Holomorphic Bundles Over Elliptic Manifolds

Holomorphic bundles over elliptic manifolds

John W. Morgan

Department of Mathematics, Columbia University,

2990 Broadway, 509 Mathematics Building, New York NY 10027, USA

Lecture given at the

School on Algebraic Geometry

Trieste, 26 July { 13 August 1999

LNS001004

[email protected]

Contents

1 Lie Groups and Holomorphic Principal G -bundles 139

1.1 Generalities on ro ots, the Weyl group, etc...... 140

1.2 Classi cation of simple groups ...... 142

1.3 Groups of C -typ e ...... 145

1.4 Principal holomorphic G -bundles ...... 148

1.5 Principal G-bundles over S ...... 150

1.6 Flat G-bundles over T ...... 152

1.7 Exercises ...... 152

2 Semi-Stable G -Bundles over Elliptic Curves 155

2.1 Stability ...... 155

2.2 Line bundles of degree zero over an elliptic curve ...... 156

2.3 Semi-stable SL (C)-bundles over E ...... 156

2.4 S -equivalence ...... 158

2.5 The mo duli space of semi-stable bundles ...... 159

2.6 The sp ectral cover construction ...... 161

2.7 Symplectic bundles ...... 162

2.8 Flat SU (n)-connections ...... 163

2.9 Flat G-bundles and holomorphic G -bundles ...... 166

2.10 The coarse mo duli space for semi-stable holomorphic G -

bundles ...... 169

2.11 Exercises: ...... 170

3 The Parab olic Construction 173

3.1 The parab olic construction for vector bundles ...... 173

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

3.3 Parab olics in G ...... 180

3.4 The distinguished maximal parab olic ...... 181

3.5 The unip otent subgroup ...... 183

3.6 Unip otent 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 sp ectral covering construction ...... 190

4.3 Globalization of the parab olic construction ...... 192

Holomorphic bund les over el liptic manifolds 2

4.4 The parab olic construction of vector bundles regular and semi-

stable with trivial determinantoneach ber ...... 197

4.5 Exercises: ...... 198

5 The Global Parab olic Construction for Holomorphic Princi-

pal Bundles 199

5.1 The parab olic construction in families ...... 199

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

5.3 Concluding remarks ...... 201

References 203

Holomorphic bund les over el liptic manifolds 139

Intro duction:

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

pact elliptically b ered manifolds. We shall examine constructions of such

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

metric ob jects, namely K 3-surfaces and del Pezzo surfaces.

We shall b e dealing throughout with holomorphic principal bundles with

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

Lie group and G is the asso ciated complex simple algebraic group. Of

course, in the sp ecial case G = SU (n) and hence G = SL (C), we are con-

C n

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 equipp ed with a

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

classical cases there are the nite numb er of exceptional groups. Amazingly

enough, motivated by questions in physics, muchinterest centers around the

group E and its subgroups. For these applications it do es 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 sp end a go o d deal of

time considering elliptically b ered manifolds of the simplest typ e { namely,

elliptic curves.

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

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

So c. 7 (1967) 414-452.

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

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

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

5 (1998) 97-118.

, Vector bundles over elliptic brations, J. Alg. Geom. 8 (1999) 4.

279-401.

1 Lie Groups and Holomorphic Principal G -bundles

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

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

Holomorphic bund les over el liptic manifolds 140

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 ro ots, the Weyl group, etc.

A go o d general reference for Ro ot Systems, Weyl groups, etc. is [3]. Let G

b e 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 de nition 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 mo dulo 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 exp onential mapping exp: t ! T

is a covering pro jection 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 complexi cation g decomp oses 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 complexi cation of the Lie al-

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

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

on a ro ot space is called a ro ot of G (with resp ect to T ) for this subspace.

By de nition the ro ots of G are non-zero elements of the character group

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

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

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

not semi-simple the center of G is p ositive dimensional, and the ro ots span a

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

of the center of G.

Equivalently, the ro ots can b e viewed as elements of the dual space t of

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

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

the rank of G plus the numb er of ro ots of G.

The collection of all ro ots forms an algebraic ob ject inside t called a

ro ot system. By de nition a ro ot system on a vector space V is a nite

subset V of ro ots such that for eachroota2= there is a dual coro ot

_ _

=a 2 V such that the `re ection' r : V ! V de ned by r (b)=bhb; a ia

a a

Holomorphic bund les over el liptic manifolds 141

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

and furthermore that the fr g generate a nite group. The element r is

a a2 a

called the re ection in the wall p erp endicular to a. The wall, denoted W ,

xed by r 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 W V determined by a =0. There is the dual re ection r : V ! V

a a

given by the formula r (v ) = v ha; v ia . These re ections generate the

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

ro ot

the ro ot lattice and denoted . Dually the coro ots span a lattice in V

called the coro ot lattice and denoted . Back to the case of a ro ot system of

a Lie group G with maximal torus T , since anyrootofGmust takeintegral

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

1 1

dual of the ro ot lattice. Furthermore, one can show that the coro ot lattice

iscontained in (T ).

The walls fW g divide V into regions called Weyl chamb ers. Eachcham-

b er has a set of walls. Wesay that a set of ro ots is a set of simple ro ots if (i)

the walls asso ciated with the ro ots are exactly the walls of some chamber C ,

and (ii) the ro ots are non-negative on this chamb er. It turns out that a set of

simple ro ots is in fact an integral basis for the ro ot lattice. Also, every ro ot

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

ro ots, and ro ots are called p ositive or negative ro ots dep ending on the sign

of the co ecients when they are expressed as a linear combination of the

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

ro ots.)

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

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

fashion and consider the ro ots and coro ots as lying in the same space. When

we do this the relative lengths of the simple ro ots and the angles between

their walls are recorded in a Dynkin diagram which completely classi es the

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

simple, then this inner pro duct is unique up to a p ositive scalar factor. In

general, we can use this inner pro duct to identify t and t . When wedowe

have = .

h ; i

There is one set of simple ro ots for each Weyl chamb er. 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 chamb ers, so that all sets

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

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

Holomorphic bund les over el liptic manifolds 142

identi ed with anyWeyl chamb er.

A Lie algebra is said to b e 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 ab elian subalgebras. A compact Lie group is

said to b e 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 pro duces

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 pro duct of simple

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

pro duct of simple groups.

There are a nite numb er 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 identi ed with the coro ot lattice of the

group. This group has a nite center C G which is in fact identi ed with the

dual to the ro ot lattice mo dulo the coro ot lattice. Any other group with the

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

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

the coro ot lattice and contained in the dual to the ro ot lattice. Its quotient

by the coro ot lattice is C .

Each compact simple group G emb eds as the maximal compact subgroup

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

C C

i cation of the Lie algebra of G and there are maximal complex tori of G

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

the complexi cation of SU (n) is SL (C ) whereas the complexi cation of

SO(n) is the complex sp ecial 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 Classi cation of simple groups

Let us lo ok at the classi cation of such ob jects.

Holomorphic bund les over el liptic manifolds 143

1.2.1 Groups of A -typ e

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

of A -typ e. The maximal torus of SU (n + 1) is usually taken to b e the group

of all diagonal matrices with entries in S and the pro duct of the entries b eing

n+1

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

1 n+1 i

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 ro ot space g ;1 i; j n; i 6= j consists of matrices

with non-trivial entry only in the ij p osition. The ro ot asso ciated to this

ro ot space is denoted and is given by ( ;::: ; )= . Writing

ij ij 1 n+1 i

things additively, we identify t with f(z ;::: ;z )j z = 0g and then

1 n+1 i

= e e where e is the linear map which is pro jection onto the i

ij i j i

co ordinate. The usual choice of simple ro ots are ; ;::: ; . With

12 23 nn+1

this choice the p ositive ro ots are the where i < j . When i < j we

have = + + . The Weyl group is the symmetric group

i(i+1) (j1)j

n+1

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

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

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

n+1

R to t is Weyl invariant. Also, it is easy to see that all the ro ots are

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

and meets the previous simple ro ot and the succeeding simple ro ot (in the

obvious ordering) in 1, and is p erp endicular to all other simple ro ots. All

this information is recorded in the Dynkin diagram for A . Since eachroot

has length two, under the induced identi cation of t with t every ro ot is

identi ed with its coro ot . Thus, in this case, and as we shall see, in all

other simply laced cases, one can identify the ro ots and coro ots 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 ro ot 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.

Holomorphic bund les over el liptic manifolds 144

1.2.2 Groups of B -typ e

For any n 3, the group S pin(2n +1) is a simple group of typ e B . The

standard maximal torus of this group is the subgroup of matrices that pro ject

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

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

for each SO(2), and we let e b e the pro jection of t onto the i -factor. The

Lie algebra is all skew symmetric matrices. For any1i; j n let g ; 1

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

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

the Lie algebra of SP in(2n + 1). Thus, there are four ro ots asso ciated with

this space, they are e e . There are also subspaces g ;1 i n, where

i j

g has non-zero entries only in p ositions (2i 1; 2n +1) and (2i; 2n +1) as

well as the symmetric lower diagonal p ositions. The two ro ots asso ciated to

g are e . Thus, S pin(2n + 1) is of rank n. We can identify the Lie algebra

of its maximal torus with R in suchaway that the ro ots are e e for

i j

i 6= j and e , where e is the pro jection onto the i co ordinate. The dual

i i

coro ots to these ro ots are e e and 2e , so that the coro ot lattice

i j i

for S pin(2n + 1) is the even integral lattice in R , whereas the fundamental

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

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

The Weyl group of S pin(2n + 1) is the group generated by re ections in the

simple ro ots. It is easy to see that these re ections generate the group of all

p ermutations of the co ordinates and also all sign changes of the co ordinates.

Thus, abstractly the group is (f1g . Clearly, the standard metric on

R is a Weyl invariant metric. Notice that something new happ ens here {

not all the ro ots have the same length. In our normalization e e has

i j

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

the ro ots 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 S pin(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 typ e B is

1 2 2 2 2 2

Holomorphic bund les over el liptic manifolds 145

1.3 Groups of C -typ e

Let J b e the 2n 2n matrix of blo ck22 diagonal entries

0 1

1 0

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

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

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

consists of all matrices A such that

tr 1

A = JAJ = JAJ:

For any n 2 the symplectic group Sympl(2n) is a group of typ e C 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

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

1 1

diagonal entries ( ; ;::: ; ; ). For example, S y mpl (2) is identi ed

1 n

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

symplectic group S y mpl (2n). It is given as the group of quaternion linear

transformations of H , so that as one would exp ect, the compact form of

S y mpl (2) is SU (2). The maximal torus of this group is again a pro duct

of circles so that t is again identi ed with R . The ro ots are e e for

i j

1 i

Its Weyl group is the same as the Weyl group of B . The coro ots dual to

the ro ots are e e and e so that the coro ot lattice is the integral

i j i

lattice Z . The dual to the ro ot lattice consists of all fx ;::: ;x g such that

1 n

x 2 (1=2)Z for all i and x x (mo d Z) for all i; j . This lattice contains

i i j

the coro ot 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 typ e C is

1 1 1 1 1

It turns out that S pin(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)

whichiswhywe b egin the C -series at n =2.

Holomorphic bund les over el liptic manifolds 146

1.3.1 Groups of D -typ e

For any n 4 the group S pin(2n) is a group of typ e D . The usual maximal

torus for S pin(2n) is the subgroup that pro jects onto SO(2) SO(2)

SO(2n). Thus, t is identi ed with R with the factors b eing tangent to

the factors in this decomp osition. The ro ots of S pin(2n) are e e for

i j

1 i

inner pro duct all ro ots have length 2. Thus, we can identify the ro ots with

their dual coro ots in this case. The coro ot lattice is then the even integral

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

integral lattice which contains the coro ot lattice with index two re ecting

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

consists of all p ermutations of the co ordinates and all even sign changes of

the co ordinates. This is a simply laced group.

The Dynkin diagram of typ e D is

1 ¡

¡¡

2 2 2 2 1 ¡ ¡¡

¡¡

C CC CC

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 E ;E ;E ;G and F . The subscript is the rank of the

6 7 8 2 4

group. There are natural inclusions D E E E . The fundamental

5 6 7 8

group of E is a cyclic group of order 9 r . Both G and F are simply

r 2 4

connected. Here are their Dynkin diagrams:

Holomorphic bund les over el liptic manifolds 147

1 2 3 2 1

1 2 3 4 3 2

2 3 4 5 6 4 2

2 4 3 2

3 2

We shall not say to o much ab out these groups now, but let me give

the lattices E ;E ;E . These are viewed as the fundamental group of the

6 7 8

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

lattices with an inner pro duct. This is the Weyl invariant inner pro duct. In

all cases these groups are simply laced and the coro ots are the elements in the

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

2 2

consider the inde nite integral quadratic form q (x; a ;::: ;a )= a x

1 r

i=1

Holomorphic bund les over el liptic manifolds 148

r +1

on Z . We let k =(3;1;1;::: ;1). Then q (k )=r9<0. For 6 r 8,

r +1

the lattice E is the orthogonal subspace in Z of k . It is of rank r and has

the induced quadratic form which is easily seen to b e even, p ositive de nite,

and of discriminant9r. For lower values of r it turns out that the lattice

de ned this way is the lattice of a classical group: E = D , E = A ,

5 5 4 4

E = A A .

3 2 1

1.4 Principal holomorphic G -bundles

Let G b e a complex linear algebraic group and let X b e a complex mani-

fold. A holomorphic principal G -bundle over X is determined byanopen

covering fU g of X and transition functions g : U \ U ! G . The tran-

i ij i j C

sition functions are required to be holomorphic and to satisfy the co cycle

conditions: g = g and g (z) g (z )=g (z ) for all z 2 U \ U \ U . As

ji ij i j

jk ik k

usual, we can use the g as gluing data to glue U G to U G along

ij i C j C

u G ,by the rule (z; g) 2 U G maps to (z; g (z ) g )inU G pro-

ij C i C ij j C

vided that z 2 U \ U . The co cycle condition tells us that the triple gluings

i j

are compatible so that wehave de ned an equivalence relation and the result

of gluing E is a Hausdor space. The pro jection mappings U G ! U

i C i

then t together to de ne a continuous map p: E ! X . The fact that the g

are holomorphic implies that the natural complex structures on the U G

i C

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

a holomorphic submersion with each b er isomorphic to G . The complex

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

the pro jection. There is a natural (right) free, holomorphic G -action on E

such that p is the quotient pro jection of this action. Twoopencoverings and

gluing functions de ne the isomorphic principal G -bundles if the resulting

total spaces are biholomorphic by a G -equivariant mapping commuting

with the pro jections to X .

