ñòð. 7 |

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

extension of OE where each extension is non-trivial. Show that H 1 (E; Ir ) is

rank one. The iterated extensions give an increasing ltration jcalF of Ir

by subbundles so that for each s r, we have Fs =Fs 1 = OE . Show that

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

any endomorphism of Ir .

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

in this case the number of linearly independent 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 two vector bundle over E H 1 (E; OE ) whose

restriction to E f0g is OE OE and whose restriction to any other ber

E fxg, x 6= 0 is isomorphic to I2 .

7. By a coarse moduli space of equivalence classes of bundles of a certain

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

bijection between the points of X and the equivalence classes of the bundles

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

the restriction Vy of V to each slice Efyg is of the type under consideration,

then the map Y ! X dened by sending y to the point of X corresponding to

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

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Holomorphic bundles over elliptic manifolds

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

there cannot be a coarse moduli space for isomorphism classes of semi-stable

bundles over E.

8. Show that H 0 (E; O(q p0 )

O(p0 ) is one-dimensional and that any non-

zero section of this bundle vanishes to order one at q. Show that H 0 (E; Ir

O(q p0)

O(p0)) has rank r and that every section of this bundle vanishes

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

order r at q.

9. Show that points (e1 ; : : : ; er ) in E are the zerosP a meromorphic function

of

f: E ! P 1 with a pole only at p0 if and only if

i ei = 0 in the group law

of E.

10. Let E be embedded in P2 so that it is given by an equation in Weierstrass

form:

y2 = 4x3 + g2 x + g3 :

Show that any meromorphic function on E with pole only at innity in this

ane model is a polynomial expression in x and y. Show that a meromorphic

function with pole only at innity which is invariant under e 7! e is a

polynomial expression in x.

11. Let g: T ! jnp0 j be the n-sheeted ramied covering constructed in

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

the Poincar line bundle over E E. Show that if (e1 ; : : : ; en ) are distinct

e

points of E then the restriction of (gId) L to fe1 ; : : : ; en gE is isomorphic

to i OE (ei p0). Show that if e1 = e2 but otherwise the ei are distinct then

(g Id) (L restricted to fe1 ; : : : ; en g E is isomorphic to OE (e1 p0 )

I2 n O(ei p0 )

i=3

12. Show that the Pontrjagen dual of a homomorphism 1 (E) ! T is a

homomorphism ! Hom(1 (E); S 1 ). Show that the choice of an origin p0

allows us to identify E with Hom(1 (E); S 1 ).

13. Show that the quotient of E by e e is P1 . Show that the quotient

=

1 )n under the action of the symmetric group on n letters is Pn .

of (P

14. Show that a weighted projective space with positive weights is a compact

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

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

gularities are modeled by quotients of nite linear group actions on a vector

space.

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Holomorphic bundles over elliptic manifolds

3 The Parabolic Construction

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

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

vector bundles of degree zero.

3.1 The parabolic construction for vector bundles

Lemma 3.1.1 Let E be an elliptic curve with p0 as origin. Then for each

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

the following properties:

1. rank(Wd ) = d.

2. det(Wd ) = O(p0 ).

3. Wd is stable.

Furthermore, H 0 (E; Wd ) is of dimension one.

Proof. The proof is by induction on d. If d = 1, then it is clear that

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

second item, namely O(p0 ). Since this bundle is stable, we have established

the existence and uniqueness when d = 1. Since O( p0 ) is of negative

degree, it has no holomorphic sections and hence by RR, H 1 (E; O( p0 ))

is one-dimensional. By Serre duality, it follows that H 0 (E : O(p0 )) is also

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

Suppose inductively for d 2, there is a unique Wd 1 as required. By

the inductive hypothesis and Serre duality we have H 1 (E; Wd 1 ) is one-

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

0 ! O ! Wd ! Wd 1 ! 0:

Clearly, using the inductive hypothesis we see that Wd is of rank d and its

determinant line bundle is isomorphic to O(p0 ). In particular, the degree

of Wd is one. Suppose that Wd has a destabilizing subbundle U. Then

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

non-positive degree. Thus, the image p(U) of U in Wd 1 has positive degree,

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

of U is at most the rank of Wd 1 , it follows that (p(U)) (Wd 1 ). Since

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Holomorphic bundles over elliptic manifolds

p(U) is non-trivial, it follows that p(U) = Wd 1 . Thus, the rank of U is either

d 1 or d. If it is of rank d, the U = Wd and does not destabilize. Thus, it

must be of rank d 1. This means that p: U ! Wd 1 is an isomorphism and

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

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

is stable.

A direct cohomology computation using the given exact sequence shows

that H 0 (E; Wd ) has rank one, completing the proof.

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

quotients except the last one being O. The last one is O(p0 ).

(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 point

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

q.

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

determinant O( p0 ).

Corollary 3.1.2 The automorphism group of Wd is C.

Proof. Suppose that ': Wd ! Wd is an endomorphism of Wd. Then we

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 Wd are multiplication by scalars. The

result follows.

Now we are ready to construct semi-stable vector bundles of rank n and

degree zero.

Proposition 3.1.3 Let E be an elliptic curve and p0 2 E an origin for the

group law on E. Fix integers d; n; 1 d n 1. Let Wd and Wn d be the

bundles of the last lemma. Then any vector bundle V over E which ts in a

non-trivial extension

0 ! Wd ! V ! Wn d ! 0

is semi-stable.

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Holomorphic bundles over elliptic manifolds

Proof. Clearly, any bundle V as above has rank n and degree zero. Suppose

that U V is a destabilizing subbundle. Then deg(U) 1. Since Wd is

stable, the intersection U \ Wd has negative degree (or is trivial). In either

case, it is not all of U. This means that the image of U in Wn d is nontrivial

and has degree at least one. Since Wn d is stable, this means the image of U

in Wn d is all of Wn d . Hence, the degree of U is one more than the degree

of U \ Wd . Since U \ Wd has negative degree or is trivial, the only way

that U can have positive degree is for U \ Wd = 0, which would 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 Wn d by Wd.

Then for any line bundle over E of degree zero, we have Hom(V; ) has

rank either zero or one.

Proof. The bundle Hom(Wn d ; ) is of degree 1 and is semi-stable. Thus,

it has no sections. Similarly, Hom(Wd ; ) has a one-dimensional space of

sections. From the long exact sequence

0 ! Hom(Wn d ; ) ! Hom(W; ) ! Hom(Wd ; ) !

we see that Hom(V; ) has rank at most one.

Corollary 3.1.5 If V is a non-trivial extension of Wn d by Wd, then V is

isomorphic to a direct sum of bundles of the form O(qi p0 )

Iri for distinct

points qi 2 E.

Proof. If V has two irreducible factors of the form O(q p0)

Ir1 and O(q

p0)

Ir2 then Hom(V; O(q p0 ) would be rank at least two, contradicting

the previous result.

3.2 Automorphism group of a vector bundle over an elliptic

curve

The following is an easy direct exercise:

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Holomorphic bundles over elliptic manifolds

Lemma 3.2.1 Let E be an elliptic curve and let L and M be non-isomorphic

line bundles of degree zero over E. Then Hom(L; M) = 0. Also, Hom(L; L) =

C.

Let V be a semi-stable bundle of degree zero over E. The support of V is

the subset of points e 2 E at which some non-zero section of V vanishes. As

we have seen in Atiyah's theorem, every semi-stable vector bundle of degree

zero over E decomposes as a direct sum of bundles with support a single

point.

Corollary 3.2.2 Let q; q0 be distinct points of E and let Vq and Vq00 be vec-

tor bundles of degree zero over E supported at q and q0 respectively. Then

Hom(Vq ; Vq00 ) = 0.

