df df df df

( z + z )+( z + z )=

f f f f

γ1 γ3 γ2 γ4

i i i i

df df θ df df

1 1 θ

=( z’ (z + θ) ) + ( z ’ (z + 1) )

f f f f

0 0 0 0

√ √ √

= θ2π ’1n + 2π ’1m ∈ 2π ’1Λ,

for some integers m and n, because in the last two integrals it is like

we integrate over a loop in the torus, yielding the winding number of f

√

around that loop, multiplied by 2π ’1. Besides, we used the residue

theorem in the ¬rst equality. We conclude that i zi vpi (f ) ∈ Λ, and

projecting this to X we have i pi vpi (f ) = 0.

Step 2: Ker(A) ‚ P (X). What we shall show is that if D = i mi [pi ]

is a divisor on X of degree 0 and such that i mi pi = O in X then

there exists a meromorphic function f on X such that div(f ) = D. The

right tool for this is the so-called theta function on C. By de¬nition,

√ √

+∞

n (n+1/2)2

θ(z) = ’1 (’1) q exp((2n + 1)π ’1z),

’∞

√

where q = exp(2π ’1θ) and hence |q| < 1. This beautiful holomorphic

function has deep geometric and arithmetic signi¬cance. Note that the

2

convergence of the series is easy to verify as |q (n+1/2) | = exp(’2π(n +

1/2)2 Im„ ) decreases very rapidly as n ’ ∞. The theta function θ(z)

has many nice properties some of which we list here. The theta function

is quasi doubly periodic, meaning that θ(z + 1) = θ(z) and θ(z + θ) =

√

q ’1 exp(’2π ’1z)θ(z). In addition, this function is odd and hence

θ(0) = 0, also the equality θ(z) = 0 implies that z ∈ Λ (so θ(z) does

not have zeros (or poles) inside the fundamental parallelogram P ).

Now having a divisor D = i mi [pi ] of degree 0 and such that

i mi pi = O on X we can pick representatives zi of pi on C such that

i mi zi = 0. De¬ne a function

θ(z ’ zi )mi .

f (z) =

i

Claim. The function f (z) is doubly periodic.

30 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES

It is immediately clear that f (z + 1) = f (z). Further, due to the

√

identity θ(z ’ zi + θ) = q ’1 exp(’2π ’1(z ’ zi ))θ(z ’ zi ) we get

√

f (z + θ)

= q Σi mi exp(’2π ’1( mi z ’ mi zi )) = 1.

f (z) i i

So, f (z) projects to a meromorphic function on X and, moreover,

div(f ) = i mi [pi ], as required.

In fact, for any compact Riemann surface of genus g we have P ic(X)

Z • J(X), where J(X) is a complex torus of dimension g. Moreover,

J(X) is always a projective manifold. In general, P ic(X) = (discrete

group) • (connected complex Lie group).

Next, recall that for X compact we have a map Cl(X) ’ H2d’2 (X, Z),

where d stands for the complex dimension of X. When X is non-

compact this map no longer exists, but instead we have a map Cl(X) ’

ˇ

H 2 (X, Z) and we give a Cech homology description of a cocycle repre-

senting this class in H 2 (X, Z).

The map P ic(X) ’ H 2 (X, Z) can be described for smooth line

bundles. Assume that we have a good cover X = ∪i Ui of X and a line

ˇ

bundle L over X with transition cocycle gij : Uij ’ C— . A Cech class

cijk ∈ Z is a cocycle if the identity cjkl ’ cikl + cijl ’ cijk = 0 holds. We

ˇ

can write gij = exp(hij ) on Uij , where hij : Uij ’ C is a Cech cocycle.

The condition on transition cocycles implies that exp(hij ’hik +hjk ) = 1

√

or (2π ’1)’1 [hij ’ hik + hjk ] ∈ Z. So, we de¬ne this to be cijk and

it represents a class in H 2 (X, Z). To see that it is well-de¬ned, we

change hij to hij + mij , where mij ∈ Z. One sees that cijk will change

ˇ

to cijk + (‚m)ijk , which is the sum of cijk and the Cech coboundary of

m. In the commutative diagram

Cl(X) P ic(X)

H 2 (X, Z)

the map on the left sends a divisor to its cohomology class and the map

on the right is de¬ned as above using a transition cocycle.

Chapter 2

Cohomology of vector

bundles

Making somebody learn unnecessary

things is as harmful as feeding him

sawdust. Bernard Shaw

ˇ

2.1 Cech cohomology for holomorphic vec-

tor bundles and applications.

