Formulate and prove the corresponding result for semi-

stable vector bundles of negative degree.

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