Corollary 3.2.3 Let V be a semi-stable bundle of degree zero over and el-

liptic curve E and let V = q2E Vq be its decomposition into bundles with

support at single points. Then Hom(V; V ) = q2E Hom(Vq ; vq ).

Now let us analyze the individual terms in this decomposition.

Lemma 3.2.4 With Ir as in Lecture 1, Hom(Ir ; Ir ) is an abelian algebra

C[t]=(tr+1 ) of dimension r.

Proof. Recall that Ir comes equipped with a ltration F0 F1

Fr = Ir with associated quotients O. The quotient of Ir =Fs is identied

with Ir s. The rst thing to prove is that this ltration is preserved under

any endomorphism. It suces by an straightforward inductive argument to

show that F1 is preserved by an endomorphisms. But F1 is the image of the

unique (up to scalar multiples) non-zero section of Ir .

Let t: Ir ! Ir be the map of the form Ir ! Ir =F1 = Ir 1 = Fr 1 Ir .

Clearly, the image of tk is contained in Fr k so that tr+1 = 0. We claim that

every endomorphism of Ir is a linear combination of f1; t; t2 ; : : : ; tr g. Suppose

that f: Ir ! Fr s Ir . Then there is an induced mapping Ir =Ir 1 !

Ir s=Ir s 1. Since both of these quotients are isomorphic to O, this map

and some multiple of the map between these quotients induced by ts are

equal. Subtracting this multiple of ts from f we produce a map Ir ! Fr s 1 .

Continuing inductively proves the result.

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Holomorphic bundles over elliptic manifolds

Notice that an endomorphism is an automorphism if and only if its image

is not contained in Ir 1 if and only if it is not an element of the ideal

generated by t.

From this description the following is easy to establish.

Corollary 3.2.5 Let V be a semi-stable bundle of degree zero and rank n

over an elliptic 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 subbundle Vq V

supported by q is of the form O(q p0 )

Ir(q) for some r(q) 0.

Proof. We decompose V into a direct sum of bundles Vi which them-

selves are indecomposable under direct sum. Thus, each Vi is of the form

O(q p0)

Ir for some q 2 E and some r 1. Thus, by the previous

result the automorphism group of Vi has dimension equal to the rank of Vi .

Clearly, then the automorphism group of V preserving this decomposition

has dimension equal to the rank of V . This will be the entire automorphism

group if and only if Hom(Vi ; Vj ) = 0 for all i 6= j. This will be the case if

and only if the Vi have disjoint support.

Note that if, in the above notation, two or more of the Vi have the same

support then the automorphism group has dimension at least two more than

the rank of V .

Denition 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 position to prove the main result along these lines.

Theorem 3.2.7 Fix n; d with 1 d < n. A vector bundle V of rank n can

be written as a non-trivial extension

0 ! Wd ! V ! Wn d ! 0

if and only if

1. The determinant of V is trivial.

2. V is semi-stable.

3. V is regular.

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Holomorphic bundles over elliptic manifolds

Proof. The rst condition is obviously necessary and in Proposition 3.1.3

and Corollary 3.1.5 we established that the second and third are also neces-

sary.

Suppose that V satises all the conditions. Condition (iii) says that

V = q2E Vq where each Vq is of the form O(q p0 )

Ir(q) . We claim

that there is a map Wd ! O(q p0 )

Ir(q) whose image is not contained in

O(q p0)

Ir(q) 1 . The reason for this is that Wd

O(q p0)

Ir is stable and

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

Wd ! O(q p0 )

Ir(q) which does not factor through O(q p0)

Ir(q) 1 .

Claim 3.2.8 If F is a subsheaf of V of degree zero and if the projection of

F onto each Vq is not contained in a proper subsheaf, then F = V .

Proof. Let W be 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 Vq

and Vq0 for q 6= q0 , it follows that any subbundle of Vq is in fact a direct

sum of its intersections with the various Vq . If the image of the subbundle

under projection to Vq is all of Vq then its intersection with Vq is all of Vq .

The claim now follows.

Let Wd ! V be a map whose projection onto each Vq is not contained

in any proper subbundle of Vq . Let us consider the image of this map. It is

a subsheaf of V which is proper since d < n. This means it has degree at

most zero. By the above it cannot be 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 possibility contradicts the stability of Wd . This shows that

the map is an isomorphism onto its image; that is to say it is an embedding

of Wd V .

Next, let us consider the cokernel X. We have already seen that the

cokernel is a bundle. Clearly, its determinant is O(p0 ) and its rank is n d.

To show that it is Wn d we need only see that it is stable. Suppose that

˜

U X is destabilizing. Then the degree of U is positive. 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 Wd , the previous claim implies

that it is all of V , which implies that U is all of Wn d , contradicting the

assumption that U was destabilizing.

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Holomorphic bundles over elliptic manifolds

Now we shall show that, given V , there is only one such extension with

V in the middle, up to automorphisms of V .

Proposition 3.2.9 Let V be a semi-stable rank n-vector bundle with trivial

determinant. Suppose that the group of automorphisms of V has dimension

n. Then V can be written as an extension

0 ! Wd ! V ! Wn d ! 0:

This extension is unique up to the action of the automorphism group of V .

Proof. Suppose we have an extension as above for V . First notice that if

the image of Wd were contained in a proper subsheaf of degree zero of V ,

then this subsheaf would project into Wn d destabilizing it. Thus, the only

subsheaf of degree zero of V that contains the image of Wd is all of V . That

is to say the image of Wd in each Vq is not contained in a lower ltration

level, i.e. a proper subsheaf of degree zero. Now we need to show that all

maps Wd ! Vq which are not contained in a proper subsheaf of degree zero

are equivalent under the action of the automorphisms of Vq . This is easily

established from the structure of the automorphism sheaf of Vq given above.

Corollary 3.2.10 For any 1 d < n, the projective space of

H 1(E; Hom(Wn d ; Wd )) = H 1 (E; Wn d

Wd ) is identied with the space

of isomorphism classes of regular semi-stable vector bundles of rank n and

trivial determinant.

Notice that it is not apparent, a priori, that for dierent d < n that the

above projective spaces can be identied in some natural manner.

The association to each regular semi-stable vector bundle of rank n and

trivial determinant of its S-equivalence class then induces a holomorphic

map from P(H 1 (E; Wn d

Wd )) to the coarse moduli space P(O(np0 )) of

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 projective spaces it is in fact a holomorphic isomorphism. Thus, for

any d; 1 d < n, we can view the projective space of H 1 (E; Wn d

Wd) as

yet another description of the coarse moduli space of S-equivalence classes

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Holomorphic bundles over elliptic manifolds

of semi-stable rank n vector bundles of trivial determinant. Notice that the

actual bundles produced by this construction are the same as those produced

by the spectral covering construction since both are families of regular semi-

stable bundles. On the other hand, using the Narasimhan-Seshadri result

gives a dierent set of bundles { namely the direct sum of line bundles. These

families agree generically, but dier along the codimension-one subvariety of

the parameter space where two or more points come together. We have seen

two constructions of holomorphic families of semi-stable vector bundles {

the spectral covering construction and the parabolic construction and both

create regular semi-stable bundles. It is not clear that the

at connection

point of view can be carried out holomorphically in families (indeed it can-

not). A hint to this fact is that it is producing dierent bundles and these

do not in general t together to make holomorphic families.

The reason for the name `parabolic' will become clear after we extend

to the general semi-simple group. Before we can give this generalization we

need to discuss parabolic subgroups of a semi-simple group.

3.3 Parabolics in GC

Let GC be a complex semi-simple group. A Borel subgroup of GC is a con-

nected complex subgroup whose Lie algebra contains a Cartan subalgebra

(the Lie algebra of a maximal complex torus) together with the root spaces

of all positive roots with respect to some basis of simple roots. All Borel

subgroups in GC are conjugate. By denition a parabolic subgroup is a

connected complex proper subgroup of GC that contains a Borel subgroup.

