Proof. Because we are working on a Riemann surface, each bundle has

a meromorphic section. (As we mentioned earlier, every bundle over

any projective manifold always has a meromorphic section.) Let s be

a non-zero meromorphic section of V . Let s have a pole of order nj at

pj . Consider the divisor D = l nj [pj ] and let LD be a line bundle

i=1

which has a holmorphic section σ such that div(σ) = D. Then the

bundle V — LD has a holomorphic section s — σ.

For us it is enough to prove our lemma for V — LD , because after-

wards we can tensor every bundle by L’D . So, we may assume that V

has a global holomorphic section s. Next we look at the zeros of s and

they de¬ne a divisor D = p vp (s)[p]. We observe that furthermore

s de¬nes a nowhere vanishing holomorphic section of L’D — V . Thus

we have a subbundle F ‚ L’D — V spanned by s and hence an exact

sequence 0 ’ F ’ L’D — V ’ Q ’ 0.

Note that in the course of proving Lemma 3.3 we have established

the following:

Fact. If V is holomorphic vector bundle over CP1 then there exists a

divisor D such that “(V — LD ) = 0. Hence there exists an integer N

such that for any n ≥ N one has “(V — L—n ) = 0 simply because one

may take N = deg(D).

COROLLARY 2.1.4 For every holomorphic vector bundle V over a

compact Riemann surface there exists a ¬‚ag of vector subbundles 0 ‚

V1 ‚ · · · ‚ Vr = V , where Vj has rank j.

Proof. Let us say r = 3, then by the above lemma one has an exact

p

sequences 0 ’ V1 ’ V ’ Q ’ 0 and 0 ’ L1 ’ Q ’ R ’ 0. Take

V2 = p’1 (L1 ), then the sequence 0 ’ V1 ’ V2 ’ L1 ’ 0 is exact and

we have our ¬‚ag.

36 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

COROLLARY 2.1.5 (Vanishing of cohomology.) There exists an in-

teger N such that for any n ≥ N one has H 1 (CP1 , V — L—n ) = 0.

Proof. Using exact sequences one observes that if for all j one has

H 1 (CP1 , (Vj /Vj’1 ) — L—n ) = 0

then H 1 (CP1 , V —L—n ) = 0 too. It can be seen using the exact sequence

H 1 (CP1 , Vj’1 —L—n ) ’ H 1 (CP1 , Vj —L—n ) ’ H 1 (CP1 , (Vj /Vj’1 )—L—n ) = 0

and induction by j.

The major problem we are facing is that if we have an extension of

holomorphic vector bundles 0 ’ L ’ E ’ Q ’ 0 for a line bundle

ˇ

L, this extension need not be split. We will attach a Cech cohomology

class to the extension. First we consider the dual extension : 0 ’ Q— ’

E — ’ L— ’ 0. The cohomological invariant of the extension which we

denote by κ lies in H 1 (CP1 , L — Q— ). And if κ = 0 then the extension

is split, i.e. E L • Q. We describe the construction of κ. Let us

tensor L with the dual sequence. We obtain another exact sequence of

holomorphic vector bundles 0 ’ Q— — L ’ E — — L ’ 1CP1 ’ 0 which

amounts to the cohomology exact sequence

‚

0 ’ H 0 (Q— — L) ’ H 0 (E — — L) ’ H 0 (1CP1 ) ’ H 1 (Q— — L).

Using the isomorphism H 0 (1CP1 ) C let us de¬ne κ = ‚(1). Now

assume κ = 0. There exists a non-vanishing holomorphic section σ of

E — —L which maps to the section 1 of 1CP1 . Thus E — —L = (Q— —L)•F ,

where F is a line subbundle of E — — L spanned by σ (hence trivial). So

the sequence is split.

Proof of the theorem. We start with the case r = 2. Let N =

max(deg div(s)), where s ranges over the set of non zero meromor-

phic sections of V . (We know that N < ∞ because N has the property

that H 0 (CP1 , V — L—’m ) = 0 for all m > N and N is just the mini-

mal number having this property.) Then as we saw we have an exact

sequence of vector bundles over CP1 : 0 ’ L—N ’ V ’ L—b ’ 0

for some integer b. If this sequence is split we are done and if it is

not split then it follows that κ = 0. This implies in particular that

ˇ

2.1. CECH COHOMOLOGY FOR VECTOR BUNDLES 37

H 1 (CP1 , L—’b — L—N ) = H 1 (CP1 , L—N ’b ) = 0. It means that N ¤ b ’ 2.

We will show that in fact H 0 (V — L—’N ’1 ) = 0 and that leads to an

immediate contradiction. After tensoring the sequence with L—’N ’1

we get a sequence 0 ’ L’1 ’ V — L—’N ’1 ’ L—b’N ’1 ’ 0. ¿From

the long exact sequence in cohomology we conclude that

0 = H 0 (L—’1 ) ’ H 0 (V — L—’N ’1 ) H 0 (L—b’N ’1 ) ’ H 1 (L—’1 ) = 0.