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

C C

is a complex linear representation, then there is an asso ciated holomorphic

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

is V where G acts on V via the representation .

G C

Example: Let G 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 asso ciated to and are isomorphic as holo-

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

Holomorphic bund les over el liptic manifolds 149

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

be identi ed with the complement of the zero section in the corresp onding

holomorphic line bundle.

Along the same lines, let G be SL (C). Then using the de ning com-

C n

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

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

bundle has the prop erty that its determinant line bundle ^ V 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 de ne the asso ciated bundle of sp ecial linear frames in V . The b er

over x 2 X consists of all bases ff ;::: ;f g for V such that f ^ ^ f

1 n x 1 n

is identi ed with 1 2 C under the given trivialization of ^ (V ). The lo-

cal trivialization of the vector bundle, pro duces a lo cal 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

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

cipal SL (C)-bundle. This sets up a bijection b etween isomorphism classes

of holomorphic principal SL (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 b er. A holomorphic

principal Symp(2n)-bundle over X is identi ed 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 b ers

which is non-degenerate on each b er.

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

simple groups: SL (C), SO(n; C), and S y mpl (2n; C).

Asso ciated to any complex group G there is the adjoint representation

of G on its Lie algebra g . Thus, asso ciated to any holomorphic principal

C C

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

C C

as a group. In the case of E 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: G has a seven dimensional representation

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

E has a 27 dimensional representation (whichwe discuss later), and E has

6 7

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

Holomorphic bund les over el liptic manifolds 150

study principal bundles over these groups is to lo ok at the vector bundles

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

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

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

structure of that bundle.

1.5 Principal G-bundles over S

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

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

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

tro duction of a at connection will b e explained in the next lecture.) The

holonomyofAaround the base circle is an elementofG, determined up to

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

sets up an isomorphism b etween the space of conjugacy classes of elements

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

connections over S . 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-understo o d 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 lo oks like. To answer this question weintro duce the ane Weyl

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

of G. Foraroot and k 2 Z we denote by W the co dimension-one ane-

linear subspace of t determined by the equation f = k g. By de nition, the

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

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

exact sequence of groups

0 ! ! W ! W ! 0:

(Recall that t is the coro ot lattice.) Here, the map W ! W is the

di erential or linearization of the ane map. (Recall that is the coro ot

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

generated by the re ections in the walls W , but the action of W on

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

semi-direct pro duct W with the natural action of W on .

Holomorphic bund les over el liptic manifolds 151

The set of walls W is a lo cally nite set and divides t into (an in nite

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 chamb er. 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

chamb er containing it together with one more. This extra wall is given byan

equation of the form ~ = 1 for some ro ot determined by the Weyl chamber

(or equivalently by the set of simple ro ots f ;::: ; g determined by the

1 r

Weyl chamb er). It turns out that ~ is a p ositive linear combination of the

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

linear combination of the simple ro ots with the prop erty that its sum with

any simple ro ot is not a ro ot. This ro ot ~ is called the highest ro ot of G.

The numb ers displayed on the Dynkin diagrams ab ove are the co ecients

of the simple ro ots in their unique linear combination which is the highest

ro ot. If is a set of simple ro ots for G, then the asso ciated set [ ~ is

denoted by and is called the extended set of simple ro ots.

As in the case of the Weyl group, it is a nice geometric argumenttoshow

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

t=W is identi ed 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 identi ed with an

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

exp(t) 2 T .

Pro of. Every g 2 G is conjugate to a p oint t 2 T . Two p oints 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 identi ed with T=W . Since G is simply connected, T = t= and hence

T=W = t=W whichwehave just seen is identi ed with an alcove.

Example: If G = SU (n + 1), then the alcove is the subset (t ;::: ;t ) 2

1 n+1

n+1

R satisfying t =0 and t 0 and t t for all 1 j n +1.

j j j j +1

Every elementinSU (n + 1) is conjugate to a diagonal matrix with diagonal

entries ( ;::: ; ), the b eing elements of S . We can do a further

1 n+1 j

conjugation under = exp(2it for 0 t 1. The determinant condition

j j j

Q P

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

j j

Holomorphic bund les over el liptic manifolds 152

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

n+1

isomorphism b etween the simplex in R and the space of conjugacy classes

in SU (n+ ).

1.6 Flat G-bundles over T

Let G b e a compact, simply connected group.

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

0 0

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

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

0 0

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

0 0

(x ;y ).

The pro of is an exercise.

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

space of isomorphism classes of principal G-bund les with at connections

over T is identi ed 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 to o surprising since for G simply connected, all G-bundles

over T are top ologically trivial.

1.7 Exercises

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

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

elementofGis contained in a maximal torus.

2. Show all maximal tori of G are conjugate.

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

in G is the torus itself.

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

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

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

( ) = where t is the coro ot lattice, where t is the ro ot

ro ot ro ot

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

ro ot

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

Holomorphic bund les over el liptic manifolds 153

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 ro ots for groups of A , B , C , and D typ e and

n n n n

determine the dimension of each of these groups.

r +1

6. For any r; 3 r 8, let Z b e the free ab elian group of rank r + 1 with

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

1 r

1, Q(e ) = 1; 1 i r , and the basis b eing mutually orthogonal. Let

k = 3h e . Then k is a lattice of rank r with a p ositive de nite

i=1

pairing of determinant9r. Show that a basis for k is e e ;::: e

1 2 r1

e ;H e e e and that these vectors are all of square 2. Count the

r 1 2 3

number of vectors in this lattice of square two. Show that for eachvector 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 coro ot lattice of A A ,

2 1

for r =4, the lattice is the coro ot lattice of A , for r =5 the lattice is the

coro ot lattice for D . In all cases the coro ots 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 coro ot lattice of E . The statements ab out the ro ots, coro ots and the

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

dimensions of the Lie groups E ;E ;E .

6 7 8

8. Let E b e an elliptic curve. Give a 1-co cycle which represents the generator

of H (E ; O )). Give a 1-co cycle that represents the principal C -bundle

O (q p ).

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 supp orts a non-degenerated quadratic form

instead.

10. De ne the compact form of Sympl (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 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 identi ed

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 b oth x and y . [Hint: Show

that the centralizer of x, Z (x) is connected.] Show that this fails to b e true G

Holomorphic bund les over el liptic manifolds 154

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

X and X 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 are classi ed up to top ological equivalence by a single characteristic class

2 2

w 2 H (T ; (G)) = (G). Show that if a G-bundle admits a at connec-

2 1 1

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

istic class in (G) G is computed as follows. One cho oses 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 pro jection to G, i.e., lies in (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 has two comp onents. Show that one of these comp onents

has dimension 2 and the other is a single p oint.

Holomorphic bund les over el liptic manifolds 155

2 Semi-Stable G -Bundles over Elliptic Curves

In this lecture we intro duce 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 mo duli 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 G -bundles. Lastly, we discuss Lo oijenga's theorem which, in

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

these mo duli spaces in terms of the coro ot integers for the group G.

2.1 Stability

Let C be a compact complex curve. The slop e 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 . Avector bundle V ! C is said to b e stable if

for every prop er subbundle W V wehave (W)<(V). The bundle V is

said to be semi-stable if (W ) (V ) for every prop er 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 whichisin-

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

b e a subbundle of V , but rather is a subsheaf of its sheaf of lo cal holomorphic

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

W under the given mapping with the prop erty that W mo dulo the image

of W is supp orted at a nite set of p oints (a sky-scrap er sheaf ). In this case

0 0 0

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

sky-scrap er sheaf. It follows that (W ) (W ) so that if W destabilizes or

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

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

to the case of curves.

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

can b e 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.

Holomorphic bund les over el liptic manifolds 156

Over high dimensional manifolds X we need a couple of mo di cations.

First of all we need a Kahler class ! so that we can de ne the degree of a

bundle V , deg (V ), to b e ! ^ c (V ). (Of course, the degree, and hence

the slop e dep ends on the choice of Kahler class.) Then as indicated ab ove

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

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

by the same formula as ab ove. With these mo di cations one de nes slop e

stability and slop e semi-stability exactly as b efore. There are more re ned

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

dimensional bases, esp ecially if one hop es to obtain compact mo duli spaces.

2.2 Line bundles of degree zero over an elliptic curve

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

complex (smo oth) curveofgenus 1, and wehave xed a p oint p 2 E . This

determines an ab elian group lawonE for which p is the identity element.

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

Ro ch,

1 0

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

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

if we consider L O(p ), then RR tells us that this bundle has at least one

holomorphic section. Let : O ! L O(p ) be such a section. Of course,

since the degree of L O(p )is1, vanishes once, say atapointq 2 E. This

means that factors to givea holomorphic mapping : O (q ) ! L O(p )

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

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

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

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

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

which is also written as O (q p ). It is easy to see that asso ciating to L the

p oint q sets up an isomorphism b etween the space of line bundles of degree

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

2.3 Semi-stable SL (C)-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-p ositive degree. Let us

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

Holomorphic bund les over el liptic manifolds 157

pro ceed in the same manner as with line bundles. The bundle V O(p )is

also easily seen to be semi-stable and of slop e 1. By RR V O(p ) has n

holomorphic sections. Let H b e the space of these sections. Then evaluation

determines a vector bundle mapping from the trivial bundle H O to

V O (p ). If no non-trivial section of V O (p )vanished at any p oint,

E 0 E 0

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

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

section of V O (p )vanishing at some p oint. Consider the cokernel of this

E 0

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

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

line bundle L tting into an exact sequence

0 !O !L !T ! 0

and an extension of to a map : L ! V O(p ) whose quotient is a

T 0

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

of L is the total length of T . By the fact that the slop e of V O (p )

T E 0

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 p oint q 2 E .

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

O (q p ) ! 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 asso ciate to V , the n p oints that are

identi ed with these line bundles. Let us now think ab out the extensions.

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

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

bundles of degree zero then any extension

0 ! L ! V ! L ! 0

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

cessive extension of line bundles L of degree zero and W as a successive

i 2

extension of line bundles M of degree zero, and no L is isomorphic to any

j i

M , then any extension

0 ! W ! V ! W ! 0

1 2

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

q 2E q q

successive extension of line bundles isomorphic to O (q p ). Lastly, there 0

Holomorphic bund les over el liptic manifolds 158

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

0 !O(qp )!V !O(qp )! 0:

0 0

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

a unique nontrivial extension by O (q p ) 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 b est we can hop e for in higher

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

see, there are stable bundles of p ositive degree.

This allows us to establish the following theorem rst proved byAtiyah:

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

morphic to a direct sum of bund les of the form O (q p ) I where the I

0 r r

are de ned inductively as fol lows; I = O and I is the unique non-trivial

1 r

extension of I by O .

r 1

r (q )

Clearly, the determinantof O(qp ) I is O (q p ) ,

q2E 0 q 2E 0

r(q)

or under our identi cation of line bundles of degree zero with p oints of E ,

r (q )q , where the sum is taken the determinantof V is identi ed with

q 2E

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

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

2.4 S -equivalence

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

curve in the sense that weenumerated the isomorphism classes. But wehave

not pro duced a mo duli 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 exempli ed

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

E fag is the bundle which is the extension of O by O given by the extension

class a. This bundle is isomorphic to I for all fag6= 0 and is isomorphic to

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

isomorphism classes of bundles I and OO must b e identi ed.

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 if there is a family

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

c 2 C the bundle VjC fcg is isomorphic to V and so that there is c 2 C 0

Holomorphic bund les over el liptic manifolds 159

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

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

relation generated by this relation.

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

zero the space of S -equivalence classes. Since I is S -equivalentto O, it

k k

follows immediately from Atiyah's theorem that:

Theorem 2.4.1 The set of S -equivalence classes of semi-stable bund les of

rank n is identi ed with the set of unordered n-tuples of points (e ;::: ;e )

1 n

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

n-tuples (e ;::: ;e ) for which e =0 in the group law of E .

1 n i

i=1

2.5 The mo duli space of semi-stable bundles

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

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

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

actually a coarse mo duli 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 corresp ondence between the p oints of this symmetric pro duct

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

identi cation 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)=S which asso ciates to

each x 2 X the p ointof the symmetric pro duct whichcharacterizes the S -

equivalence class of VjE fxg. Of course, this do es de ne a function from X

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

To establish this we need a direct algebraic construction which go es from

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

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

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

whichwe can view as a map from the trivial bundle H (E ; V O(p )) O

0 E

to V O(p ). Taking the determinants we get a map of line bundles

n 0

^ H (E ; V O) O ! detV O(np ):

E 0

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.

Holomorphic bund les over el liptic manifolds 160

Taking the supp ort of this torsion mo dule, 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 p oints in

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

that the sections of I O(q) all vanish to rst order at p so that such a

r 0

factor pro duces a zero of order r at q in the ab ove determinant map.

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

stable vector bund les of rank n and degree zero over an el liptic curve E is

identi ed with (E E)=S . The coarse moduli space of those with trivial

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

to zero in the group law of E .

Pro of. To each semi-stable vector bundle of degree zero we asso ciate the

unordered n-tuple of p oints in E which corresp onds to the set of line bundles

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

classi cation we see that this is a well-de ned function. Clearly, S -equivalent

semi-stable bundles are mapp ed to the same p oint and in light of Atiyah's

classi cation, two bundles which map to the same p oint are S -equivalent.

It remains to see that if V!EX 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)=S is holomorphic. To see this let : E X ! X be

the pro jection and consider the cohomology of along the b ers of V O(p ).