Up to conjugation parabolic subgroups of GC are classied by proper (and

possibly empty) subdiagrams of the Dynkin diagram of G. Fix a maximal

torus of GC and a set of simple roots f1 ; : : : ; n g. A subdiagram is given

simply by a subset f1 ; : : : ; r g of the set of simple roots. The Lie algebra

of the parabolic is the Cartan subalgebra tangent to the maximal torus, to-

gether with all the positive root spaces and all the root spaces associated

with negative linear combinations of the f1 ; : : : ; r g. Thus, a Borel sub-

group corresponds to the empty subdiagram. The full diagram gives GC

and hence is not a parabolic subgroup. Up to conjugation a parabolic P is

contained in a parabolic P 0 if and only if the diagram corresponding to P

is a subdiagram of that corresponding to P 0 . It follows that the maximal

parabolic subgroups of GC up to conjugation are in one-to-one correspon-

dence with the subdiagrams of the Dynkin diagram of G obtained by deleting

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Holomorphic bundles over elliptic manifolds

a single vertex. This sets up a bijective correspondence between conjugacy

classes of maximal parabolic subgroups of GC and vertices of the Dynkin

diagram, or equivalently with the set of simple roots for GC.

A parabolic subgroup P has a maximal unipotent subgroup U whose Lie

algebra is the sum of the roots spaces of positive roots whose negatives are

not roots 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 product U L. The derived subgroup of L is a semi-

simple group whose Dynkin diagram is the subdiagram that determined P

in the rst place. A maximal torus of P is the original maximal torus of

G. If P is a maximal parabolic then the character group of P is isomorphic

to the integers, and the component 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 coroot associated with the simple root i that is omitted from

the Dynkin diagram in order to create the subdiagram that determines P.

The value of this weight on any root is simply the coecient of i in the

linear combination of the simple roots which is . The root spaces of the

Lie algebra of P are those ones which the character is non-negative, and the

Lie algebra of the unipotent radical is the sum of the root spaces of roots on

which this character is positive.

Example: The maximal parabolics of SLn(C) correspond to nodes of

its diagram. Counting from one end we index these by integers 1 d < n.

The parabolic subgroup corresponding to the integer d is the subgroup of

block diagonal matrices with the lower left d (n d) block being zero.

The Levi factor is the block diagonal matrices or equivalently pairs (A; B) 2

GLd (C)GLn d(C) with det(A) = det(B). A vector bundle with structure

reduced to this parabolic 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 unipotent subgroup is a vector group

Hom(Cn d; Cd).

3.4 The distinguished maximal parabolic

For all simple groups except those of An type we shall work with a distin-

guished maximal parabolic. It is described as follows: If the group is simply

laced, then the node of the Dynkin diagram that is omitted is the trivalent

one. If the group is non-simply laced, then either vertex which is omitted

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Holomorphic bundles over elliptic manifolds

is the long one connected to the multiple bond. It is easy to see that in all

cases the Levi factor of this parabolic is written as the subgroup of a product

of GLki of matrices with a common determinant.

Examples: (i) For a group of type Cn, there is a unique long root.

The Levi factor of the corresponding subgroup is GLn(C) and the unipo-

tent radical is the self-adjoint maps Cn to its dual. In terms of complex

symplectic 2n-dimensional bundles, a reduction of structure group of V 2n

to this parabolic means the choice of a self-annihilating n-dimensional sub-

bundle W n. This bundle has structure group the Levi factor GLn(C) of

the parabolic. The quotient of the bundle by this subbundle is simply the

dual bundle Wn . The extension class that determines the bundle and its

symplectic form is an element in H 1 (E; SymHom(W; W )), where SymHom

means the self-adjoint homomorphisms.

(ii) For a group of type Bn the distinguished maximal parabolic has

Levi factor the subgroup of GLn 1(C) GL2(C) consisting of matrices of

the same determinant. Let us consider the orthogonal group instead of

the spin group. Then a reduction in the structure group of an orthogonal

bundle V 2n+1 to this parabolic is a self-annihilating subspace W1 V 2n+1

of dimension n. This produces a three term ltration W1 W2 W3

where W2 = W1?. Under the orthogonal pairing W1 and W3 =W2 are dually

paired and W2 =W1 , which is three-dimensional, has a self-dual pairing and

is identied with the adjoint of the bundle over the GL2(C)-factor. The

subbundle W1 is the bundle over the GLn(C)-factor of the Levi. There are

two levels of extension data one giving the extension comparing W1 W2

which is an element of H 1 (E; (W2 =W1 )

W1 )) and the other an extension

class in H 1 (E; SkewHom(W3 =W2 ; W1 )), where SkewHom refers to the anti-

self adjoint mappings under the given pairing.

(iii) There is a similar description for D2n . Here the Levi factor of the dis-

tinguished parabolic is the subgroup of matrices in GLn 2 (C) GL2 (C)

GL2(C) consisting of matrices with a common determinant. This time a

reduction of the structure group to P corresponds to a self-annihilating sub-

space W1 of dimension n 2, it is the bundle over the GLn 2(C)-factor of

the Levi. The quotient W2 =W1 is four-dimensional and self-dually paired. It

is identied with the tensor product of the bundle over one of the GL2(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 W1 W2 and the other a self-dual extension class for W3 =W2 by

W1.

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Holomorphic bundles over elliptic manifolds

(iv) In the case of Er , r = 6; 7; 8 the Levi factor is the subgroup of

matrices in GL2(C) GL3(C) GLr 3 (C) with the same determinants.

It is more dicult to describe what a reduction of the structure group to

this parabolic means since we have no standard linear representation to use.

For E6 there is the 27-dimensional representation, which would then have a

three-step ltration with various properties.

In the case of SLn (C) we began with a particular, minimally unstable

vector bundle Wd Wn d whose structure group has been reduced to the

Levi factor of the parabolic subgroup. We then considered extensions

0 ! Wd ! V ! Wn d ! 0:

These extensions have structure group the entire parabolic. They also have

the property that modulo the unipotent subgroup they become the unstable

bundle Wd Wn d with structure group reduced to the Levi factor L of this

maximal parabolic.

3.5 The unipotent subgroup

Let us consider the unipotent subgroup of a maximal parabolic group. Fix

a maximal torus T of GC and a set of simple roots f1 ; : : : ; n g. Suppose

that this parabolic is the one determined by deleting the simple root i . We

begin with its Lie algebra. Consider the direct sum of all the root spaces

gu associated with positive roots whose negatives are not roots of P. These

are exactly the positive roots which, when expressed as a linear combination

of the simple roots have a positive coecient times i . Clearly, these roots

form a subset which is closed under addition, in the sense that if the sum of

two roots of this type is a root, then that root is also of this type. This means

˜

that the sum U of the root spaces for these roots makes a Lie subalgebra of

gC . Furthermore, there is an integer k > 0 such that any sum of at least

k roots of this type is not a root. (The integer k can be taken to be the

largest coecient of i in any root of gC.) This means that the Lie algebra

˜

U is in fact nilpotent of index of nilpotency at most k. It follows that the

˜

restriction of the exponential map to U is a holomorphic isomorphism from

˜

U to a unipotent subgroup U GC. The dimension of this group is equal

to the number of roots with positive coecient on i . Furthermore, U is

ltered by a chain of normal subgroups f1g Uk Uk 1 U1 where

Ui is the unipotent subgroup whose Lie algebra is the root spaces of roots

whose i -coecient is at least i. Clearly, Uk is contained in the center of the

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Holomorphic bundles over elliptic manifolds

group and Ui =Ui 1 is contained in the center of U=Ui . The entire structure

of the unipotent group can be directly read o from the set of roots with

positive coecient on i together with the information about which sums of

roots are roots.