Because of the fact that b ’ N ’ 1 ≥ 1 we got the desired result. So,

V = L—N • L—b . In the general rank case we have an exact sequence

like 0 ’ L—N ’ V ’ Q ’ 0. We make the inductive assumption that

Q = L—b1 • · · · • L—br’1 for some integers b1 ≥ b2 ≥ · · · ≥ br’1 . If the

extension is not split then H 1 (Q— — L—N ) = 0 where the number N is

de¬ned as before. The splitting of Q into a sum of line bundles allows

us to conclude that H 1 (L—N ’bj ) = 0 for some j and hence N ¤ bj ’2 ¤

b1 ’ 2. By the method analogous to the one used in the case r = 2 one

may conclude that H 0 (V — L—’N ’1 ) = 0 and that again implies the

contradiction with the assumption of the sequence not being split.

So, now if V = L—a1 • · · · • L—ar with a1 ≥ · · · ≥ ar then we intend

to show that the sequence (a1 , ..., ar ) is unique. by L—’ar +1 with no

harm to our arguments. We claim that the sequence (a1 , ..., ar ) may

be uniquely recovered from the function n ’ dim H 0 (E — L—n ). More

precisely, n < ’ar if and only if f (n) ’ f (n ’ 1) < r. We have the

following formula for f (n) in which H(x) denotes the Heaviside function

H(x) = 1 if x ≥ 0, H(x) = 0 if x < 0.

r

f (n) = H(n + aj )(n + aj + 1)

j=1

from which we see that f (n) ’ f (n ’ 1) is the cardinality of the set of

j™s such that n ≥ ’aj .

The sequence (a1 , ..., ar ) is a holomorphic invariant of V . The only

two topological invariants of V are of course r and a1 + a2 + · · · + ar .

The above theorem says that in terms of our matrix-valued transi-

tion function g01 : U01 = C— ’ GL(n, C) we may assume that

« a1

z 0 ··· 0

¬0 ··· 0 ·

z a2

g01 = ¬ ·

¬ ·

· · ··· ·

· · · z ar

0 0

38 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

This matrix-valued function for the vector bundle E gives the compari-

son between trivializing frames (e1 , · · · , er ) over U0 and (f1 , · · · , fr ) over

U1 . More precisely we have the equality of row-vectors of section of E:

(f1 , · · · , fr ) = (e1 , · · · , er )g01

over U01 . The ¬rst version of this theorem which was proved by Birkho¬

said that if M = M (z) : C— ’ GL(n, C) is a matrix valued holomorphic

function then there exist entire functions A(z), B(z) : C ’ GL(n, C)

and integers (a1 , ..., ar ) such that

« a1

z 0 ··· 0

¬0 ··· 0 · 1

z a2

M (z) = A(z) ¬ ·

· B( ).

¬

· · ··· · z

· · · z ar

0 0

The matrix-valued functions A(z) and B(z) represent changes of trivial-

ization over U0 and U1 and this decomposition also appears as Birkho¬-

Bruhat decomposition for loop groups (see the book Loop Groups [47]

by Pressley and Segal).

2.2 Extensions of vector bundles

Given a holomorphic vector bundle F over CP1 and a non-zero class

ˇ

κ ∈ H 1 (CP1 , F ) there always exists a vector bundle extension

p

i

0 ’ F ’ E ’ 1CP1 ’ 0

ˇ

such that the corresponding class in H 1 (CP1 , F ) is equal to κ. Further-

more this extension is non-trivial meaning that it is not split.

In general, if we have a sequence as above for an arbitrary manifold

X, we can pick an open covering (Ui ) of X and a holomorphic section

si of E over Ui such that p(si ) = 1. Then fij = sj ’ si is a holomorphic

section of E over Uij and it is a 1-cocycle. If it de¬nes zero cohomology

class, then fij = σj ’σi for σi ∈ “(Ui , F ). It follows from sj ’si = σj ’σi

that sj ’ σj de¬ne a global section s of E as they agree on intersections

and such that p(s ) = 1. In this case we have managed to split the

sequence and E = F • 1, where 1 is spanned by s .

2.2. EXTENSIONS OF VECTOR BUNDLES 39

Let us make an explicit cocycle construction of bundle extension.

A vector bundle F of rank r over X is given by transition unctions

gij : Uij ’ GL(r, C) that satisfy the cocycle condition gij gjk = gik .