Let p: E X ! E b e the pro jection to the other comp onent. Since wehave

already seen that for each x 2 X , the cohomology H (E ; V j(E fxg O(p ))

has rank n, it follows that the cohomology along the b ers R (V p O (p ))

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

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

evaluation mapping induces a map from ev : L ! ^ V p O (np ), which

b er-by- b er in X is the map we considered ab ove. In particular, the zero

lo cus of ev is a subvarietyof EX whose pro jection to X is an n-sheeted

rami ed covering. Its intersection (counted with multiplicity) with each E

fxg gives the unordered n-p oints in E asso ciated with the bundle VjE fxg.

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

(This argumentworks in either the classical analytic top ology or in the

Zariski top ology.)

Holomorphic bund les over el liptic manifolds 161

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 bund les with trivial determinant on an el liptic curve E is identi-

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

Pro of. Given n p oints e ;::: ;e in E whose sum is zero there is a mero-

1 n

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

tiplicities) with a p ole only at p . Of course, the order of this p ole is at

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

is distinct from p . (Our evaluation mapping constructed sucha function.)

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

zeros and p oles.

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

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

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

on this moduli space is unchanged.

2.6 The sp ectral cover construction

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

P(H (E ; O (p )), whichto simplify notation we denote by P . There is a

E 0 n

covering T ! P de ned as follows: a p oint of T consists of a pair ([f ] 2

jO (np )j;e) where e 2 E is a p oint of the supp ort of the zero lo cus of f .

E 0

In other words, letting S S be the stabilizer of the rst p oint, T =

n1 n

=S . Clearly, T ! P is an n-sheeted rami ed covering. (E E)

n1 n

{z } |

ntimes

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

p oint e. We let L b e pullback of the Poincare bundles O ( E fp g)to

E E 0

TE under g Id. Then the pushforward V =(gId) (L)ofLover T E

to P E is a rank n vector bundle. A generic p ointof P has n distinct

n n

preimages in T and at such p oints V restricts to b e a sum of n distinct line

bundles of degree zero { the sum of bundles asso ciated with the n-p oints in

the preimage. If one asks what happ ens at p oints where g rami es, it turns

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

0 r

Holomorphic bund les over el liptic manifolds 162

when r p oints come together. Thus, in fact, this construction pro duces

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

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

have that (g Id) (L q M) is also a rank n vector bundle on P E whichis

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

bundles can b e quite di erent, and for example have di erentcharacteristic

classes over P E .

This construction is universal in the following sense, whichwe 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 isarank

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

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

U n

~ ~

the S -equivalence class of Uj . Let S = S T and let '~: S ! S and

fsgE

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

is isomorphic to

(' Id) (( Id) L p M);

~ ~

where p : S E ! S is the projection.

2.7 Symplectic bundles

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

same lines as the analysis for SL . 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 whichisskew-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, L this bundle will supp ort a skew symmetric pairing

i=1

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

2i1

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

two pairings, the individual rank two pairings pairing L and L via the

2i1 2i

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

dle with trivial determinant supp orts a symplectic form then the asso ciated

p oints (e ;::: ;e ) in E are invariant (up to p ermutation) under the map

1 2n

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

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

p oints in E=fe eg. Of course, E=fe eg is the pro jective line P .

= =

Holomorphic bund les over el liptic manifolds 163

Then the space of n unordered p oints in P is the n-fold symmetric pro duct

of the pro jective line, whichis well-known to be pro jective n-space P . In

terms of linear systems, n unordered p oints on P is the pro jective space of

0 1

H (P ; O (n)).

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

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

of a p olynomial of degree at most n in p. This p olynomial has a p ole only

at p and that p ole has order at most 2n.

Once again one can make a sp ectral covering construction. We de ne

T as the space of pairs (f 2jO (2np )f (x)=f(x);e) where e is

Symp E 0

in the supp ort of the zero lo cus 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 pro jective space of even functions on E with p ole less

than 2np 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 mo duli space to

the pro jective space of a linear series can b e made. For the other groups the

coarse mo duli space is not a pro jective space, but rather a space of a typ e

whichis a slight generalization called a weighted pro jective space. For the

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

make a similar construction pro ducing a pro jective space, but it is somewhat

delicate and I shall not give it here.

2.8 Flat SU (n)-connections

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

phic vector bundles to SU (n)-bundles equipp ed with a at connection. This

is a variant of the famous Narasimhan-Seshadri theorem.

Supp ose that W ! E is an SU (n)-bundle equipp ed with a connection A.

Consider the complex vector bundle asso ciated to the de ning n-dimensional

n 0 1

representation: W = W C . Let d : (E;W ) ! (E;W ) be

C A C C

SU (n)

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

0 0;1

(0; 1)-part of the covariant derivative @ : (E;W ) ! (E;W ). Since

A C C

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

ishes, and hence it de nes a holomorphic structure on W . 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.

Holomorphic bund les over el liptic manifolds 164

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

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

bund le W ! E and a at SU (n)-connection on W such that the induced

holomorphic structure on the bund le W is S -equivalent to V . This bund le

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

morphism classes of at SU (n)-bund les is identi ed with the moduli space

of S -equivalence classes of semi-stable holomorphic vector bund les of rank n

with trivial determinant.

Remarks: (1) As wehave already p ointed 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 opp osed to prop erly 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 asso ciating to a at

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

morphism b etween the space of the space of conjugacy classes of irreducible

representations of (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 prop erly semi-stable bundles.

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

(E ) ! SU (n) up to conjugation. Of course, cho osing a basis for H (E ), or

1 1

equivalently cho osing a pair of one-cycles on E whichintersect transversely in

a single p oint, with +1 interesection at that p oint, identi es (E ) with a free

ab elian 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 elementofSU (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 mo duli space of

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

Holomorphic bund les over el liptic manifolds 165

(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 . Still we have one more step to complete this

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

action on t. Wehave not yet identi ed (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= (E ) and T = t= where

t R is the subspace of p oints whose co ordinates sum to zero, and where

the coro ot lattice is the intersection of this subspace with the integral

lattice. A homomorphism (E ) ! T dualizes under Pontrjagin dualitytoa

1 1

homomorphism Hom (T; S ) ! Hom ( (E );S ). The rst group is identi ed

with the dual to the lattice and the second is identi ed with the dual

curve E to E . Since wehave chosen a p oint p on E wehave identi ed E

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

is a map ! E , or equivalently an elementof E. We have the exact

sequence

0 ! ! Z ! Z ! 0

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

0 ! E ! E !E

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

E is identi ed with the subset (e ;::: ;e ) 2 E of p oints which sum to

1 n n

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

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

Weyl action on Hom( (E );T) b ecomes the p ermutation action on the space

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

Hom ( (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 mo duli space. This structure agrees with the usual functorial one of

the coarse mo duli space. Notice that since, as wehave already seen by dif-

ferent metho ds, the quotient is a pro jective space, in the end, the complex

structure is indep endent of the complex structure on E .

Holomorphic bund les over el liptic manifolds 166

2.9 Flat G-bundles and holomorphic G -bundles

The ab ove result for vector bundles generalizes to an arbitrary complex

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

form G. Supp ose that W ! E is a principal G-bundle equipp ed with a

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

sets fU g. Then for each i, there is a trivialization of W jU in which A

i i

is the pro duct connection. This induces a trivialization of W G jU .

G C i

The overlap functions on g : U \ U ! G in the given trivializations for W

ij i j

are lo cally constant and hence holomorphic functions when viewed as maps

of U \ U into G G . Thus, we have pro duced a holomorphic bundle

i j G C

structure. (A b etter argument shows that any G-connection pro duces a

holomorphic bundle structure on the asso ciated G -bundle.)

De nition 2.9.1 A holomorphic G -bundle over E is semi-stable if its ad-

joint bundle is semi-stable. We say that two semi-stable G -bundles V ;V

C 1 2

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

stable G -bundles on E containing bundles isomorphic to each of V and

C 1

V .

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

morphic principal G -bundles over an elliptic curve.

Theorem 2.9.2 Let E be an el liptic curve and let G be a complex semi-

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

holomorphic G -bund le. Then there is a at G-bund le W ! E such that

the induced holomorphic G -bund le structure on W G is S -equivalent

C G C

to V . This at G-bund le is uniquely determined up to isomorphism.

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

two and establishes an isomorphism b etween the space of conjugacy classes

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

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

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

C 1

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 waywe did when G is

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

Holomorphic bund les over el liptic manifolds 167

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

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

only if they are conjugate bya 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-bund les with at connections is identi ed

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

E )=W where W acts trivial ly 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 coro ot lattice is the integral

n n

lattice in R and the Weyl group is the group (1) o S with the 1's

acting as sign changes of the various co ordinates and S p ermuting the

co ordinates. Thus, E is identi ed with E and the action of the Weyl

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

n 1

E=(1) is P and the symmetric group acts on this to pro duce a

n n

quotient naturally identi ed with P . This recaptures what wesaw directly

in terms of holomorphic bundles.

Theorem 2.9.3 do es 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 p erp endicular planes

commute but cannot b e 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-

mentofSU (2). If we could put b oth 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 b e lifts which commuted. A similar phenomenon o ccurs in any

non-simply connected group G=C . It is an interesting problem to determine

the dimension of the space of representations of (E ) ! G=C which pro-

duce bundles of a given nontrivial top ological typ e. 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 mo duli space of bundles

over a non-simply connected semi-simple complex group. This description

is quite interesting but much more complicated.

Holomorphic bund les over el liptic manifolds 168

2.9.1 Lo oijenga's theorem

Wehave seen that in the cases of G = SU (n) and G = Sympl(2n) the space

of S -equivalence classes of semi-stable principal G -bundles or equivalently

the space of conjugacy classes of representations of (E ) ! G is a pro jective

space. In fact, in each case we explicitly identi ed the mo duli space with the

pro jective space of a linear system either on the elliptic curveorontheP

quotient of the elliptic curveby1. Wenow give a theorem which determines

the nature of these mo duli spaces for an arbitrary simply connected and

simple group G.

First, let us recall the notion of a weighted pro jective space. Supp ose

that V is a k -dimensional complex vector space with a linear action of C .

Of course, this action can b e 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! for some

integer r , and in this fashion the characters of C are identi ed 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

p ositive, then we say that the action is an action with p ositiveweights. In

this case, there is a nice compact quotient space: V f0g=C which is called

a weighted pro jective space. For any set (g ;g ;::: ;g ) of p ositiveintegers,

0 1

the symbol P(g ;g ;::: ;g ) denotes the quotient of the action of C with

0 1

weights (g ;g ;::: ;g ). This quotient space is compact and of dimension k .

0 1

Indeed, is nitely covered in a rami ed fashion by an ordinary pro jective

space P . A weighted pro jective space is not in general a smo oth complex

variety, since the action of C has nite cyclic isotropy groups along certain

subspaces. Rather, the quotient space has cyclic orbifold-typ e singularities.

(The quotient space is lo cally isomorphic to the quotient of C 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 el liptic curve. Then the spaceofconjugacy classes of homor-

phisms (E ) ! G has a natural complex structure and with this structure

is isomorphic to a weightedprojective space P(1;g ;::: ;g ) where g ;::: ;g

1 r 1 r

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

(E ) ! G is identi ed with ( E )=W . Since is abstractly a free ab elian 1

Holomorphic bund les over el liptic manifolds 169

group of rank r , E is E and hence has a natural complex structure

inherited from that of E . Clearly, the Weyl action is holomorphic, and thus

there is a p ossibly singular complex structure on the quotient space. This is

the one referred to in Lo oijenga's theorem.

2.10 The coarse mo duli space for semi-stable holomorphic

G -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

mo duli space (one has to worry whether or not sucha coarse mo duli space

even exists). As a rst step in constructing the coarse mo duli space for

semi-stable holomorphic G -bundles, we claim that there is a holomorphic

family of G -bundles over E parametrized by E. Actually, this family

is a holomorphic family of T -bundles over E . To construct this family

we cho ose an integral basis for , hence identifying it with Z and hence

identifying E with E . This also identi es the complexi cation T of

r C

the maximal torus T of G with a pro duct C . Then we let P ! E E

b e the Poincare line bundle O ( E fp g) (where is the divisor given

by the diagonal emb edding of E ). This is a family of line bundles of degree

zero over the second factor parametrized by the rst factor. Over ( E ) E

we form p P where p :( E) E ! E E is given by the pro duct

i r

i=1 i

of pro jection onto the i -comp onent in the rst factor and the identity in

the second factor. This sum of line bundles is then equivalent to a family

of principal C -bundles, and though our identi cation, with a family of

T -bundles. It is easy to trace through the identi cations and see that the

resulting family of T -bundles is indep endent of the choice of basis for .

The Weyl group W acts, as wehave already used several times, in the ob-

vious way on the parameter space E . It also acts as outer automorphism

of the T and hence changes one T -principal bundle into a di erent one.

C C

The family is equivariant under these actions: the T -bundle parametrized

by e is transformed by w 2 W acting on the set of T -bundles to the

T -bundle parametrized by w () e. Thus, when we extend the structure

group from T to G the bundles parametrized by p oints in the same Weyl

C C

orbit are isomorphic. Thus, our family of G -bundles is equivariant under

the Weyl action. Consequently, if there is a coarse mo duli space M for

S -equivalence classes of semi-stable holomorphic G -bundles over E , then

we obtain a Weyl invariant holomorphic mapping E ! M , and

G C

Holomorphic bund les over el liptic manifolds 170

hence holomorphic mapping ( E )=W ! M . Since the set of p oints

of ( E )=W are identi ed with the set of S -equivalence classes of such

bundles, this map would then b e a bijection. In fact there is such a mo duli

space (as can b e 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 natural ly identi ed

with the set of S -equivalence classes of semi-stable G -bund les over E . If

W ! E X is a holomorphic family of semi-stable G -bund les over E

parametrizedbyX, then the function X ! ( E )=W induced by associating

to each x 2 X the point of ( E )=W identi ed with the S -equivalence class

of W jE fxg is a holomorphic mapping. This makes ( E )=W the coarse

moduli space for S -equivalence classes of semi-stable G -bund les over E .