Examples: (i) For SLn (C) and any maximal parabolic the ltration is

trivial U1 = U; U2 = f1g. The reason is of course that all positive roots are

linear combinations of the simple roots with coecients 0; 1 only. Thus, U

is a vector group. It is Hom(Cd ; Cn d ).

(ii) For groups of type Cn and maximal parabolics obtained by deleting

the vertex corresponding to the unique long root, again U = U1 ; U2 = f1g

(all roots have coecient 1 or 0 on this simple root). The unipotent radical

U is the vector group ^2 Cn. For all other maximal parabolics of groups of

type Cn , the unipotent radical has a two-step ltration and is not abelian.

(iii) For groups of type Bn and for maximal parabolics obtained by delet-

ing the simple root i which is long and which corresponds to a vertex of the

double bond in the Dynkin diagram, the ltration is f1g U2 U1 = U.

The dimension of U2 is (n 1)(n 2)=2 and the dimension of U1 =U2 is

2(n 1). The Lie bracket mapping U1 =U2

U1 =U2 ! U2 is onto, so that the

unipotent group is not a vector group, i.e., it is unipotent but not abelian.

(iv) For groups of type Dn and maximal parabolics obtained by deleting

the simple root corresponding to the trivalent vertex, once again the ltration

is of length 2: we have f1g U2 U1 = U. The dimension of U2 is

(n 2)(n 3)=2 and the dimension of U1 =U2 is 4(n 2). Once again the

bracket mapping U1 =U2

U1 =U2 ! U2 is onto, and hence the group is non-

abelian.

(iv) Once we leave the classical groups, the ltrations become more com-

plicated. For E6 the ltration of the unipotent subgroup of the distinguished

maximal parabolic is f1g U3 U2 U1 where the dimension of U3 is two,

the dimension of U2 =U3 is 9 and the dimension of U1 =U2 is 18. For E7 and

the distinguished maximal parabolic, the ltration of the unipotent radical

begins at U4 which is three-dimensional and descends to U1 with U1 =U2 be-

ing 24 dimensional. For E8 and the distinguished maximal parabolic, the

ltration begins at U6 which is ve-dimensional and descends all the way to

U1 with U1=U2 being of dimension 30.

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Holomorphic bundles over elliptic manifolds

3.6 Unipotent cohomology

Fix a simple group GC and a maximal parabolic subgroup P, and x a

splitting P = U L. Also, x a holomorphic principal bundle L ! 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 ! L. Let us

choose a covering of the elliptic curve by small analytic open subsets fUi g.

The bundle L is described by a cocycle nij : Ui \ Uj ! L. A bundle and

isomorphism =U ! L is given by maps uij : Ui \ Uj ! U satisfying the

cocycle condition:

uij nij ujk njk = uik nik :

Since the fnij g are already a cocycle, we can rewrite this condition as

uij unij = uik ;

jk

where un = nun 1 for u 2 U and n 2 L. This is the twisted cocycle con-

dition associated with the bundle L and the action of L (by conjugation)

on the unipotent subgroup U. A zero cochain is simply a collection of holo-

morphic maps vi : Ui ! U. Varying a twisted cocycle fuij g by replacing the

coboundary of this zero cochain means replacing it by vi uij (vj 1 )nij .

The set of cocycles modulo the equivalence relation of coboundary makes

a set, denoted H 1 (E; U(L )). In fact, it is a pointed set since we have

the trivial cocycle: uij = 1 for all i; j. In the case when U is abelian,

associated to L and the action of L on U (which is linear), there is a vector

bundle U(L ). The twisted cocycles modulo coboundaries are exactly the

usual Cech cohomology of this vector bundle, H 1 (E; U(L )), 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 associated

gradeds being vector groups, we can lter the twisted cohomology and the

associated gradeds are naturally the usual cohomology of the vector bundles

H 1(E; (Ui =Ui 1 (L )). In this situation, the entire cohomology H 1 (E; U(L ))

can be given the structure on an ane space which has an origin, and which

is ltered with associated gradeds being vector bundles.

The center of P is C (more precisely, the component of the identity of

the center of P) and hence acts on U and on H 1 (E; U(L )). This action

preserves the origin, and the ltration and on each associated graded is a

linear action of homogeneous weight. That weight is given by the index of

that ltration level (weight i on Ui =Ui 1 ). It is a general theorem that since

all these weights of the C -action are positive, there is in fact an isomorphism

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Holomorphic bundles over elliptic manifolds

of this ane space with a vector space in such a way that the C -action

becomes linearized. In particular, the quotient of H 1 (E; U(L )) f0g =C

is isomorphic as a projective variety to a weighted projective space. The

dimension of the subprojective space of weight i is equal to one less that the

dimension of H 1 (E; Ui =Ui 1 )).

This is a fairly formal construction and it is not clear that it has anything

to do with stable GC -bundles. Here is a theorem that tells us that using

the distinguished maximal parabolic identied above and a special unstable

bundle with structure group the Levi subgroup of this parabolic in fact leads

to semi-stable GC-bundles. It is a generalization of what we have established

directly for vector bundles.

Theorem 3.6.1 Let GC be a simply connected simple group and let P GC

be the distinguished parabolic subgroup as above. Then the Levi factor of

Q

P is isomorphic to the L i GLn i consisting of all fAi 2 GLni gi

such that det(Ai ) = det(Aj ) for all i; j. Let Wni be the unique stable

bundle of rank ni and determinant O(p0 ). Then L = i Wni is natu-

rally a holomorphic principal L-bundle over E. Every principal P-bundle

which is obtained from a non-trivial cohomology class in H 1 (E; U(L )) be-

comes semi-stable when extended to a GC-bundle. Cohomology classes in

the same C -orbit determine isomorphic GC-bundles. This sets up an iso-

morphism between H 1 (E; U(L )) f0g =C and the coarse moduli space of

S-equivalence classes of semi-stable GC-bundles over E. Every GC-bundle

constructed this way is regular in the sense that its GC-automorphism group

has dimension equal to the rank of G, and any regular semi-stable GC -bundle

arises from this construction. Any non-regular semi-stable GC -bundle has

automorphism group of dimension at least two more than the rank of G.

This result gives a dierent proof of Looijenga's theorem. It identies

the coarse moduli space as weighted projective space associated with a non-

abelian cohomology space. It is easy to check given the information about

the roots and their coecients over the distinguished simple root that the

weights of this weighted projective space are as given in Looijenga's theorem.

3.7 Exercises:

1. Suppose that we have a non-trivial extension

0 ! O ! X ! Wd 1 ! 0;

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Holomorphic bundles over elliptic manifolds

where Wd 1 is as in the rst lemma of this lecture. Show that H 0 (E; X) is

one-dimensional and hence that H 1 (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 identied with End(V ) = H 0 (V

V ).

3. Show that if V1 and V2 are semi-stable bundles over E, then so is V1

V2 .

Compute the degree of V1

V2 in terms of the degrees and ranks of V1 and

V2 .

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 which is

not regular, then the dimension of the automorphism group of V is at least

r + 2.

6. Let Vqi be semi-stable vector bundles over E of degree zero and disjoint

support. Show that any subbundle of degree zero in Vqi is in fact a direct

sum of subbundles of the Vqi .

7. Show that if any two homomorphisms Wd ! Ir which have image not

contained in Ir 1 dier by an automorphism of Ir .

8. Show that a Borel subgroup of GC is determined by a choice of a maximal

torus for GC and a choice of simple roots for that torus. Show all Borel

subgroups of GC are conjugate.

9. Up to conjugation, describe explicitly all parabolic subgroups of SLn(C).

10. Let GC be a semi-simple group. Show that the character group of a

maximal parabolic subgroup of GC is isomorphic to Z. Show that the center

of a maximal parabolic subgroup of GC is one-dimensional.

