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4. Show that there is a unique non-trivial extension I2 of OE by OE . Show
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 de ned by sending y to the point of X corresponding to
the equivalence class of Vy is a holomorphic mapping. Show that if a coarse
172
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 in nity in this
ane model is a polynomial expression in x and y. Show that a meromorphic
function with pole only at in nity which is invariant under e 7! e is a
polynomial expression in x.
11. Let g: T ! jnp0 j be the n-sheeted rami ed 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 di erent 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 satis es 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 identi ed
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 .
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 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 satis es 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 identi ed 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 di erent d < n that the
above projective spaces can be identi ed 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 di erent set of bundles { namely the direct sum of line bundles. These
families agree generically, but di er 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 di erent 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 de nition a parabolic subgroup is a
connected complex proper subgroup of GC that contains a Borel subgroup.
Up to conjugation parabolic subgroups of GC are classi ed by proper (and
possibly empty) subdiagrams of the Dynkin diagram of G. Fix a maximal
torus of GC and a set of simple roots f 1 ; : : : ; n g. A subdiagram is given
simply by a subset f 1 ; : : : ; 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 f 1 ; : : : ; 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 identi ed 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 identi ed 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 f 1 ; : : : ; 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.
185
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
186
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 identi ed 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 di erent proof of Looijenga's theorem. It identi es
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;
187
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 identi ed 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 di er 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 )).
188
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 de nes a cubic curve in the projective plane with homogeneous
coordinates (x; y; z). The point at in nity, 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 in nity 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 de nitions
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
189
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 de nes 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 de nition 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 de ne 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,  T bers , is isomorphic to L 1 . Also important

for us will be the relative dualizing sheaf. It is  T bers . (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 compacti cation 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 identi ed
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 de nition 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:
191
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 identi es 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. De ne 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 rami ed 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 rami ed 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
192
Holomorphic bundles over elliptic manifolds

rami ed 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 de ne 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 identi ed 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) ! :
193
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
194
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 identi ed 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 .
195
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 di ers 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).
196
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

197
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 identi ed 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 in nity. 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 identi cation 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 identi cation 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 di er 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
198
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 de ned in the lecture,
vanishes. Show that the point at in nity is always a smooth point.
2. Show that for any Weierstrass cubic the usual geometric law de nes a
group structure on the subset of smooth points with the point at in nity 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 )).
199
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 de nition 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
identi cations =U = L . Just as in the case of a single elliptic curve, these
deformations are classi ed by equivalence classes of twisted cocycles, which
200
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
identi ed 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 identi ed with the space
of sections of a sum of line bundles over B, and hence the space of extensions
is identi ed 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.
201
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.
202
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, classi cation, 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 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 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.
203
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

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