This completes the problem of understanding semi-stable holomorphic

G -bundles over E . There is a coarse mo duli space which is ( E )=W

with the identi cation of its p oints with bundles as given ab ove. Further-

more, by Lo oijenga's theorem, this complex space is a weighted pro jective

space with p ositive weights given by the co ecients of the simple coro ots

in the linear combination which is the coro ot dual to the highest ro ot. The

only cases when this weighted pro jective space is in fact an honest pro jec-

tive space is when all the weights are 1 and this o ccurs only for the groups

SU (n) of A-typ e and the groups Sympl(2n) of C -typ e. In these two cases

we directly identi ed with pro jective space as b eing asso ciated with an ap-

propriate linear series.

In the next lecture, we will give a construction which will describ e the

other mo duli spaces in terms of a C -action on an ane space. In the two

sp ecial cases given ab ove this ane space will b e a linear space and the action

will b e the usual C -action hence pro ducing a quotient pro jective space. In

the other cases, we will show that the ane action can be linearized to

pro duce a quotient whichis a weighted pro jective space. This will provide

a pro of of Lo oijenga's theorem, di erent from his original pro of.

2.11 Exercises:

1. Show that if V; W are vector bundles over a smo oth curve and that if

W ! V is a holomorphic map which is one-to-one on the generic b er, then

there is a subbundle W V which contains the image of W and so that

Holomorphic bund les over el liptic manifolds 171

^ ^

W=W is a sky-scrap er sheaf. Show that the degree of W is the degree of W

plus the total length of W=W.

2. Let E be a smo oth pro jective curve over the complex numb ers of genus

one. Show that the universal covering of E is analytically isomorphic to

C and that the fundamental group of E is identi ed with a lattice C.

Show that xing a p oint p 2 E there is a holomorphic group lawonE which

is ab elian and for which p is the origin. Show this group law is unique given

p . Using the Weierstrass p-function asso ciated with this lattice show that

E can b e emb edded as a cubic curveinP with equation of the form

2 3

y =4x +g x+g

2 3

for appropriate constants g ;g .

2 3

3. Using the RR theorem show that if V is a semi-stable vector bundle of

1 0

p ositive degree over E , then H (E ; V ) = 0 and H (E ; V ) has rank equal

to the degree of V . Formulate and prove the corresp onding result for semi-

stable vector bundles of negative degree.

4. Show that there is a unique non-trivial extension I of O by O . Show

2 E E

more generally that there is a unique bundle I of rank r which is an iterated

extension of O where each extension is non-trivial. Show that H (E ; I )is

E r

rank one. The iterated extensions give an increasing ltration jcal F of I

by subbundles so that for each s r , we have F =F = O . Show that

s s1 E

this ltration is stable under any automorphism of I and is preserved under

any endomorphism of I .

5. Show that if V is a vector bundle over any base and if every nonzero

section of V vanishes nowhere then the union of the images of the sections of

V pro duce a trivial subbundle of V with torsion-free cokernel. In particular,

in this case the numb er of linearly indep endent sections is at most the rank

of V and if it is equal to the rank of V , then V is a trivial bundle.

6. Show that there is a rank twovector bundle over E H (E ; O ) whose

restriction to E f0g is O O and whose restriction to any other b er

E E

E fxg, x 6= 0 is isomorphic to I .

7. By a coarse mo duli space of equivalence classes of bundles of a certain

typ e we mean the following: we have a reduced analytic space X and a

bijection b etween the p oints of X and the equivalence classes of the bundles

in question. Furthermore, if V ! E Y is a holomorphic bundle such that

the restriction V of V to each slice E fyg is of the typ e under consideration,

then the map Y ! X de ned by sending y to the p ointof X corresp onding to

the equivalence class of V is a holomorphic mapping. Show that if a coarse y

Holomorphic bund les over el liptic manifolds 172

mo duli space exists, then it is unique up to unique isomorphism. Show that

there cannot b e a coarse mo duli space for isomorphism classes of semi-stable

bundles over E .

8. Show that H (E ; O (q p ) O(p ) is one-dimensional and that any non-

0 0

zero section of this bundle vanishes to order one at q . Show that H (E ; I

O (q p ) O(p )) has rank r and that every section of this bundle vanishes

0 0

to order one at q . Thus, the determinant map for this bundle has a zero of

order r at q .

9. Show that p oints (e ;::: ;e )inE are the zeros of a meromorphic function

1 r

f : E ! P with a p ole only at p if and only if e =0 in the group law

0 i

of E .

10. Let E be emb edded in P so that it is given by an equation in Weierstrass

form:

2 3

y =4x +g x+g :

2 3

Show that any meromorphic function on E with p ole only at in nity in this

ane mo del is a p olynomial expression in x and y . Show that a meromorphic

function with p ole only at in nity which is invariant under e 7! e is a

p olynomial expression in x.

11. Let g : T ! jnp j be the n-sheeted rami ed covering constructed in

Section 2.6. Let L b e the line bundle over T E obtained by pulling back

the Poincare line bundle over E E . Show that if (e ;::: ;e ) are distinct

1 n

p oints of E then the restriction of (g Id) L to fe ;::: ;e gE is isomorphic

1 n

to O (e p ). Show that if e = e but otherwise the e are distinct then

i E i 0 1 2 i

(g Id) (L restricted to fe ;::: ;e gE is isomorphic to O (e p )

1 n E 1 0

I O (e p )

2 i 0

i=3

12. Show that the Pontrjagen dual of a homomorphism (E ) ! T is a

homomorphism ! Hom ( (E );S ). Show that the choice of an origin p

1 0

allows us to identify E with Hom( (E );S ).

13. Show that the quotient of E by e e is P . Show that the quotient

1 n n

of (P ) under the action of the symmetric group on n letters is P .

14. Show that a weighted pro jective space with p ositiveweights is a compact

complex variety. Show that it is nitely covered by an ordinary pro jective

space. Show that in general these varieties are singular, but that their sin-

gularities are mo deled by quotients of nite linear group actions on a vector space.

Holomorphic bund les over el liptic manifolds 173

3 The Parab olic Construction

In this section we are going to give a completely di erent construction of

semi-stable bundles over an elliptic curve. We b egin rst with the case of

vector bundles of degree zero.

3.1 The parab olic construction for vector bundles

Lemma 3.1.1 Let E be an el liptic curve with p as origin. Then for each

integer d 1 there is, up to isomorphism, a unique vector W over E with

the fol lowing properties:

1. rank(W )=d.

2. det(W )=O(p ).

3. W is stable.

Furthermore, H (E ; W ) is of dimension one.

Pro of. The pro of is by induction on d. If d = 1, then it is clear that

there is exactly one bundle, up to isomorphism, which satis es the rst and

second item, namely O (p ). Since this bundle is stable, wehave established

the existence and uniqueness when d = 1. Since O (p ) is of negative

degree, it has no holomorphic sections and hence by RR, H (E ; O (p ))

is one-dimensional. By Serre duality, it follows that H (E : O (p )) is also

one-dimensional. This completes the pro of of the result for d =1.

Supp ose inductively for d 2, there is a unique W as required. By

the inductive hyp othesis and Serre duality we have H (E ; W ) is one-

dimensional. Thus, there is a unique non-trivial extension

0 !O !W !W !0:

d d1

Clearly, using the inductivehyp othesis we see that W is of rank d and its

determinant line bundle is isomorphic to O (p ). In particular, the degree

of W is one. Supp ose that W has a destabilizing subbundle U . Then

d d

deg (U ) > 0. The intersection of U with O is a subsheaf of O and hence has

non-p ositive degree. Thus, the image p(U )ofU in W has p ositive degree,

and hence degree at least one. In particular, it is non-zero. Since the rank

of U is at most the rank of W , it follows that (p(U )) (W ). Since

d1 d1

Holomorphic bund les over el liptic manifolds 174

p(U ) is non-trivial, it follows that p(U )=W . Thus, the rank of U is either

d 1ord. If it is of rank d, the U = W and do es not destabilize. Thus, it

must b e of rank d 1. This means that p: U ! W is an isomorphism and

hence that U splits the exact sequence for W . This is a contradiction since

the sequence was a nontrivial extension. This contradiction proves that W

is stable.

A direct cohomology computation using the given exact sequence shows

that H (E ; W ) has rank one, completing the pro of.

Note (1): We have seen that W is given as successive extensions with all

quotients except the last one b eing O . The last one is O (p ).

(2): There is a one-parameter family of stable bundles of rank d and

degree 1. The determinant of such a bundle is of the form O (q ) for some p oint

q 2 E and the isomorphism class of the bundle is completely determined by

q .

(3) The bundle W is then a stable bundle of degree minus one and

determinant O (p ).

Corollary 3.1.2 The automorphism group of W is C .

Pro of. Supp ose that ': W ! W is an endomorphism of W . Then we

d d d

see that if ' is not an isomorphism, then deg (Ker(')) = deg (Coker (')).

But by stability, deg (Ker(')) 0 or ' = 0. Similarly, stability implies

that deg (Coker (')) 1 or ' is trivial. We conclude that either ' is an

isomorphism or ' =0. If ' is an endomorphism and is an eigenvalue of

', then applying the previous to ' Id we conclude that ' = Id.

This shows that all endomorphisms of W are multiplication by scalars. The

result follows.

Nowwe are ready to construct semi-stable vector bundles of rank n and

degree zero.

Prop osition 3.1.3 Let E be an el liptic curve and p 2 E an origin for the

group law on E . Fix integers d; n;1 d n 1. Let W and W be the

d nd

bund les of the last lemma. Then any vector bund le V over E which ts in a

non-trivial extension

0 ! W ! V ! W ! 0

is semi-stable.

Holomorphic bund les over el liptic manifolds 175

Pro of. Clearly,any bundle V as ab ove has rank n and degree zero. Supp ose

that U V is a destabilizing subbundle. Then deg (U ) 1. Since W is

stable, the intersection U \ W has negative degree (or is trivial). In either

case, it is not all of U . This means that the image of U in W is nontrivial

and has degree at least one. Since W is stable, this means the image of U

in W is all of W . Hence, the degree of U is one more than the degree

nd nd

of U \ W . Since U \ W has negative degree or is trivial, the only way

d d

that U can have p ositive degree is for U \ W = 0, whichwould mean that

U splits the sequence. This proves that all nontrivial extensions of the form

given are semi-stable bundles.

Lemma 3.1.4 Suppose that V is a non-trivial extension of W by W .

Then for any line bund le over E of degree zero, we have Hom(V; ) has

rank either zero or one.

Pro of. The bundle Hom(W ;) is of degree 1 and is semi-stable. Thus,

it has no sections. Similarly, Hom(W ;) has a one-dimensional space of

sections. From the long exact sequence

0 ! Hom (W ;) ! Hom (W;) ! Hom(W ;) !

we see that Hom(V; ) has rank at most one.

Corollary 3.1.5 If V is a non-trivial extension of W by W , then V is

isomorphic to a direct sum of bund les of the form O (q p ) I for distinct

i 0 r

points q 2 E .

Pro of. If V has two irreducible factors of the form O (q p ) I and O (q

0 r

p ) I then Hom (V; O(q p ) would be rank at least two, contradicting

0 r 0

the previous result.

3.2 Automorphism group of a vector bundle over an elliptic

curve

The following is an easy direct exercise:

Holomorphic bund les over el liptic manifolds 176

Lemma 3.2.1 Let E be an el liptic curve and let L and M be non-isomorphic

line bund les of degree zero over E . Then Hom(L; M )=0. Also, Hom(L; L)=

Let V b e a semi-stable bundle of degree zero over E . The supp ort of V is

the subset of p oints e 2 E at which some non-zero section of V vanishes. As

wehave seen in Atiyah's theorem, every semi-stable vector bundle of degree

zero over E decomp oses as a direct sum of bundles with supp ort a single

p oint.

0 0

Corollary 3.2.2 Let q; q be distinct points of E and let V and V be vec-

tor bund les of degree zero over E supported at q and q respectively. Then

Hom(V ;V )=0.

Corollary 3.2.3 Let V be a semi-stable bund le of degree zero over and el-

liptic curve E and let V = V be its decomposition into bund les with

q 2E q

support at single points. Then Hom(V; V )= Hom(V ;v ).

q2E q q

Now let us analyze the individual terms in this decomp osition.

Lemma 3.2.4 With I as in Lecture 1, Hom (I ;I ) is an abelian algebra

r r r

r +1

C[t]=(t ) of dimension r .

Pro of. Recall that I comes equipp ed with a ltration F F

r 0 1

F = I with asso ciated quotients O . The quotient of I =F is identi ed

r r r s

with I . The rst thing to prove is that this ltration is preserved under

r s

any endomorphism. It suces by an straightforward inductive argumentto

show that F is preserved by an endomorphisms. But F is the image of the

1 1

unique (up to scalar multiples) non-zero section of I .

Let t: I ! I b e the map of the form I ! I =F = I = F I .

r r r r 1 r 1 r 1 r

k r +1

Clearly, the image of t is contained in F so that t =0. We claim that

r k

2 r

every endomorphism of I is a linear combination of f1;t;t ;::: ;t g. Supp ose

that f : I ! F I . Then there is an induced mapping I =I !

r r s r r r 1

I =I . Since b oth of these quotients are isomorphic to O , this map

r s r s1

and some multiple of the map between these quotients induced by t are

equal. Subtracting this multiple of t from f we pro duce a map I !F .

r rs1

Continuing inductively proves the result.

Holomorphic bund les over el liptic manifolds 177

Notice that an endomorphism is an automorphism if and only if its image

is not contained in I if and only if it is not an element of the ideal

r 1

generated by t.

From this description the following is easy to establish.