11. For E6 ; E7 ; E8 ; G2 ; F4 work out the dimensions of the various ltration

levels in the unipotent subgroups associated with the distinguished maximal

parabolic subgroups.

12. Check that in the formula given for the action by a coboundary on a

twisted cocylce that the resulting one-cochain is still a twisted cocycle.

13. For groups of type Bn and Dn and the distinguished parabolic and the

given bundle L over the Levi factor, compute the cohomology vector spaces

H 1 (E; U2 (L )) and H 1 (E; U1 =U2 (L )).

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Holomorphic bundles over elliptic manifolds

4 Bundles over Families of Elliptic Curves

In this lecture we will generalize the constructions for the case of vector

bundles over an elliptic curve to vector 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 best choice for our context is a family of Weierstrass

cubic curves. Recall that a single Weierstrass cubic is an equation of the

form

y2 = 4x3 + g2 x + g3 ;

or written in homogeneous coordinates is given by:

zy2 = 4x2 + g2 xz 2 + g3 z 3 :

This equation denes a cubic curve in the projective plane with homogeneous

coordinates (x; y; z). The point at innity, i.e., the point with homogeneous

coordinates (0; 1; 0) is always a smooth point of the curve. In the case when

the curve is itself smooth, this point is taken to be the identity element of the

group law on the curve. More generally, there are only two types of singular

curves which can occur as Weierstrass cubics { a rational curve with a single

3 2

node { which occurs when (g2 ; g3 ) = 0 where (g2 ; g3 ) = g2 + 27g3 is the

discriminant, and the cubic cusp when g2 = g3 = 0. In each of these cases

the subvariety of smooth points of the curve forms a group (C in the nodal

case and C in the cuspidal case), and again we use the point at innity as

the origin of the group law on the subvariety of smooth points.

Now suppose that we wish to study a family of such cubic curves para-

metrized by a base B which we take to be a smooth variety. Then we x a

line bundle L over B. We interpret the variables x; y; z as follows: let E be

the three-plane bundle OB L2 L3 over B; z: E ! OB , x: E ! L2 , and

y: E ! L3 are the natural projections. Furthermore, g2 is a global section of

L4 and g3 is a global section of L6 . With these denitions

zy2 (4x3 + g2 xz 2 + g3 z3 )

is a section of Sym3 (E )

L6. Its vanishing locus projectivizes to give a sub-

variety Z P(E) over B, which ber-by-ber is the elliptic curve (possible

singular) given by trivializing the bundle L over the point b 2 B in question

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Holomorphic bundles over elliptic manifolds

and viewing g2 (b) and g3 (b) as complex numbers so that the above cubic

equation with values in L6 becomes an ordinary cubic equation depending

on b. While the actual equation associated to b will of course depend on

the trivialization of Ljfbg , the homogeneous cubic curve it denes will be

independent of this choice.

Thus, as long as the sections g2 and g3 are generic enough so as not to

always lie in the discriminant locus, Z ! B is an elliptic bration (which

by denition is a

at family of curves over B whose generic member is an

elliptic curve). This family of elliptic curves comes equipped with a choice of

base point, i.e., there is a given section of Z ! B. It is the section given by

fz = x = 0g or equivalently, by the section [L3 ] 2 P(OB L2 L3). (This

is the globalization of the point (0; 1; 0) in a single Weierstrass curve.) This

does indeed dene a section of Z ! B. The image of this section is always a

smooth point of the ber. If we use local berwise coordinates (u = x=y; v =

z=y) near this section, then the local equation is v = 4u3 + g2uv2 + g3 v3 ,

and its gradient at the point (0; 0) points 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 coordinate x=y, these lines t together to

form the line bundle L2

(L3 ) 1 = L 1 , which then is the normal bundle of

in Z. This bundle is also of course the relative tangent bundle of the bers

along the section . Since the tangent bundle of each ber is trivialized, it

follows that the pushforward, Tbers , is isomorphic to L 1 . Also important

for us will be the relative dualizing sheaf. It is Tbers . (Of course, as I have

presented it, we are working only at smooth bers. But because the singular

curves have suciently mild singularities the relative dualizing sheaf is still

a line bundle, and in fact is the bundle L.)

We have proved:

Lemma 4.1.1 Let be the normal bundle of in Z. Let : Z ! B be the

natural projection. Then = OZ ()j and (OZ ()j ) = ( ) = L 1 :

The bundle L is the relative dualizing line bundle.

N.B. The subvariety of 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 g2 and g3 the codimension of these subvarieties are one and two,

respectively.

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Holomorphic bundles over elliptic manifolds

4.2 Globalization of the spectral 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 be to try to nd the analogue for (E {z E) =Sn . The obvious can-

| }

n times

didate is (Z B {z B Z) =Sn . This works ne as long as Z is smooth over B

| }

n times

but does not give a good result at the singular bers. There is in fact a way

to globalize this construction, at least across the nodes. It involves consid-

ering Zreg B B Zreg , where Zreg is the open subvariety of points regular

in their bers, and then given an appropriate toroidal compactication at

the nodal bers. I shall not discuss this construction here.

There is however another way to view n points on E which sum to

zero, up to permutation. Namely, as we have already seen, these points

are naturally the points of the projective space H 0 (E; OE (np0 )). Thus, a

better way to globalize is to replace O(p0 ) by OZ () and thus consider

R0 (OZ (n)). This is a vector bundle of rank n on B. Its associated

projective space bundle is then a locally trivial Pn 1 bundle over B. The

ber of this projective bundle over a point b 2 B is canonically identied

with the projective bundle of the linear system jnp0 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 bundle R0(OZ (n)) is naturally split as a sum of line

bundles: OB L 2 L 3 L n.

Proof. By denition we are considering the bundle whose sections over an

open subset U B are the analytic functions on ZjU with poles only along

\(ZjU ) and those being of order at most n. We have already at our disposal

functions with this property: 1; x; x2 ; : : : ; x[n=2] ; y; xy; : : : ; x[(n 3)=2] y. Given

any function with this property over U, we can subtract (uniquely) a multiple

of one of these basic functions, xa or xa y, so that the order of the pole is

reduced by at least one. The multiple will have a coecient which is a section

of the line bundle L 2a in the rst case and L 2a+3 in the second. In this

way we identify the sections of our vector bundle over U with expressions of

the form

a0 + a1 x + + a[n=2]x[n=2] + b0 y + + b[(n 3)=2] x[(n 3)=2]y:

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Holomorphic bundles over elliptic manifolds

The coecient of xa lies in L a and the coecient of xa y lies in L (2a+3) .

This identies the space of sections with the sum OB L 2 L 3 L n .

Notice that a section of this n-plane bundle is then a family of S-

equivalence classes of semi-stable bundles on the bers of Z==B, but that

it is not yet a vector bundle on Z. Nevertheless, the spectral covering con-

struction generalizes to produce a vector bundle. Let Pn be the bundle

of projective spaces associated to the vector bundle R0 (OZ (n)). This

is the bundle whose ber over b 2 B is the projective space of the linear

system OEb (np0 ). Consider the natural map OZ (n) ! OZ (n). It is

surjective and we denote by E its kernel which is a vector bundle of rank

n 1. Dene T = P(E). A point of OZ (n) consists of an element

f 2 jOEb (n(b))j together with a point z 2 Eb. The ber E consists of all

pairs for which f(z) = 0. The bundle T is a Pn 2 -bundle over Z whose ber

over any z 2 Eb is the projective space of the linear system OEb (n(b) z) on

Eb. The composition of the inclusion T ! Pn B Z followed by the projec-

tion onto Pn is a ramied n-sheeted covering denoted g, which ber-by-ber

is the map we constructed before for a single elliptic curve.

