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vector bundles 0 ’ E ’ V ’ Q ’ 0, where E is a line bundle.

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

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