Corollary 3.2.5 Let V be a semi-stable bund le of degree zero and rank n

over an el liptic curve E . Then the dimension of the automorphism group is

at least n. It is exactly n if and only if for each q 2 E , the subbund le V V

supported by q is of the form O (q p ) I for some r (q ) 0.

r (q )

Pro of. We decomp ose V into a direct sum of bundles V which them-

selves are indecomp osable under direct sum. Thus, each V is of the form

O (q p ) I for some q 2 E and some r 1. Thus, by the previous

0 r

result the automorphism group of V has dimension equal to the rank of V .

i i

Clearly, then the automorphism group of V preserving this decomp osition

has dimension equal to the rank of V . This will b e the entire automorphism

group if and only if Hom(V ;V ) =0 for all i 6= j . This will b e the case if

i j

and only if the V have disjoint supp ort.

Note that if, in the ab ove notation, two or more of the V have the same

supp ort then the automorphism group has dimension at least two more than

the rank of V .

De nition 3.2.6 A semi-stable vector bundle over an elliptic curve whose

automorphism group has dimension equal to the rank of the bundle is called

a regular bundle.

We are now in a p osition to prove the main result along these lines.

Theorem 3.2.7 Fix n; d with 1 d

be written as a non-trivial extension

! V ! W ! 0 0 ! W

if and only if

1. The determinant of V is trivial.

2. V is semi-stable.

3. V is regular.

Holomorphic bund les over el liptic manifolds 178

Pro of. The rst condition is obviously necessary and in Prop osition 3.1.3

and Corollary 3.1.5 we established that the second and third are also neces-

sary.

Supp ose that V satis es all the conditions. Condition (iii) says that

V = V where each V is of the form O (q p ) I . We claim

q 2E q q 0

r (q )

that there is a map W !O(qp ) I whose image is not contained in

r(q)

O (q p ) I . The reason for this is that W O(q p ) I is stable and

0 0 r

r (q )1

has degree r . Thus, its space of sections has dimension exactly r . Applying

this with r = r (q ) and r = r (q ) 1 we see that there is a homomorphism

W !O(qp ) I which do es not factor through O (q p ) I .

0 0

r(q) r (q )1

Claim 3.2.8 If F is a subsheaf of V of degree zero and if the projection of

F onto each V is not contained in a proper subsheaf, then F = V .

Pro of. Let W b e the smallest subbundle of V containing F . It has degree

at least zero and is equal to F if and only if its degree is zero. By stability

it has degree zero and hence is equal to F . This shows that F is in fact a

subbundle. Since there are no non-trivial maps between subbundles of V

and V for q 6= q , it follows that any subbundle of V is in fact a direct

sum of its intersections with the various V . If the image of the subbundle

under pro jection to V is all of V then its intersection with V is all of V .

q q q q

The claim now follows.

Let W ! V be a map whose pro jection onto each V is not contained

in any prop er subbundle of V . Let us consider the image of this map. It is

a subsheaf of V which is prop er since d < n. This means it has degree at

most zero. By the ab ove it cannot b e of degree zero. Thus, it is of degree at

most 1. Hence the kernel of the map is either trivial or has degree at least

0. This latter p ossibility contradicts the stability of W . This shows that

the map is an isomorphism onto its image; that is to sayitisanemb edding

of W V .

Next, let us consider the cokernel X . We have already seen that the

cokernel is a bundle. Clearly, its determinantisO(p ) and its rank is n d.

To show that it is W we need only see that it is stable. Supp ose that

U X is destabilizing. Then the degree of U is p ositive. Let U V be

the preimage of U . It has degree one less than U and hence has degree at

least zero. But since it contains the image of W , the previous claim implies

that it is all of V , which implies that U is all of W , contradicting the

assumption that U was destabilizing.

Holomorphic bund les over el liptic manifolds 179

Nowwe shall show that, given V , there is only one such extension with

V in the middle, up to automorphisms of V .

Prop osition 3.2.9 Let V be a semi-stable rank n-vector bund le with trivial

determinant. Suppose that the group of automorphisms of V has dimension

n. Then V can be written as an extension

0 ! W ! V ! W ! 0:

This extension is unique up to the action of the automorphism group of V .

Pro of. Supp ose wehave an extension as ab ove for V . First notice that if

the image of W were contained in a prop er subsheaf of degree zero of V ,

then this subsheaf would pro ject into W destabilizing it. Thus, the only

subsheaf of degree zero of V that contains the image of W is all of V . That

is to say the image of W in each V is not contained in a lower ltration

level, i.e. a prop er subsheaf of degree zero. Now we need to show that all

maps W ! V which are not contained in a prop er subsheaf of degree zero

are equivalent under the action of the automorphisms of V . This is easily

established from the structure of the automorphism sheaf of V given ab ove.

Corollary 3.2.10 For any 1 d

1 1

H (E ; Hom(W ;W )) = H (E ; W W ) is identi ed with the space

d nd d

of isomorphism classes of regular semi-stable vector bund les of rank n and

trivial determinant.

Notice that it is not apparent, a priori, that for di erent d

ab ove pro jective spaces can b e identi ed in some natural manner.

The asso ciation to each regular semi-stable vector bundle of rank n and

trivial determinant of its S -equivalence class then induces a holomorphic

)) to the coarse mo duli space P(O (np )) of W map from P(H (E ; W

d nd

S -equivalence classes of such bundles. It follows immediately from Atiyah's

theorem that each S -equivalence class contains a unique regular represen-

tative up to isomorphism, so that this map is bijective. Since it is a map

between pro jective spaces it is in fact a holomorphic isomorphism. Thus, for

)as W any d; 1 d

d nd

yet another description of the coarse mo duli space of S -equivalence classes

Holomorphic bund les over el liptic manifolds 180

of semi-stable rank n vector bundles of trivial determinant. Notice that the

actual bundles pro duced by this construction are the same as those pro duced

by the sp ectral covering construction since b oth are families of regular semi-

stable bundles. On the other hand, using the Narasimhan-Seshadri result

gives a di erent set of bundles { namely the direct sum of line bundles. These

families agree generically, but di er along the co dimension-one subvarietyof

the parameter space where two or more p oints come together. Wehave seen

two constructions of holomorphic families of semi-stable vector bundles {

the sp ectral covering construction and the parab olic construction and b oth

create regular semi-stable bundles. It is not clear that the at connection

p oint of view can b e carried out holomorphically in families (indeed it can-

not). A hintto this fact is that it is pro ducing di erent bundles and these

do not in general t together to make holomorphic families.

The reason for the name `parab olic' will b ecome clear after we extend

to the general semi-simple group. Before we can give this generalization we

need to discuss parab olic subgroups of a semi-simple group.

3.3 Parab olics in G

Let G b e a complex semi-simple group. A Borel subgroup of G is a con-

C C

nected complex subgroup whose Lie algebra contains a Cartan subalgebra

(the Lie algebra of a maximal complex torus) together with the ro ot spaces

of all p ositive ro ots with resp ect to some basis of simple ro ots. All Borel

subgroups in G are conjugate. By de nition a parab olic subgroup is a

connected complex prop er subgroup of G that contains a Borel subgroup.

Up to conjugation parab olic subgroups of G are classi ed by prop er (and

p ossibly empty) sub diagrams of the Dynkin diagram of G. Fix a maximal

torus of G and a set of simple ro ots f ;::: ; g. A sub diagram is given

C 1 n

simply bya subset f ;::: ; g of the set of simple ro ots. The Lie algebra

1 r

of the parab olic is the Cartan subalgebra tangent to the maximal torus, to-

gether with all the p ositive ro ot spaces and all the ro ot spaces asso ciated

with negative linear combinations of the f ;::: ; g. Thus, a Borel sub-

1 r

group corresp onds to the empty sub diagram. The full diagram gives G

and hence is not a parab olic subgroup. Up to conjugation a parab olic P is

contained in a parab olic P if and only if the diagram corresp onding to P

is a sub diagram of that corresp onding to P . It follows that the maximal

parab olic subgroups of G up to conjugation are in one-to-one corresp on-

dence with the sub diagrams of the Dynkin diagram of G obtained by deleting

Holomorphic bund les over el liptic manifolds 181

a single vertex. This sets up a bijective corresp ondence b etween conjugacy

classes of maximal parab olic subgroups of G and vertices of the Dynkin

diagram, or equivalently with the set of simple ro ots for G .

A parab olic subgroup P has a maximal unip otent subgroup U whose Lie

algebra is the sum of the ro ots spaces of p ositive ro ots whose negatives are

not ro ots of P . This subgroup is normal and its quotient is a reductive group

called the Levi factor L of P . There is always a splitting so that P can be

written as a semi-direct pro duct U L. The derived subgroup of L is a semi-

simple group whose Dynkin diagram is the sub diagram that determined P

in the rst place. A maximal torus of P is the original maximal torus of

G. If P is a maximal parab olic then the character group of P is isomorphic

to the integers, and the comp onent of the identity of the center of P is

C , and any nontrivial character of P is non-trivial on the center. On the

level of the Lie algebra the generating character of P is given by the weight

dual to the coro ot asso ciated with the simple ro ot that is omitted from

the Dynkin diagram in order to create the sub diagram that determines P .

The value of this weightonanyroot is simply the co ecientof in the

linear combination of the simple ro ots which is . The ro ot spaces of the

Lie algebra of P are those ones which the character is non-negative, and the

Lie algebra of the unip otent radical is the sum of the ro ot spaces of ro ots on

which this character is p ositive.

Example: The maximal parab olics of SL (C) corresp ond to no des of

its diagram. Counting from one end we index these by integers 1 d

The parab olic subgroup corresp onding to the integer d is the subgroup of

blo ck diagonal matrices with the lower left d (n d) blo ck b eing zero.

The Levi factor is the blo ck diagonal matrices or equivalently pairs (A; B ) 2

GL (C) GL (C) with det(A) = det(B ). Avector bundle with structure

d nd

reduced to this parab olic is simply a bundle with a rank d subbundle, or

equivalently a bundle written as an extension of a rank d bundle by a rank

(n d) bundle. In this case, the unip otent subgroup is a vector group

nd d

Hom(C ; C ).

3.4 The distinguished maximal parab olic

For all simple groups except those of A typ e we shall work with a distin-

guished maximal parab olic. It is describ ed as follows: If the group is simply

laced, then the no de of the Dynkin diagram that is omitted is the trivalent

one. If the group is non-simply laced, then either vertex which is omitted

Holomorphic bund les over el liptic manifolds 182

is the long one connected to the multiple b ond. It is easy to see that in all

cases the Levi factor of this parab olic is written as the subgroup of a pro duct

of GL of matrices with a common determinant.

Examples: (i) For a group of typ e C , there is a unique long ro ot.

The Levi factor of the corresp onding subgroup is GL (C) and the unip o-

tent radical is the self-adjoint maps C to its dual. In terms of complex

symplectic 2n-dimensional bundles, a reduction of structure group of V

to this parab olic means the choice of a self-annihilating n-dimensional sub-

bundle W . This bundle has structure group the Levi factor GL (C) of

the parab olic. The quotient of the bundle by this subbundle is simply the

dual bundle W . The extension class that determines the bundle and its

symplectic form is an elementinH (E; SymHom (W;W )), where SymHom

means the self-adjoint homomorphisms.

(ii) For a group of typ e B the distinguished maximal parab olic has

Levi factor the subgroup of GL (C) GL (C) consisting of matrices of

n1 2

the same determinant. Let us consider the orthogonal group instead of

the spin group. Then a reduction in the structure group of an orthogonal

2n+1 2n+1

bundle V to this parab olic is a self-annihilating subspace W V

of dimension n. This pro duces a three term ltration W W W

1 2 3

where W = W . Under the orthogonal pairing W and W =W are dually

2 1 3 2

paired and W =W , whichis three-dimensional, has a self-dual pairing and

2 1

is identi ed with the adjoint of the bundle over the GL (C)-factor. The

subbundle W is the bundle over the GL (C)-factor of the Levi. There are

1 n

two levels of extension data one giving the extension comparing W W

1 2

which is an elementofH (E;(W =W ) W )) and the other an extension

2 1 1

class in H (E ;SkewHom (W =W ;W )), where SkewHom refers to the anti-

3 2 1

self adjoint mappings under the given pairing.

(iii) There is a similar description for D . Here the Levi factor of the dis-

tinguished parab olic is the subgroup of matrices in GL (C) GL (C)

n2 2

GL (C) consisting of matrices with a common determinant. This time a

reduction of the structure group to P corresp onds to a self-annihilating sub-

space W of dimension n 2, it is the bundle over the GL (C)-factor of

1 n2

the Levi. The quotient W =W is four-dimensional and self-dually paired. It

2 1

is identi ed with the tensor pro duct of the bundle over one of the GL (C)-

factor with the inverse of the bundle over the other. Once again the coho-

mology describing the extension data is two step { one giving the extension

which is W W and the other a self-dual extension class for W =W by

1 2 3 2

W . 1

Holomorphic bund les over el liptic manifolds 183

(iv) In the case of E , r = 6; 7; 8 the Levi factor is the subgroup of

matrices in GL (C) GL (C) GL (C) with the same determinants.

2 3 r 3

It is more dicult to describ e what a reduction of the structure group to

this parab olic means since wehave no standard linear representation to use.

For E there is the 27-dimensional representation, whichwould then havea

three-step ltration with various prop erties.

In the case of SL (C) we b egan with a particular, minimally unstable

vector bundle W W whose structure group has b een reduced to the

d nd

Levi factor of the parab olic subgroup. We then considered extensions

0 ! W ! V ! W ! 0:

These extensions have structure group the entire parab olic. They also have

the prop erty that mo dulo the unip otent subgroup they b ecome the unstable

bundle W W with structure group reduced to the Levi factor L of this

maximal parab olic.