Using this map we can construct a family of vector bundles over Z semi-

stable on each ber. Namely, we consider the pullback to T B Z of the

diagonal 0 Z B Z. Then we have a line bundle

L = OT B Z ( T B ):

The pushforward (g B Id) (L) is a rank n vector bundle on Z which is

regular semi-stable and of trivial determinant on each ber. Analogous to

our result for a single curve we have the following universal property for this

construction.

Theorem 4.2.2 Let U ! Z be a vector bundle which is regular, semi-stable

with trivial determinant on each ber of Z==B. Then associating to each

b 2 B the class of UjEb determines a section sA: B ! Pn . Let TA be the

pullback of T ! Pn via this section. Then the natural projection TA ! B is

an n-sheeted ramied covering. Let LA be the pullback to TA B Z of the line

bundle L over T B Z by sA B Id. Then there is a line bundle M over TA

such that U is isomorphic to (g Id) (LA

p M), where p1 : TA B Z ! TA.

1

Notice that there are in essence two ingredients in this construction: the

rst is a section A of Pn ! B and the second is a line bundle over the induced

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Holomorphic bundles over elliptic manifolds

ramied covering TA of B. The section A is equivalent to the information of

the isomorphism class of the bundle on each ber of Z==B. The line bundle

over TA gives us the allowable twists of the bundle on Z which do not change

the isomorphism class on each ber.

This completes the spectral covering construction. It has the advantage

that it produces all vector bundles over Z which are regular and semi-stable

with trivial determinant on each ber. Its main drawback is that it does

not easily generalize to other simple groups. The construction that does

generalize easily is the parabolic construction to which we turn now.

4.3 Globalization of the parabolic construction

It turns out that (except in the case of E8 -bundles and cuspidal bers) that

the parabolic construction of vector bundles globalizes in a natural way.

The rst step in establishing this is to globalize the bundles Wd which are

an essential part of the construction, both for vector bundles and for more

general principal G-bundles.

4.3.1 Globalization of the bundles Wd

We dene inductively the global versions of the bundles Wd . The globaliza-

tion of W1 = OE (p0 ) is of course W1 = OZ (), so that the way we have

chosen to globalize curves has already given us a natural globalization of

W1 . Clearly, the restriction of this line bundle to any ber E of Z==B is

the bundle OE (p0 ). (Notice that even if the ber is singular, p0 is a smooth

point of it, so that OE (p0 ) still makes sense as a line bundle.)

Claim 4.3.1 There is, up to non-zero scalar multiples, a unique non-trivial

extension

0 ! L ! X ! W1 ! 0:

The restriction of X to any ber is isomorphic to W2 of that ber.

Proof. Let us compute the global extension group Ext1(OZ (); L). Since

both the terms are vector bundles, the extension group is identied with

the cohomology group H 1 (Z; OZ ()

L). The local-to-global spectral

sequence produces an exact sequence

0 ! H 1 (B; (OZ ()

L) ! H 1 (Z; OZ ()

L)

! H 0 (B; R1OZ ()

L) ! H 2(B; OZ ()

L) ! :

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Holomorphic bundles over elliptic manifolds

Since the restriction of OZ () to each ber is semi-stable of negative degree,

it follows that the rst term and the fourth term are both zero, and hence

we have an isomorphism

H 1 (OZ ()

L) ! H 0 (B; R1 ( OZ ()

L)) = H 0 (B; R1(OZ ())

L):

But we have already seen that R1 (OZ ( )) = L 1 , so that we are con-

sidering H 0 (B; (L 1

L) = H 0 (B; OB ) = C. Since any non-trivial section

of this bundle is nonzero at each point, any non-trivial extension class has

non-trivial restriction to each ber and hence any non-trivial extension of

the form 0 ! L ! W1 ! 0 restricts to each ber Eb to give a nontrivial

restriction of W1 by OEb and hence restricts to each ber to give a bundle

isomorphic to W2 on that ber.

Now let us continue this construction. The following is easily established

by induction.

Proposition 4.3.2 For each integer n 1 there is a bundle Wn over Z

with the following properties:

1. W1 = OZ ()

2. For any n 2 we have a non-split exact sequence

0 ! Ln 1 ! Wn ! Wn 1 ! 0:

3. R1 Wn = L n.

4. R0 Wn = 0.

For these bundles the restriction of Wn to any ber of Z==B is isomorphic

to the bundle Wn of that Weierstrass cubic curve.

Proof. The proof is by induction on d, with the case d = 1 being the last

claim. Suppose inductively we have constructed Wd 1 as required. Since

Wd 1 is semi-stable of negative degree on each ber, and since R1 Wd 1 =

L1 d, it follows by exactly the same local-to-global spectral sequence argu-

ment as in the claim that H 1 (Wd 1

Ld 1 ) = H 0 (B; L1 d

Ld 1 ) =

H 0(B; OB ) = C. Thus, there is a unique (up to scalar multiples) nontrivial

extension of the form

0 ! Ld 1 ! X ! Wd 1 ! 0

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Holomorphic bundles over elliptic manifolds

and the restriction of this extension to each ber of Z==B is nontrivial. We

let Wd be the bundle which is such a nontrivial extension. The computations

of Ri Wd are straightforward from the extension sequence.

Notice that Wn is not the only bundle that restricts to each ber to give

Wn . Any bundle of the form Wn

M for any line bundle M on B will

also have that property. Since the endomorphism group of Wn is C , one

shows easily that these are the only bundles with that property.

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 Wd be

Ld(d 1)=2

O)Z() will determine Wd up to isomorphism.

4.3.2 Globalizing the construction of vector bundles

Lemma 4.3.3 Ext1 (Wn d; Wd ) is identied with the space of global sections

of the sheaf

R1 (Wn d ; Wd )

on B.

Proof. First of all since Wn d and Wd are vector bundles, we can iden-

tify Ext1 (Wn d ; Wd ) with H 1 (Z; Wn d

Wd ). The local-to-global spectral

sequence produces an exact sequence

0 ! H 1 (B; R0 (Wn d

Wd )) ! H 1 (Z; Wn d

Wd )

! H 0 (B; R1(Wn d

Wd )) ! H 2 (B; R0 Wn d

Wd ):

Since Wd and Wn d are both semi-stable and of negative degree on each

ber, the restriction of their tensor product to each ber has no sections. It

follows that R0 (Wn d

Wd ) is trivial. Thus, we have an isomorphism

H 1(Z; Wn d

Wd ) ! H 0 (B; R1 (Wn d

Wd ));

as claimed in the statement.

Next we need to compute the sheaf R1 (B; Wd

Wn d ) on B.

Proposition 4.3.4 R1(B; Wd

Wn d) is a vector bundle and is isomor-

phic to the direct sum of line bundles L L 1 L 2 L1 n .

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Holomorphic bundles over elliptic manifolds

First we consider a special case:

Lemma 4.3.5 R1(B; OZ ( )

Wn 1) is isomorphic to LL 1L 2

L1 n :

Proof. Let Rn 1 = R0 (OZ ()

Wn 1). Since the restriction of OZ ()

0

Wn 1 to each ber is a semi-stable bundle of degree n, Rn 1 is a vector

0

bundle of rank n over B.

The relative dualizing sheaf for Z==B is L and R1 L = OB . Thus,

relative Serre duality is a map

S: R1 (OZ ( )

Wn 1 ) ! R0 (OZ ()

Wn 1

L 1)

R1 L = Rn 1

L 1:

0

Consider the composition of S with the map

(Rn 1 )

L 1 A n 1 Rn 1

det(Rn 1 ) 1

L 1

V

!