For an open set V ‚ X a section of F over V is a family (vi ) of

functions V © Ui ’ Cr such that over the intersection Uij one has

ˇ

vi = gij vj . . Any class in H 1 (X, F ) may be described by a 1-cocycle

hij : Uij ’ Cr using the trivialization over Ui . The cocycle condition

means that hik = hij + gij hjk , where the multiple gij stands for the

change of trivialization. Now the matrix-valued function

gij hij

Mij : Uij ’ GL(r + 1, C) : Mij =

0 1

clearly satis¬es Mik = Mij Mjk and hence de¬nes a holomorphic rank

two vector bundle E over X. Because the matrices Mij are upper-

triangular we have an inclusion of the vector bundle F into E (corre-

sponding to the upper left corner of the transition matrices Mij . The

presence 1 in the lower right corner indicates that the quotient line bun-

dle is trivial: 0 ’ F ’ E ’ 1 ’ 0. The corresponding cohomology

ˇ

class in H 1 (X, F ) is the class of the cocycle hij and when it is not a

coboundary, the extension is not split.

ˇ

Example. T — CP1 = L—’2 and hence H 1 (CP1 , T — CP1 ) is one-dimensional

and there is a distinguished non-zero element κ in this group with the

corresponding non-trivial bundle extension 0 ’ T — CP1 ’ P ’ 1 ’ 0.

The rank two bundle P is called the bundle of order 1 jets of holomor-

phic functions. As we know from the Birkho¬-Grothendieck theorem

the bundle P can be decomposed as L—a • L—b with a ≥ b. Those num-

bers can be determined from the condition that H 0 (P — L—’a ) = 0, but

H 0 (P — L—’a’1 ) = 0. Let us take a = 0 ¬rst. We get from the short

exact sequence of bundles the “long” sequence like

‚

ˇ

0 = H 0 (T — CP1 ) ’ H 0 (P) ’ (H 0 (1) = C) ’ H 1 (T — CP1 ).

The map ‚ is injective because of the assumption ‚(1) = κ = 0, so we

conclude that H 0 (P) = 0. Another “long” exact sequence correspond-

ing to the previous short sequence multiplied by L is

0 = H 0 (T — CP1 — L) ’ H 0 (P — L) H 0 (L) ’ H 1 (T — CP1 — L) = 0.

40 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

So, we have dim H 0 (P — L) = 2, because dim H 0 (L) = 2. It follows

that a = ’1. Besides b = ’1 too because the magnitude of the jump

in dimension is 2.

Another interesting example is an example of 1-parameter family

of rank 2 holomorphic vector bundles over CP1 parametrized by » and

expressed in terms of the transition matrices as

gij »hij

E» = .

0 1

For » = 0 the bundle E» is always the same

L—’1 • L—’1 .

E» E1 = P

But when » = 0 then clearly E0 = L—’2 • 1. This is an example of

so-called degeneration of bundles. More precisely, we can construct a

holomorphic vector bundle E over CP1 —C using a parameter » on C. We

use the open covering of CP1 — C by the Ui — C. The transition matrices

gij »hij

then describe E. When we restrict to CP1 — {»} ‚ CP1 — C

0 1

we get the bundle E» .

Let us now take a compact Riemann surface X and let S be a ¬nite

number of marked points in X. Then the manifold X \ S is Stein.

Let us further take two non-empty and non-intersecting ¬nite sets of

marked points S0 and S1 and let U0 = X \ S0 and U1 = X \ S1 . We

ˇ

want to construct the natural bijection H 1 (U, T — X) ’ C. Any 1 -

cocycle is a holomorphic 1 - form ω over U01 . Let us make an extra

fairly little assumption that ω is meromorphic on X. Then our map is

ω ’ l(ω) = p∈S0 Resp (ω). Next we show that this map gives rise to

a well-de¬ned map in cohomology. We have to establish the fact that

all coboundaries are in the kernel of this map. If ω is a coboundary

then over U01 one has ω = ω1 ’ ω0 , where ωi is a holomorphic 1 -

form over Ui for i = 0, 1. Then one makes a simple observation that

l(ω1 ) = p∈S0 Resp (ω1 ) = 0, because of holomorphicity of ω1 over U1 .

Furthermore

l(ω0 ) = Resp (ω0 ) = Resp (ω0 ) = 0,

p∈S0 p∈X

2.3. COHOMOLOGY OF PROJECTIVE SPACE 41

due to the well-known fact that the sum of all residues of a meromorphic

1-form is 0. This implies that l induces a linear map

ˇ

H 1 (X, T — X) ’ C,

which is in fact an isomorphism. In the case of CP1 the generator of

ˇ

the cohomology group H 1 (CP1 , T — CP1 ) is given by the 1-form ’dz/z.

2.3 Cohomology of projective space

We will start our computations of cohomology groups of projective

space CPn with coe¬cients in a line bundle L—m (m ∈ Z) with recollec-

ˇ

tion of some classical results concerning Cech cohomology groups.

THEOREM 2.3.1 (H. Cartan, J.-P. Serre). Given a compact