3.5 The unip otent subgroup

Let us consider the unip otent subgroup of a maximal parab olic group. Fix

a maximal torus T of G and a set of simple ro ots f ;::: ; g. Supp ose

C 1 n

that this parab olic is the one determined by deleting the simple ro ot . We

b egin with its Lie algebra. Consider the direct sum of all the ro ot spaces

g asso ciated with p ositive ro ots whose negatives are not ro ots of P . These

are exactly the p ositive ro ots which, when expressed as a linear combination

of the simple ro ots have a p ositive co ecient times . Clearly, these ro ots

form a subset which is closed under addition, in the sense that if the sum of

two ro ots of this typ e is a ro ot, then that ro ot is also of this typ e. This means

that the sum U of the ro ot spaces for these ro ots makes a Lie subalgebra of

g . Furthermore, there is an integer k > 0 such that any sum of at least

k ro ots of this typ e is not a ro ot. (The integer k can be taken to be the

largest co ecientof in anyrootof g .) This means that the Lie algebra

i C

U is in fact nilp otent of index of nilp otency at most k . It follows that the

restriction of the exp onential map to U is a holomorphic isomorphism from

U to a unip otent subgroup U G . The dimension of this group is equal

to the number of ro ots with p ositive co ecient on . Furthermore, U is

ltered by a chain of normal subgroups f1g U U U where

k k 1

U is the unip otent subgroup whose Lie algebra is the ro ot spaces of ro ots

whose -co ecient is at least i. Clearly, U is contained in the center of the

i k

Holomorphic bund les over el liptic manifolds 184

group and U =U is contained in the center of U=U . The entire structure

i i1 i

of the unip otent group can be directly read o from the set of ro ots with

p ositive co ecienton together with the information ab out which sums of

ro ots are ro ots.

Examples: (i) For SL (C) and any maximal parab olic the ltration is

trivial U = U ; U = f1g. The reason is of course that all p ositive ro ots are

1 2

linear combinations of the simple ro ots with co ecients 0; 1 only. Thus, U

d nd

is a vector group. It is Hom (C ; C ).

(ii) For groups of typ e C and maximal parab olics obtained by deleting

the vertex corresp onding to the unique long ro ot, again U = U ; U = f1g

1 2

(all ro ots have co ecient 1 or 0 on this simple ro ot). The unip otent radical

2 n

U is the vector group ^ C . For all other maximal parab olics of groups of

typ e C , the unip otent radical has a two-step ltration and is not ab elian.

(iii) For groups of typ e B and for maximal parab olics obtained by delet-

ing the simple ro ot which is long and which corresp onds to a vertex of the

double b ond in the Dynkin diagram, the ltration is f1g U U = U .

2 1

The dimension of U is (n 1)(n 2)=2 and the dimension of U =U is

2 1 2

2(n 1). The Lie bracket mapping U =U U =U ! U is onto, so that the

1 2 1 2 2

unip otent group is not a vector group, i.e., it is unip otent but not ab elian.

(iv) For groups of typ e D and maximal parab olics obtained by deleting

the simple ro ot corresp onding to the trivalentvertex, once again the ltration

is of length 2: we have f1g U U = U . The dimension of U is

2 1 2

(n 2)(n 3)=2 and the dimension of U =U is 4(n 2). Once again the

1 2

bracket mapping U =U U =U ! U is onto, and hence the group is non-

1 2 1 2 2

ab elian.

(iv) Once we leave the classical groups, the ltrations b ecome more com-

plicated. For E the ltration of the unip otent subgroup of the distinguished

maximal parab olic is f1gU U U where the dimension of U is two,

3 2 1 3

the dimension of U =U is 9 and the dimension of U =U is 18. For E and

2 3 1 2 7

the distinguished maximal parab olic, the ltration of the unip otent radical

b egins at U which is three-dimensional and descends to U with U =U be-

4 1 1 2

ing 24 dimensional. For E and the distinguished maximal parab olic, the

ltration b egins at U whichis ve-dimensional and descends all the wayto

U with U =U b eing of dimension 30.

1 1 2

Holomorphic bund les over el liptic manifolds 185

3.6 Unip otent cohomology

Fix a simple group G and a maximal parab olic subgroup P , and x a

splitting P = U L. Also, x a holomorphic principal bundle ! E with

structure the Levi factor L of P . We wish to study holomorphic bundles

! E with structure group P with a given isomorphism =U ! . Let us

cho ose a covering of the elliptic curve by small analytic op en subsets fU g.

The bundle is describ ed by a co cycle n : U \ U ! L. A bundle and

L ij i j

isomorphism =U ! is given by maps u : U \ U ! U satisfying the

L ij i j

co cycle condition:

u n u n = u n :

ij ij

jk jk ik ik

Since the fn g are already a co cycle, we can rewrite this condition as

u u = u ;

n 1

where u = nun for u 2 U and n 2 L. This is the twisted co cycle con-

dition asso ciated with the bundle and the action of L (by conjugation)

on the unip otent subgroup U . A zero co chain is simply a collection of holo-

morphic maps v : U ! U . Varying a twisted co cycle fu g by replacing the

i i ij

cob oundary of this zero co chain means replacing it by v u (v ) .

i ij

The set of co cycles mo dulo the equivalence relation of cob oundary makes

a set, denoted H (E ; U ( )). In fact, it is a p ointed set since we have

the trivial co cycle: u = 1 for all i; j . In the case when U is ab elian,

asso ciated to and the action of L on U (which is linear), there is a vector

bundle U ( ). The twisted co cycles mo dulo cob oundaries are exactly the

usual Cech cohomology of this vector bundle, H (E ; U ( )), and hence this

cohomology space is in fact a vector group. The general situation is not quite

this nice. But since U is ltered by normal subgroups with the asso ciated

gradeds b eing vector groups, we can lter the twisted cohomology and the

asso ciated gradeds are naturally the usual cohomology of the vector bundles

1 1

H (E ;(U =U ( )). In this situation, the entire cohomology H (E ; U ( ))

i i1 L L

can b e given the structure on an ane space which has an origin, and which

is ltered with asso ciated gradeds b eing vector bundles.

The center of P is C (more precisely, the comp onent of the identityof

the center of P ) and hence acts on U and on H (E ; U ( )). This action

preserves the origin, and the ltration and on each asso ciated graded is a

linear action of homogeneous weight. That weight is given by the index of

that ltration level (weight i on U =U ). It is a general theorem that since

i i1

all these weights of the C -action are p ositive, there is in fact an isomorphism

Holomorphic bund les over el liptic manifolds 186

of this ane space with a vector space in such a way that the C -action

b ecomes linearized. In particular, the quotientof H (E;U( )) f0g =C

is isomorphic as a pro jective variety to a weighted pro jective space. The

dimension of the subpro jective space of weight i is equal to one less that the

dimension of H (E ; U =U )).

i i1

This is a fairly formal construction and it is not clear that it has anything

to do with stable G -bundles. Here is a theorem that tells us that using

the distinguished maximal parab olic identi ed ab ove and a sp ecial unstable

bundle with structure group the Levi subgroup of this parab olic in fact leads

to semi-stable G -bundles. It is a generalization of what wehave established

directly for vector bundles.

Theorem 3.6.1 Let G be a simply connected simple group and let P G

C C

be the distinguished parabolic subgroup as above. Then the Levi factor of

GL consisting of al l fA 2 GL g P is isomorphic to the L

ni i n i

such that det(A ) = det(A ) for al l i; j . Let W be the unique stable

i j n

bund le of rank n and determinant O (p ). Then = W is natu-

i 0 L i

ral ly a holomorphic principal L-bund le over E . Every principal P -bund le

which is obtained from a non-trivial cohomology class in H (E ; U ( )) be-

comes semi-stable when extended to a G -bund le. Cohomology classes in

the same C -orbit determine isomorphic G -bund les. This sets up an iso-

morphism between H (E ; U ( )) f0g =C and the coarse moduli spaceof

S-equivalence classes of semi-stable G -bund les over E . Every G -bund le

C C

constructed this way is regular in the sense that its G -automorphism group

has dimension equal to the rank of G, and any regular semi-stable G -bund le

arises from this construction. Any non-regular semi-stable G -bund le has

automorphism group of dimension at least two more than the rank of G.

This result gives a di erent pro of of Lo oijenga's theorem. It identi es

the coarse mo duli space as weighted pro jective space asso ciated with a non-

ab elian cohomology space. It is easy to check given the information ab out

the ro ots and their co ecients over the distinguished simple ro ot that the

weights of this weighted pro jective space are as given in Lo oijenga's theorem.

3.7 Exercises:

1. Supp ose that wehave a non-trivial extension

0 !O !X !W !0;

Holomorphic bund les over el liptic manifolds 187

where W is as in the rst lemma of this lecture. Show that H (E ; X ) is

one-dimensional and hence that H (E ; X ) is also of dimension one.

2. Let V be a holomorphic vector bundle. Show that Aut(V ) is a complex

Lie group and that its Lie algebra is identi ed with End(V )=H (V V ).

3. Show that if V and V are semi-stable bundles over E , then so is V V .

1 2 1 2

Compute the degree of V V in terms of the degrees and ranks of V and

1 2 1

V .

4. Prove Lemma 3.2.1 and Corollary 3.2.2.

5. Show that if V is a semi-stable vector bundle of rank n over E whichis

not regular, then the dimension of the automorphism group of V is at least

r +2.

6. Let V b e semi-stable vector bundles over E of degree zero and disjoint

supp ort. Show that any subbundle of degree zero in V is in fact a direct

sum of subbundles of the V .

7. Show that if any two homomorphisms W ! I which have image not

contained in I di er by an automorphism of I .

r 1 r

8. Show that a Borel subgroup of G is determined bya choice of a maximal

torus for G and a choice of simple ro ots for that torus. Show all Borel

subgroups of G are conjugate.

9. Up to conjugation, describ e explicitly all parab olic subgroups of SL (C).

10. Let G be a semi-simple group. Show that the character group of a

maximal parab olic subgroup of G is isomorphic to Z. Show that the center

of a maximal parab olic subgroup of G is one-dimensional.

11. For E ;E ;E ;G ;F work out the dimensions of the various ltration

6 7 8 2 4

levels in the unip otent subgroups asso ciated with the distinguished maximal

parab olic subgroups.

12. Check that in the formula given for the action by a cob oundary on a

twisted co cylce that the resulting one-co chain is still a twisted co cycle.

13. For groups of typ e B and D and the distinguished parab olic and the

n n

given bundle over the Levi factor, compute the cohomology vector spaces

1 1

H (E ; U ( )) and H (E ; U =U ( )).

2 L 1 2 L

Holomorphic bund les over el liptic manifolds 188

4 Bundles over Families of Elliptic Curves

In this lecture we will generalize the constructions for the case of vector

bundles over an elliptic curvetovector bundles over families of elliptic curves.

4.1 Families of elliptic curves

The rst thing that we need to do is to decide what we shall mean by a family

of elliptic curves. The b est choice for our context is a family of Weierstrass

cubic curves. Recall that a single Weierstrass cubic is an equation of the

form

2 3

y =4x +g x+g ;

2 3

or written in homogeneous co ordinates is given by:

2 2 2 3

zy =4x +g xz + g z :

2 3

This equation de nes a cubic curve in the pro jective plane with homogeneous

co ordinates (x; y ; z ). The p oint at in nity, i.e., the p oint with homogeneous

co ordinates (0; 1; 0) is always a smo oth p oint of the curve. In the case when

the curve is itself smo oth, this p oint is taken to b e the identity elementofthe

group law on the curve. More generally, there are only twotyp es of singular

curves which can o ccur as Weierstrass cubics { a rational curve with a single

3 2

no de { which o ccurs when (g ;g ) = 0 where (g ;g )=g +27g is the

2 3 2 3

2 3

discriminant, and the cubic cusp when g = g =0. In each of these cases

2 3

the subvariety of smo oth p oints of the curve forms a group (C in the no dal

case and C in the cuspidal case), and again we use the p ointat in nityas

the origin of the group law on the subvariety of smo oth p oints.

Now supp ose that we wish to study a family of such cubic curves para-

metrized by a base B whichwe take to b e a smo oth variety. Then we xa

line bundle L over B . We interpret the variables x; y ; z as follows: let E be

2 3 2

the three-plane bundle O L L over B ; z : E ! O , x: E ! L , and

B B

y : E ! L are the natural pro jections. Furthermore, g is a global section of

4 6

L and g is a global section of L . With these de nitions

2 3 2 3

zy (4x + g xz + g z )

2 3

is a section of Sym (E ) L . Its vanishing lo cus pro jectivizes to give a sub-

variety Z P(E ) over B , which b er-by- b er is the elliptic curve (p ossible

singular) given by trivializing the bundle L over the p oint b 2 B in question

Holomorphic bund les over el liptic manifolds 189

and viewing g (b) and g (b) as complex numb ers so that the ab ove cubic

2 3

equation with values in L b ecomes an ordinary cubic equation dep ending

on b. While the actual equation asso ciated to b will of course dep end on

the trivialization of Lj , the homogeneous cubic curve it de nes will be

fbg

indep endent of this choice.

Thus, as long as the sections g and g are generic enough so as not to

2 3

always lie in the discriminant lo cus, Z ! B is an elliptic bration (which

by de nition is a at family of curves over B whose generic member is an

elliptic curve). This family of elliptic curves comes equipp ed with a choice of

base p oint, i.e., there is a given section of Z ! B . It is the section given by

3 2 3

fz = x =0g or equivalently,by the section [L ] 2 P(O L L ). (This

is the globalization of the p oint(0;1;0) in a single Weierstrass curve.) This

do es indeed de ne a section of Z ! B . The image of this section is always a

smo oth p oint of the b er. If we use lo cal b erwise co ordinates (u = x=y ; v =

3 2 3

z=y) near this section, then the lo cal equation is v = 4u + g uv + g v ,

2 3

and its gradient at the p oint(0;0) p oints in the direction of the v -axis. This

means that along the surface Z is tangent to the u-axis. Since what we

are calling the u-axis actually has co ordinate x=y , these lines t together to

2 3 1 1

form the line bundle L (L ) = L , which then is the normal bundle of

in Z . This bundle is also of course the relative tangent bundle of the b ers

along the section . Since the tangent bundle of each b er is trivialized, it

follows that the pushforward, T , is isomorphic to L . Also imp ortant

b ers

for us will b e the relative dualizing sheaf. It is T . (Of course, as I have

b ers

presented it, we are working only at smo oth b ers. But b ecause the singular

curves have suciently mild singularities the relative dualizing sheaf is still

a line bundle, and in fact is the bundle L.)