0 0 0

ev

Id

Id 0

! R (det(OZ ()

Wn 1))

det(Rn 1) 1

L 1

0

= R0 (OZ (n))

L(n 1)(n 2)=2

det(Rn 1 ) 1

L 1 ;

0

where the map A is induced by taking adjoints from the natural pairing

n1

^

n 1

Rn 1 ! det(Rn 1 );

R0 0 0

and ev is the map

n1 n1

^

0 ( ^ O

(OZ (

Wn 1 )!R Z ()

Wn 1 )

R0

ev:

obtained by evaluating sections. Clearly, both 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 Proposition 3.13 in Vector Bundles over Elliptic Fibrations.

Assuming this result, we see that the vector bundle we are interested in

computing diers from R0 (OZ (n) by twisting by the line bundle L 1

0

detRn 1 .

According to Claim 4.2.1 R0 (OZ (n)) splits as a sum of line bundles

O L 2 L 3 L1 n. Now to complete the evaluation of R1(Wd

Wn d) we need only to compute the line bundle detR0(OZ (n)

Wn 1).

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Holomorphic bundles over elliptic manifolds

Claim 4.3.6 detR0(OZ (n)

Wn 1) is equal to L(n 2)(n 1)=2 2 .

Proof. In computing the determinants we can assume that all sequences

split. This allows us to replace Wn 1 by OZ () L L2 Ln 2 . Since

OZ (2) sits in an exact sequence

0 ! OZ () ! OZ () ! OZ ()j ! 0

and since R0 OZ ()) = L and R0 OZ ()j = L 1 , and R0 (OZ ()

La ) = La 1, the result follows easily.

Putting all this together we see that

R1 (B; OZ ( )

Wn 1 ) = R0 (B; OZ (n))

L:

This completes the proof of Lemma 4.3.5

Now we are ready to complete the proof of Proposition 4.3.4. This is

done by induction on d. The case d = 1 is exactly the case covered by

Lemma 4.3.5. Suppose inductively that we have established the result for

Wd

Wn d for some d 1. We consider the commutative diagram

0 0 0

? ? ?

? ? ?

y y y

! Wd

W n ! Wd+1

Wn ! L d

Wn !0

0 d1 d1 d1

? ? ?

? ? ?

y y y

! Wd

?Wn ! Wd+1

Wn ! L d

Wn !0

0 d d d

? ?

? ? ?

y y y

L

! Wd

? 1+d ! Wd+1

L1+d ! L d

? 1+d !0

n n n

L

0

?

? ? ?

y y y

0 0 0

The natural maps R1 (Wd+1

L1+d n ) ! R1 (L1 d

L1+d n ) and

R1(L d

Wn d ) ! R1 (L d

L1+d n) are both isomorphisms. It follows

that the images of R1 (Wd+1

Wn d 1 ) and of R1 (Wd

Wn d ) in

R1 (Wd+1

Wn d) are equal to the kernel of the natural map R1 (Wd+1

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Holomorphic bundles over elliptic manifolds

Wn d ! R1(L d

L1+d n). Since all the bundles in question are semi-

stable and of negative degree on each ber, they all have trivial R0 . Thus

the maps R1 (Wd+1

Wn d 1 ) and R1 (Wd

Wn d ) to R1 (Wd+1

Wn d) are injections. It follows that R1 (Wd+1

Wn d 1) and R1(Wd

Wn d) are identied This completes the inductive step and hence the proof

of the theorem.

4.4 The parabolic construction of vector bundles regular and

semi-stable with trivial determinant on each ber

Let Z ! B be a family of Weierstrass cubic curves with the section

at innity. Fix a line bundle M on B and sections ti of L i

M for i =

0; 2; 3; 4; : : : ; n. Supposing that there is no point of B where all these sections

vanish we can construct a vector bundle as follows.

The identication of Ext1 (Wn d ; Wd ) with L L 1 L 2 L1 n

can be twisted by tensoring with M so as to produce an identication of

Ext1 (Wn d ; Wd

M) with M

L M

L 1 M

L1 n . Thus,

the sections ti determine an element of Ext1 (Wn d ; Wd

M) and hence

determine an extension

0 ! Wd

M ! V ! Wn d ! 0:

Since we are assuming that not all the sections ti vanish at the same point

of B, the restriction of V to each ber is a non-trivial extension of Wn d by

Wd. Thus, the restriction of V to each ber is in fact semi-stable, regular

and with trivial determinant.

This parabolic construction thus produces one particular vector bun-

dle associated with each line bundle M on B and each non-zero section of

R0 (OZ (n))

M. This bundle is automatically regular and semi-stable on

each ber and has trivial determinant on each ber. Conversely, given the

bundle regular and semi-stable and with trivial determinant of each ber, it

determines a section of the projective bundle Pn ! B, to which we can ap-

ply the parabolic construction. The result of the parabolic construction may

not agree with the original bundle { but they will have isomorphic restric-

tions to each ber. Thus, they will dier by twisting by a line bundle on the

spectral covering corresponding to the section. That is to say to construct

all bundles corresponding to a given section we begin with the one produced

by the parabolic construction. The section also gives us a spectral covering

T ! B. We are then free to twist the bundle constructed by the parabolic

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Holomorphic bundles over elliptic manifolds

construction by any line bundle on T, just as in the spectral covering con-

struction. Thus, the moduli space of bundles that we are constructing bers

over the projective space of H 1 (Z; Wd

Wn d ) with bers being Jacobians

of the spectral coverings T ! B produced by the section. This twisting cor-

responds to nding all bundles which agree with the given one ber-by-ber.

By general theory all such bundles are obtained by twisting with the sheaf

of groups H 1 (B; (Aut(V; V)).

4.5 Exercises:

1. Show that a Weierstrass cubic has at most one singularity, and that is

either a node or a cusp. Show that the cusp appears only if g2 = g3 =

0. Show that the node appears when (g2 ; g3 ), as dened in the lecture,

vanishes. Show that the point at innity is always a smooth point.

2. Show that for any Weierstrass cubic the usual geometric law denes a

group structure on the subset of smooth points with the point at innity be-

ing the origin for the group law. Show that this algebraic group is isomorphic

to C if the curve is nodal 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. Describe the singularities of Z B B Z at the nodes and cusps of

Z==B.

6. Show that if V ! Z is a vector bundle and for each ber Eb of Z==B we

have H i (Eb ; V jEb ) is of dimension k, show that Ri (V ) is a vector bundle

of rank k on B.

7. Let M be a line bundle over B and let V t in an exact sequence

0 ! Wd

M ! V ! Wn d ! 0:

Compute the Chern classes of V.

8. Show that if V and U are vector bundles over a smooth variety, then

Ext1 (U; V ) = H 1 (U

V ). V

9. Show that if V is a rank n vector bundle then n 1 V is isomorphic to

V

det(V ).

10. State relative Serre duality and show that it is correctly applied to

produce the map S given in the proof of Lemma 4.3.5.

11. Suppose that V is a vector bundle. Show that to rst order the defor-

mations of V are given by H 1 (Hom(V; V )).

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Holomorphic bundles over elliptic manifolds

5 The Global Parabolic Construction for Holomor-

phic Principal Bundles

In this section we wish to generalize the parabolic 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 GC .

This construction will produce holomorphic principal bundles on the total

space Z of the family of Weierstrass cubics which have the property that

they are regular semi-stable GC-bundles on each ber of Z==B. Of course,

this construction can also be viewed as a generalization of the construction

given in the third lecture for a single elliptic curve. It is important to note

that we do not give an analogue of the spectral covering construction for

GC-bundles. We do not know whether such a construction exists for groups

other than SLn (C) and Sympl(2n).