Wehave proved:

Lemma 4.1.1 Let be the normal bund le of in Z . Let : Z ! B be the

natural projection. Then = O ( )j and (O ( )j ) = ( ) = L :

Z Z

The bund le L is the relative dualizing line bund le.

N.B. The subvarietyof B consisting of b 2 B for which the Weierstrass

curve parametrized by b is singular, resp., a cuspidal curve, is a subvariety.

For generic g and g the co dimension of these subvarieties are one and two,

2 3

resp ectively.

Holomorphic bund les over el liptic manifolds 190

4.2 Globalization of the sp ectral covering construction

Having said how we shall replace our single elliptic curve by a family of

elliptic curves with a section, we now turn to globalizing the vector bundle

constructions. Our rst attempt at globalizing the previous constructions

would b e to try to nd the analogue for (E E) =S . The obvious can-

| {z }

ntimes

didate is (Z Z)=S . This works ne as long as Z is smo oth over B

B B n

{z } |

ntimes

but do es not give a go o d result at the singular b ers. There is in fact a way

to globalize this construction, at least across the no des. It involves consid-

ering Z Z , where Z is the op en subvariety of p oints regular

reg B B reg reg

in their b ers, and then given an appropriate toroidal compacti cation at

the no dal b ers. I shall not discuss this construction here.

There is however another way to view n p oints on E which sum to

zero, up to p ermutation. Namely, as we have already seen, these p oints

are naturally the p oints of the pro jective space H (E ; O (np )). Thus, a

E 0

b etter way to globalize is to replace O (p ) by O ( ) and thus consider

0 Z

R (O (n )). This is a vector bundle of rank n on B . Its asso ciated

pro jective space bundle is then a lo cally trivial P bundle over B . The

b er of this pro jective bundle over a p oint b 2 B is canonically identi ed

with the pro jective bundle of the linear system jnp j on E .

As the next result shows, this pushed-forward bundle splits naturally as

a sum of line bundles.

Claim 4.2.1 The bund le R (O (n )) is natural ly split as a sum of line

2 3 n

bund les: O L L L .

Pro of. By de nition we are considering the bundle whose sections over an

op en subset U B are the analytic functions on Z j with p oles only along

\ (Z j ) and those b eing of order at most n. Wehave already at our disp osal

2 [n=2] [(n3)=2]

functions with this prop erty: 1;x;x ;::: ;x ;y;xy;:::;x y . Given

any function with this prop ertyover U ,we can subtract (uniquely) a multiple

a a

of one of these basic functions, x or x y , so that the order of the p ole is

reduced by at least one. The multiple will have a co ecient which is a section

2a 2a+3

of the line bundle L in the rst case and L in the second. In this

waywe identify the sections of our vector bundle over U with expressions of

the form

[n=2] [(n3)=2]

a + a x + + a x + b y + + b x y:

0 1 0

[n=2] [(n3)=2]

Holomorphic bund les over el liptic manifolds 191

a a a (2a+3)

The co ecient of x lies in L and the co ecient of x y lies in L .

2 3 n

This identi es the space of sections with the sum O L L L .

Notice that a section of this n-plane bundle is then a family of S -

equivalence classes of semi-stable bundles on the b ers of Z ==B , but that

it is not yet a vector bundle on Z . Nevertheless, the sp ectral covering con-

struction generalizes to pro duce a vector bundle. Let P be the bundle

of pro jective spaces asso ciated to the vector bundle R (O (n )). This

is the bundle whose b er over b 2 B is the pro jective space of the linear

system O (np ). Consider the natural map O (n ) !O (n ). It is

E 0 Z Z

surjective and we denote by E its kernel which is a vector bundle of rank

n 1. De ne T = P(E ). A p oint of O (n ) consists of an element

f 2jO (n (b))j together with a p oint z 2 E . The b er E consists of all

pairs for which f (z )=0. The bundle T is a P -bundle over Z whose b er

over any z 2 E is the pro jective space of the linear system O (n (b) z )on

E . The comp osition of the inclusion T ! P Z followed by the pro jec-

n B

tion onto P is a rami ed n-sheeted covering denoted g , which b er-by- b er

is the map we constructed b efore for a single elliptic curve.

Using this map we can construct a family of vector bundles over Z semi-

stable on each b er. Namely,we consider the pullback to T Z of the

diagonal Z Z . Then wehave a line bundle

0 B

L = O ( T ):

T Z B

The pushforward (g Id) (L) is a rank n vector bundle on Z whichis

regular semi-stable and of trivial determinant on each b er. Analogous to

our result for a single curvewehave the following universal prop erty for this

construction.

Theorem 4.2.2 Let U ! Z beavector bund le which is regular, semi-stable

with trivial determinant on each ber of Z ==B . Then associating to each

b 2 B the class of UjE determines a section s : B ! P . Let T be the

A n A

pul lback of T ! P via this section. Then the natural projection T ! B is

n A

an n-sheetedrami edcovering. Let L be the pul lback to T Z of the line

A A B

bund le L over T Z by s Id. Then there is a line bund le M over T

B A B A

such that U is isomorphic to (g Id) (L p M ), where p : T Z !T .

A 1 A B A

Notice that there are in essence two ingredients in this construction: the

rst is a section A of P ! B and the second is a line bundle over the induced n

Holomorphic bund les over el liptic manifolds 192

rami ed covering T of B . The section A is equivalent to the information of

the isomorphism class of the bundle on each berofZ ==B . The line bundle

over T gives us the allowable twists of the bundle on Z which do not change

the isomorphism class on each b er.

This completes the sp ectral covering construction. It has the advantage

that it pro duces all vector bundles over Z which are regular and semi-stable

with trivial determinant on each b er. Its main drawback is that it do es

not easily generalize to other simple groups. The construction that do es

generalize easily is the parab olic construction to whichwe turn now.

4.3 Globalization of the parab olic construction

It turns out that (except in the case of E -bundles and cuspidal b ers) that

the parab olic construction of vector bundles globalizes in a natural way.

The rst step in establishing this is to globalize the bundles W which are

an essential part of the construction, b oth for vector bundles and for more

general principal G-bundles.

4.3.1 Globalization of the bundles W

We de ne inductively the global versions of the bundles W . The globaliza-

tion of W = O (p ) is of course W = O ( ), so that the way we have

1 E 0 1 Z

chosen to globalize curves has already given us a natural globalization of

W . Clearly, the restriction of this line bundle to any b er E of Z ==B is

the bundle O (p ). (Notice that even if the b er is singular, p is a smo oth

E 0 0

p oint of it, so that O (p ) still makes sense as a line bundle.)

E 0

Claim 4.3.1 There is, up to non-zeroscalar multiples, a unique non-trivial

extension

0 ! L !X !W !0:

The restriction of X to any ber is isomorphic to W of that ber.

Pro of. Let us compute the global extension group Ext (O ( ); L). Since

b oth the terms are vector bundles, the extension group is identi ed with

the cohomology group H (Z ; O ( ) L). The lo cal-to-global sp ectral

sequence pro duces an exact sequence

1 1

0 ! H (B ; (O ( ) L) ! H (Z ; O ( ) L)

Z Z

0 1 2

! H (B ; R O ( ) L) ! H (B ; O ( ) L) ! :

Z Z

Holomorphic bund les over el liptic manifolds 193

Since the restriction of O ( ) to each b er is semi-stable of negative degree,

it follows that the rst term and the fourth term are b oth zero, and hence

wehave an isomorphism

1 0 1 0 1

H (O ( ) L) ! H (B ; R ( O ( ) L)) = H (B ; R (O ( )) L):

Z Z Z

1 1

But we have already seen that R (O ( )) = L , so that we are con-

0 1 0

sidering H (B ;(L L)=H (B;O )=C. Since any non-trivial section

of this bundle is nonzero at each p oint, any non-trivial extension class has

non-trivial restriction to each b er and hence any non-trivial extension of

the form 0 ! L ! W ! 0 restricts to each b er E to give a nontrivial

restriction of W by O and hence restricts to each b er to give a bundle

1 E

isomorphic to W on that b er.

Now let us continue this construction. The following is easily established

by induction.

Prop osition 4.3.2 For each integer n 1 there is a bund le W over Z

with the fol lowing properties:

1. W = O ( )

1 Z

2. For any n 2 we have a non-split exact sequence

0 ! L !W !W !0:

n n1

1 n

3. R W = L .

4. R W =0.

For these bund les the restriction of W to any ber of Z ==B is isomorphic

to the bund le W of that Weierstrass cubic curve.

Pro of. The pro of is by induction on d, with the case d = 1 b eing the last

claim. Supp ose inductively we have constructed W as required. Since

is semi-stable of negative degree on each b er, and since R W = W

L , it follows by exactly the same lo cal-to-global sp ectral sequence argu-

d1 0 1d d1 1

L ) = H (B ; L L ) = ment as in the claim that H (W

H (B ; O )=C. Thus, there is a unique (up to scalar multiples) nontrivial

extension of the form

0 ! L !X !W !0

Holomorphic bund les over el liptic manifolds 194

and the restriction of this extension to each berofZ ==B is nontrivial. We

let W b e the bundle which is such a nontrivial extension. The computations

of R W are straightforward from the extension sequence.

Notice that W is not the only bundle that restricts to each b er to give

W . Any bundle of the form W M for any line bundle M on B will

n n

also have that prop erty. Since the endomorphism group of W is C , one

shows easily that these are the only bundles with that prop erty.

N.B If we assume that B is simply connected then there are no torsion

line bundles on B . In this case requiring that the determinant of W be

d(d1)=2

L O)Z() will determine W up to isomorphism.

4.3.2 Globalizing the construction of vector bundles

Lemma 4.3.3 Ext (W ; W ) is identi ed with the space of global sections

of the sheaf

R (W ; W )

nd d

on B .

Pro of. First of all since W and W are vector bundles, we can iden-

tify Ext (W ; W ) with H (Z ; W W ). The lo cal-to-global sp ectral

d nd d

sequence pro duces an exact sequence

1 0 1

0 ! H (B ; R (W W )) ! H (Z ; W W )

nd d nd d

0 1 2 0

! H (B;R (W W )) ! H (B ; R W W ):

d nd d

Since W and W are b oth semi-stable and of negative degree on each

d nd

b er, the restriction of their tensor pro duct to each b er has no sections. It

follows that R (W W ) is trivial. Thus, wehave an isomorphism

nd d

1 0 1

H (Z ; W W ) ! H (B;R (W W ));

nd d d

as claimed in the statement.

)onB. W Next we need to compute the sheaf R (B ; W

nd d

) isavector bund le and is isomor- W Prop osition 4.3.4 R (B ; W

nd d

1 2 1n

phic to the direct sum of line bund les L L L L .

Holomorphic bund les over el liptic manifolds 195

First we consider a sp ecial case:

1 1 2

Lemma 4.3.5 R (B ; O ( ) W ) is isomorphic to LL L

L :

0 0

Pro of. Let R = R (O ( ) W ). Since the restriction of O ( )

Z n1 Z

W to each b er is a semi-stable bundle of degree n, R is a vector

bundle of rank n over B .

The relative dualizing sheaf for Z ==B is L and R L = O . Thus,

relative Serre duality is a map

1 0 1

S : R (O ( ) W ) ! R (O () W L )

Z Z

1 0 1

R L = R L :

Consider the comp osition of S with the map

0 1 0 0 1 1

(R ) L ! R det(R ) L

n1 n1 n1

ev Id Id

0 0 1 1

! R (det(O ( ) W )) det(R ) L

Z n1

0 (n1)(n2)=2 0 1 1

= R (O (n )) L det(R ) L ;

where the map A is induced by taking adjoints from the natural pairing

0 0 0

R R ! det (R );

n1 n1 n1

and ev is the map

n1 n1

^ ^

0 0

ev : R (O ( W ) ! R ( O () W )

Z n1 Z n1

obtained by evaluating sections. Clearly, b oth S and A are isomorphisms.

It is not so clear, but it is still true that ev is also an isomorphism. I shall

not prove this result { it is somewhat involved but fairly straightforward. A

reference is Prop osition 3.13 in Vector Bundles over Elliptic Fibrations.

Assuming this result, we see that the vector bundle we are interested in

0 1

computing di ers from R (O (n )bytwisting by the line bundle L

detR .

According to Claim 4.2.1 R (O (n )) splits as a sum of line bundles

2 3 1n 1

OL L L . Now to complete the evaluation of R (W

)we need only to compute the line bundle det R (O (n ) W ). W

Z n1

Holomorphic bund les over el liptic manifolds 196

0 (n2)(n1)=22

Claim 4.3.6 detR (O (n ) W ) is equal to L .

Z n1

Pro of. In computing the determinants we can assume that all sequences

2 n2

split. This allows us to replace W by O ( ) L L L . Since

n1 Z

O (2 ) sits in an exact sequence

0 !O () !O () !O ()j !0

Z Z Z

0 0 1 0

and since R O ( )) = L and R O ( )j = L , and R (O ( )

Z Z Z

a a1

L )=L , the result follows easily.

Putting all this together we see that

1 0

R (B ; O ( ) W )=R (B;O (n )) L:

Z n1 Z

This completes the pro of of Lemma 4.3.5

Now we are ready to complete the pro of of Prop osition 4.3.4. This is

done by induction on d. The case d = 1 is exactly the case covered by

Lemma 4.3.5. Supp ose inductively that we have established the result for

W W for some d 1. We consider the commutative diagram

d nd

0 0 0

? ? ?

y y y

0 ! W W ! W W ! L W ! 0

d nd1 d+1 nd1 nd1

? ? ?

y y y

0 ! W W ! W W ! L W ! 0

d nd d+1 nd nd

? ? ?

y y y

1+dn 1+dn d 1+dn

0 ! W L ! W L ! L L ! 0

d d+1

? ? ?

y y y

0 0 0

1+dn 1 1d 1+dn 1

L ) ! R (L L ) and The natural maps R (W

d+1

1 d 1+dn 1 d

) ! R (L L ) are b oth isomorphisms. It follows R (L W

1 1

) in W ) and of R (W that the images of R (W W

d+1

nd d nd1

1 1

W ) are equal to the kernel of the natural map R (W R (W

d+1 d+1

Holomorphic bund les over el liptic manifolds 197

1 d 1+dn

W ! R (L L ). Since all the bundles in question are semi-

stable and of negative degree on each b er, they all have trivial R . Thus

1 1 1

the maps R (W W ) and R (W W ) to R (W

d+1

nd1 d nd d+1

1 1

W ) are injections. It follows that R (W W ) and R (W

nd d+1

nd1 d

W ) are identi ed This completes the inductive step and hence the pro of

of the theorem.

4.4 The parab olic construction of vector bundles regular and

semi-stable with trivial determinant on each b er

Let Z ! B be a family of Weierstrass cubic curves with the section

at in nity. Fix a line bundle M on B and sections t of L M for i =

0; 2; 3; 4;::: ;n. Supp osing that there is no p ointof B where all these sections

vanish we can construct a vector bundle as follows.

1 2 1n

The identi cation of Ext (W ; W ) with L L L L

can be twisted by tensoring with M so as to pro duce an identi cation of

1 1n

Ext (W ;W M) with M L M L M L . Thus,

the sections t determine an element of Ext (W ; W M ) and hence

determine an extension

0 !W M !V ! W !0:

Since we are assuming that not all the sections t vanish at the same p oint

of B , the restriction of V to each b er is a non-trivial extension of W by

W . Thus, the restriction of V to each b er is in fact semi-stable, regular

and with trivial determinant.

This parab olic construction thus pro duces one particular vector bun-

dle asso ciated with each line bundle M on B and each non-zero section of

R (O (n )) M . This bundle is automatically regular and semi-stable on

each b er and has trivial determinant on each b er. Conversely, given the

bundle regular and semi-stable and with trivial determinantofeach b er, it

determines a section of the pro jective bundle P ! B , to whichwe can ap-

ply the parab olic construction. The result of the parab olic construction may

not agree with the original bundle { but they will have isomorphic restric-

tions to each b er. Thus, they will di er bytwisting by a line bundle on the

sp ectral covering corresp onding to the section. That is to say to construct

all bundles corresp onding to a given section we b egin with the one pro duced

by the parab olic construction. The section also gives us a sp ectral covering

T ! B . We are then free to twist the bundle constructed by the parab olic

Holomorphic bund les over el liptic manifolds 198

construction by any line bundle on T , just as in the sp ectral covering con-

struction. Thus, the mo duli space of bundles that we are constructing b ers

over the pro jective space of H (Z ; W W ) with b ers b eing Jacobians

d nd

of the sp ectral coverings T ! B pro duced by the section. This twisting cor-

resp onds to nding all bundles which agree with the given one b er-by- b er.

By general theory all such bundles are obtained by twisting with the sheaf

of groups H (B ; (Aut(V ; V )).

4.5 Exercises:

1. Show that a Weierstrass cubic has at most one singularity, and that is

either a no de or a cusp. Show that the cusp app ears only if g = g =

2 3

0. Show that the no de app ears when (g ;g ), as de ned in the lecture,

2 3

vanishes. Show that the p oint at in nity is always a smo oth p oint.

2. Show that for any Weierstrass cubic the usual geometric law de nes a

group structure on the subset of smo oth p oints with the p oint at in nitybe-

ing the origin for the group law. Show that this algebraic group is isomorphic

to C if the curve is no dal and isomorphic to C if the curve is cuspidal.

3. Show that any family of Weierstrass cubics is a at family of curves over

the base.

4. Prove Lemma 4.1.1.

5. Describ e the singularities of Z Z at the no des and cusps of

B B

Z ==B .

6. Show that if V ! Z is a vector bundle and for each berE of Z ==B we

i i

have H (E ; V jE ) is of dimension k , show that R (V )isavector bundle

b b

of rank k on B .

7. Let M b e a line bundle over B and let V t in an exact sequence

0 !W M !V ! W !0:

Compute the Chern classes of V .

8. Show that if V and U are vector bundles over a smo oth variety, then

(U; V )=H Ext (U V).

9. Show that if V is a rank n vector bundle then V is isomorphic to

V det(V ).

10. State relative Serre duality and show that it is correctly applied to

pro duce the map S given in the pro of of Lemma 4.3.5.

11. Supp ose that V is a vector bundle. Show that to rst order the defor-

mations of V are given by H (Hom(V; V )).

Holomorphic bund les over el liptic manifolds 199

5 The Global Parab olic Construction for Holomor-

phic Principal Bundles

In this section we wish to generalize the parab olic construction to families

of Weierstrass cubics. In the last lecture we did this for vector bundles,

here we consider principal bundles over an arbitrary semi-simple group G .

This construction will pro duce holomorphic principal bundles on the total

space Z of the family of Weierstrass cubics which have the prop erty that

they are regular semi-stable G -bundles on each b er of Z ==B . Of course,

this construction can also b e viewed as a generalization of the construction

given in the third lecture for a single elliptic curve. It is imp ortantto note

that we do not give an analogue of the sp ectral covering construction for

G -bundles. We do not know whether such a construction exists for groups

other than SL (C) and Sympl(2n).

5.1 The parab olic construction in families

We let Z ! B be a family of Weierstrass cubics with section : B ! Z as

b efore. Let G be a simply connected simple group. Fix a maximal torus

and a set of simple ro ots for G, and let P G b e the distinguished maximal

parab olic subgroup with resp ect to these choices. Then the Levi factor L

of P is isomorphic to the subgroup of a pro duct of general linear groups

GL consisting of matrices with a common determinant. The charac-

i=1

ter group of P and of L is Z and the generator is the character that takes

the common determinant. We consider the bundle W W . This

n n

naturally determines a holomorphic principal L-bundle over Z . Viewed as

a bundle over G it is unstable since the G -adjoint bundle asso ciated with

C C

this L bundle splits into three pieces: the adjoint ad( ) of the L-bundle,

the vector bundle asso ciated with the tangent space to the unip otent radical

U ( ) and the vector bundle asso ciated to the ro ot spaces negative to those

+ L

in U , U ( ). The rst bundle has degree zero, the second has negative

+ L

degree and the third has p ositive degree. The degree of the entire bundle is

zero. This makes it clear that ad( G ) is unstable, and hence according

L L C

to our de nition that G is an unstable principal G -bundle. (Notice

C L C C

that the L-bundle is stable as an L-bundle.)

Once again we are interested in deformations of to P -bundles with

identi cations =U = . Just as in the case of a single elliptic curve, these

deformations are classi ed by equivalence classes of twisted co cycles, which

Holomorphic bund les over el liptic manifolds 200

we denote by H (Z ; U ( )). Recall that U is ltered 0 U U

L n n1

U = U where the center C acts on U and has homogeneous weight i on

U =U . Furthermore, U =U is ab elian and hence is a vector space which

i i1 i i1

lies in the center of U=U . Thus, once again we can lter the cohomology

by H (Z ; U ( )) with the asso ciated gradeds b eing ordinary cohomology

i L

of vector bundles H (Z ; U =U ( )). Since det( ) as measured with re-

i i1 L L

sp ect to the generating dominantcharacter of P is negative, it follows that

R (U =U ( )) is trivial for all i. A simply inductive argument then

i i1 L

shows that R (U ( )) is a bundle of zero dimensional ane spaces over

B , and hence H (Z ; U ( )) has only the trivial element.

A similar inductive discussion shows that R (U ( )) is ltered with

the asso ciated gradeds b eing the vector bundles R (U =U ( )). This

i i1 L

implies that R (U ( )) is in fact a bundle of ane spaces over B , with

a distinguished element { the trivial cohomology class on each b er. The

lo cal-to-global sp ectral sequence, the vanishing of the R (U ( )) and an

inductive argument shows that in fact the cohomology set H (Z ; U ( )) is

identi ed with the global sections of R (U ( )) over B .

5.2 Evaluation of the cohomology group

In all cases except G = E and over the cuspidal b ers we can in fact split

the bundle R (U ( )) of ane spaces so that it b ecomes a direct sum of

vector bundles. Under this splitting the C action b ecomes linear.

Theorem 5.2.1 Let G beacompact simply connected, simple group and let

Z ! B be a family of Weierstrass cubic curves. Assume either that G is not

isomorphic to E or that no ber of Z ==B is a cuspidal curve. Then there

1 1d

is an isomorphism R U ( ) with a direct sum of line bund les L

L i

where d =0 and d ;::: ;d are the Casimir weights associated to the group

1 2 r

G. Furthermore, the C action that produces the weighted projective space

is diagonal with respect to this decomposition and is a linear action on each

line bund le.

Corollary 5.2.2 The cohomology H (Z ; U ( )) is identi ed with the space

of sections of a sum of line bund les over B , and hence the space of extensions

is identi ed with a bund le of weighted projective spaces over B . The bers

are weighted projective spaces of type P(g ;g ;::: ;g ) where g =1 and for

0 1 r 0

i =1;::: ;r the g are the coroot integers. i

Holomorphic bund les over el liptic manifolds 201

Here is a table of the Casimir weights group ed by C -weights

Gr oup 1 2 3 4

A 0; 2; 3;::: ;n

B 0;2;4 6;8;::: ;2n

C 0;2;4;::: ;2n

D 0;2;4;n 6;8;::: ;2n 2

E 0;2;5 6;8;9 12

E 0; 2 6; 8; 10 12; 14 18

G 0; 2 6

F 0; 2 6; 8 12

Thus, with this choice of splitting for the unip otent cohomology,achoice

of a line bundle M over B and sections t of M L will determine a

section of R (U ( )) and a P -bundle over Z deforming the original L-

bundle . By construction there will be a given isomorphism from the

quotient of the deformed bundle mo dulo the unip otent subgroup back to

. Furthermore, if the sections t never all vanish at the same p oint of

L i

B , then the resulting P -bundle will extend to a G -bundle which is regular

and semi-stable on each berofZ ==B . The resulting section of the weighted

pro jective space bundle is equivalent to the data of the S -equivalence class of

the restriction of the principal G -bundle to each berofZ ==B . Of course,

since these bundles are regular, it is equivalentto the isomorphism class of

the restriction of the G -bundle to each b er.

5.3 Concluding remarks

Thus, for each collection of sections we are able to construct a G -bundle

bf C

which is regular semi-stable on each b er. The study of all bundles which

agree with one of this typ e b er-by- b er is more delicate. From the parab olic

p oint of view, it requires a study of the sheaf R (Aut( )) which can be

quite complicated, and is only partially understo o d at b est.

Even assuming this, we are far from knowing the entire story { one would

like to have control over the automorphism sheaf so as to nd all bundles

which are the same b er-by- b er. Then one would like to complete the

space of bundles by adding those which b ecome unstable on some b ers

(but remain semi-stable on the generic b er). Finally, to complete the space

it is surely necessary to add in torsion-free sheaves of some sort. All these

issues are rip e for investigation { little if anything is currently known.

Holomorphic bund les over el liptic manifolds 202

The study of these bundles is an interesting problem in its own right.

After varieties themselves bundles are probably the next most studied ob-

jects in algebraic geometry. Constructions, invariants, classi cation, mo duli

spaces are the main sources of interest. The study wehave b een describing

here ts p erfectly in that pattern. Nevertheless, from my p oint of view, there

is another completely di erent motivation for this study. That motivation

is the connection with other di erential geometric, algebro-geometric, and

theoretical physical questions.

The study of stable G-bundles over surfaces is closely related to the

study of anti-self-dual connections on G-bundles (this is a variant of the

Narasimhan-Seshadri theorem in for surfaces rather than curves and was

rst established by Donaldson [4]) and whence to the Donaldson p olynomial

invariants of these algebraic surfaces. Thus, the study describ ed here can b e

used to compute the Donaldson invariants of elliptic surfaces. These were

the rst such computations of those invariants, see [5].

More recently, there has b een a connection prop osed, see [9], between

algebraic n-manifolds elliptically b ered over a base B with E E -bundle

8 8

and algebraic (n + 1)-dimensional manifolds b ered over the same base with

b er an elliptically b ered K 3 with a section. The physics of this later set-

up is called F -theory. The precise mathematical statements underlying this

physically suggested corresp ondence are not well understo o d yet, and this

work is an attempt to clarify the relationship b etween these two seemingly

disparate mathematical ob jects. All the evidence to date is extremely p osi-

tive { the two theories E E -bundles over families elliptic curves line up

8 8

p erfectly as far as we can tell with families of elliptically b ered K 3-surfaces

with sections over the same base. Yet, there is still much that is not under-

stood in this corresp ondence. Sorting it out will lead to much interesting mathematics around these natural algebro-geometric ob jects.

Holomorphic bund les over el liptic manifolds 203

References

[1] Adams, J., Exceptional Lie Groups,

[2] Borel, A. Sous-group es commutatifs et torsion de group es de Lie com-

pacts connexes, Tohoku Journal, Ser. 2 13 (1961), 216-240.

[3] Bourbaki, N., Elements de Mathematique, group es et algebres de Lie,

Chapt. 4,5,6. Hermann, Paris, 1960 -1975.

[4] Donaldson, S., Antiselfdual Yang-Mills connections on complex algebraic

surfaces and stable bundles, Pro c. London Math. So c. 3 (1985), 1.

[5] Friedman, R. and Morgan, J., Smo oth Four-Manifolds and Complex Sur-

faces, Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Folge, vol.

27, Springer-Verlag, Berlin, Heidelb erg, New York, 1994. Friedman

[6] Lo oijenga, E., Ro ot systems and elliptic curves, Invent. Math. 38 (1997),

17.

[7] , Invariant theory for generalized ro ot systems, Invent. Math. 61

(1980), 1.

[8] Narasimhan, M. and Seshadri, C., Deformations of the mo duli space of

vector bundles over an algebraic curve, Ann. Math (2) 82 (1965), 540.

[9] Vafa, C. Evidence for F -theory, hep-th 9602065, Nucl. Phys. B 469

(1996), 403.