5.1 The parabolic construction in families

We let Z ! B be a family of Weierstrass cubics with section : B ! Z as

before. Let GC be a simply connected simple group. Fix a maximal torus

and a set of simple roots for G, and let P G be the distinguished maximal

parabolic subgroup with respect to these choices. Then the Levi factor L

of P is isomorphic to the subgroup of a product of general linear groups

Qs

i=1 GLni consisting of matrices with a common determinant. The charac-

ter group of P and of L is Z and the generator is the character that takes

the common determinant. We consider the bundle Wn1 Wns . This

naturally determines a holomorphic principal L-bundle L over Z. Viewed as

a bundle over GC it is unstable since the GC -adjoint bundle associated with

this L bundle splits into three pieces: the adjoint ad(L ) of the L-bundle,

the vector bundle associated with the tangent space to the unipotent radical

U+ (L ) and the vector bundle associated to the root spaces negative to those

in U+ , U (L ). The rst bundle has degree zero, the second has negative

degree and the third has positive degree. The degree of the entire bundle is

zero. This makes it clear that ad(L L GC) is unstable, and hence according

to our denition that C L GC is an unstable principal GC-bundle. (Notice

that the L-bundle is stable as an L-bundle.)

Once again we are interested in deformations of L to P-bundles with

identications =U = L . Just as in the case of a single elliptic curve, these

deformations are classied by equivalence classes of twisted cocycles, which

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Holomorphic bundles over elliptic manifolds

we denote by H 1 (Z; U(L )). Recall that U is ltered 0 Un Un 1

U1 = U where the center C acts on U and has homogeneous weight i on

Ui =Ui 1 . Furthermore, Ui =Ui 1 is abelian and hence is a vector space which

lies in the center of U=Ui 1 . Thus, once again we can lter the cohomology

by H 1 (Z; Ui (L )) with the associated gradeds being ordinary cohomology

of vector bundles H 1 (Z; Ui =Ui 1 (L )). Since det(L ) as measured with re-

spect to the generating dominant character of P is negative, it follows that

R0 (Ui =Ui 1 (L)) is trivial for all i. A simply inductive argument then

shows that R0 (U(L )) is a bundle of zero dimensional ane spaces over

B, and hence H 0 (Z; U(L )) has only the trivial element.

A similar inductive discussion shows that R1 (U(L )) is ltered with

the associated gradeds being the vector bundles R1 (Ui =Ui 1 (L )). This

implies that R1 (U(L )) is in fact a bundle of ane spaces over B, with

a distinguished element { the trivial cohomology class on each ber. The

local-to-global spectral sequence, the vanishing of the R0 (U(L )) and an

inductive argument shows that in fact the cohomology set H 1 (Z; U(L )) is

identied with the global sections of R1 (U(L )) over B.

5.2 Evaluation of the cohomology group

In all cases except G = E8 and over the cuspidal bers we can in fact split

the bundle R1 (U(L )) of ane spaces so that it becomes a direct sum of

vector bundles. Under this splitting the C action becomes linear.

Theorem 5.2.1 Let G be a compact simply connected, simple group and let

Z ! B be a family of Weierstrass cubic curves. Assume either that G is not

isomorphic to E8 or that no ber of Z==B is a cuspidal curve. Then there

is an isomorphism R1 U(L ) with a direct sum of line bundles i L1 di

where d1 = 0 and d2 ; : : : ; dr are the Casimir weights associated to the group

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 bundle.

Corollary 5.2.2 The cohomology H 1 (Z; U(L )) is identied with the space

of sections of a sum of line bundles over B, and hence the space of extensions

is identied with a bundle of weighted projective spaces over B. The bers

are weighted projective spaces of type P(g0 ; g1 ; : : : ; gr ) where g0 = 1 and for

i = 1; : : : ; r the gi are the coroot integers.

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Holomorphic bundles over elliptic manifolds

Here is a table of the Casimir weights grouped by C -weights

Group 1 2 3 4

An 0; 2; 3; : : : ; n

Bn 6; 8; : : : ; 2n

0; 2; 4

Cn 0; 2; 4; : : : ; 2n

Dn 0; 2; 4; n 6; 8; : : : ; 2n 2

E6 0; 2; 5 6; 8; 9 12

E7 0; 2 6; 8; 10 12; 14 18

G2 0; 2 6

F4 0; 2 6; 8 12

Thus, with this choice of splitting for the unipotent cohomology, a choice

of a line bundle M over B and sections ti of M

L1 di will determine a

section of R1 (U(L )) and a P-bundle over Z deforming the original L-

bundle L . By construction there will be a given isomorphism from the

quotient of the deformed bundle modulo the unipotent subgroup back to

L. Furthermore, if the sections ti never all vanish at the same point of

B, then the resulting P-bundle will extend to a GC-bundle which is regular

and semi-stable on each ber of Z==B. The resulting section of the weighted

projective space bundle is equivalent to the data of the S-equivalence class of

the restriction of the principal GC-bundle to each ber of Z==B. Of course,

since these bundles are regular, it is equivalent to the isomorphism class of

the restriction of the GC-bundle to each ber.

5.3 Concluding remarks

Thus, for each collection of sections we are able to construct a GbfC -bundle

which is regular semi-stable on each ber. The study of all bundles which

agree with one of this type ber-by-ber is more delicate. From the parabolic

point of view, it requires a study of the sheaf R1 (Aut()) which can be

quite complicated, and is only partially understood at best.

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 ber-by-ber. Then one would like to complete the

space of bundles by adding those which become unstable on some bers

(but remain semi-stable on the generic ber). Finally, to complete the space

it is surely necessary to add in torsion-free sheaves of some sort. All these

issues are ripe for investigation { little if anything is currently known.

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Holomorphic bundles over elliptic manifolds

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, classication, moduli

spaces are the main sources of interest. The study we have been describing

here ts perfectly in that pattern. Nevertheless, from my point of view, there

is another completely dierent motivation for this study. That motivation

is the connection with other dierential 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 polynomial

invariants of these algebraic surfaces. Thus, the study described here can be

used to compute the Donaldson invariants of elliptic surfaces. These were

the rst such computations of those invariants, see [5].

More recently, there has been a connection proposed, see [9], between

algebraic n-manifolds elliptically bered over a base B with E8 E8 -bundle

and algebraic (n + 1)-dimensional manifolds bered over the same base with

ber an elliptically bered K3 with a section. The physics of this later set-

up is called F-theory. The precise mathematical statements underlying this

physically suggested correspondence are not well understood yet, and this

work is an attempt to clarify the relationship between these two seemingly

disparate mathematical objects. All the evidence to date is extremely posi-

tive { the two theories E8 E8 -bundles over families elliptic curves line up

perfectly as far as we can tell with families of elliptically bered K3-surfaces

with sections over the same base. Yet, there is still much that is not under-

stood in this correspondence. Sorting it out will lead to much interesting

mathematics around these natural algebro-geometric objects.

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Holomorphic bundles over elliptic manifolds

References

[1] Adams, J., Exceptional Lie Groups,

[2] Borel, A. Sous-groupes commutatifs et torsion de groupes de Lie com-

pacts connexes, Tohoku Journal, Ser. 2 13 (1961), 216-240.

[3] Bourbaki, N., Elments de Mathmatique, groupes et algbres de Lie,

e e e

Chapt. 4,5,6. Hermann, Paris, 1960 -1975.

[4] Donaldson, S., Antiselfdual Yang-Mills connections on complex algebraic

surfaces and stable bundles, Proc. London Math. Soc. 3 (1985), 1.

[5] Friedman, R. and Morgan, J., Smooth Four-Manifolds and Complex Sur-

faces, Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Folge, vol.

27, Springer-Verlag, Berlin, Heidelberg, New York, 1994. Friedman

[6] Looijenga, E., Root systems and elliptic curves, Invent. Math. 38 (1997),

17.

, Invariant theory for generalized root systems, Invent. Math. 61

[7]

(1980), 1.

[8] Narasimhan, M. and Seshadri, C., Deformations of the moduli 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.

Degenerations of the moduli spaces of

vector bundles on curves

C.S. Seshadri

Chennai Mathematical Institute,

92 G.N. Chetty Road, Chennai-600 017, India

ñòð. 7 |