One would like to consider a problem of classi¬cation of all holomorphic

vector bundles over CP1 . We denote by “(X, V ) = “(V ) the space of

global holomorphic sections of a holomorphic vector bundle V over a

complex manifold X. It is a module over the algebra H(X) of holo-

morphic functions on X. Let us have an exact sequence of holomorphic

vector bundles over X: 0 ’ S ’ E ’ Q ’ 0. Contrary to the smooth

case, such a sequence as we saw before is not always split, and if we

take the corresponding spaces of global sections then we get a shortened

version of the previous exact sequence

±

0 ’ “(X, S) ’ “(X, E) ’ “(X, Q),

where the map ± is not necessarily surjective. In these cases people say

that the global section functor is left exact, which is a weaker condition

31

32 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

than exact. We have to require some additional properties from X to

guarantee that the above sequence may be continued to the right by

zero that is the surjectivity of ±. For instance, it happens if X is a so-

called Stein manifold. (By a theorem of Cartan and Oka.) We de¬ne

the notion of Stein manifold later and now we just say that the main

example of Stein manifolds is given by the common vanishing locus X

in Cn of a ¬nite number of holomorphic functions in n variables. If

all of the functions are actually polynomials then X is called an a¬ne

algebraic variety. It is a general fact that a closed complex submanifold

of a Stein manifold is again Stein.

If we have a connected open set U ‚ Cn then U is Stein if and only

if U is a domain of holomorphy. It means that for any point p ∈ ‚U and

for any neighbourhood V ‚ Cn of p there exists a holomorphic function

on U which does not extend to a holomorphic function on U ∪ V .

For instance, every convex open U is a domain of holomorphy. More

generally, if Y ‚ Cn is a complex analytic subvariety and U is open in

Y , then U is Stein if and only if U inside Y is a domain of holomorphy.

Examples of Stein manifolds involve C n itself, C— , (C— )k — Cn . More

generally, any product of Stein manifolds is again Stein. Any open

subset of Cn given as the non-vanishing locus of a holomorphic function

is also an example of Stein manifold. We saw before that for n ≥ 2,

the space Cn \ { origin } is not Stein.

ˇ

Now if X is any complex manifold one can de¬ne a Cech cohomology

for an open covering U = (Ui ) of X. This covering will have good

properties i f each of Ui are Stein. (Then any intersection Ui0 i1 ...ip is

Stein too as it is a closed complex submanifold of the product manifold

ˇ

Ui0 — · · · Uip .) We de¬ne a Cech cochain complex for a holomorphic

vector bundle V which will de denoted

p

ˇ ˇ

‚

· · · ’ C p (U, V ) ’ C p+1 (U, V ) ’ · · · .

ˇ ˇ

The space C p (U, V ) is the space of Cech p-cochains. A p-cochain is a

ˇ

family fi0 ...ip of sections of V over Ui0 © ... © Uip . A Cech 0-cochain

is a family (fi ), where fi ∈ “(Ui , V ). A 1-cochain is a family (fij ∈

“(Uij , V )). The coboundary map ‚ p is then

p+1

p

(’1)j (fi0 ...iˆ ...ip+1 )|Ui0 ...ip+1 ,

(‚ f )i0 i1 ...ip+1 = j

j=0

ˇ

2.1. CECH COHOMOLOGY FOR VECTOR BUNDLES 33

where as usual we put a hat over the index that is omitted. One

easily checks that ‚ p+1 ‚ p = 0, so that the above complex is actually

ˇ

a complex. Its cohomology groups H p (U, V ) = Ker(‚ p )/Im(‚ p’1 ) are

ˇ

called the Cech cohomology groups of the cover U with coe¬cient in

V.

In practice, one uses alternating cochains (such that fσ(i0 )σ(i1 )...σ(ip ) =

(σ)fi0 i1 ...ip , where σ is an element of the permutation group Σp+1 and

(σ) is its sign) and there is a theorem, which asserts that one gets

the same cohomology groups if one uses alternating cochains. The ad-

vantage of alternating cochains is obvious: if our cover consists of k

open sets then the above complex is ¬nite and has length ¤ k which

ˇ

immediately implies that H p (U, V ) = 0 for p ≥ k.

The zeroth cohomology group has an explicit interpretation. It is

the space of global sections “(X, V ) of the vector bundle V . Returning

to our exact sequence 0 ’ S ’ E ’ Q ’ 0, we have a short exact

sequence of cochain complexes

ˇ ˇ ˇ

0 ’ C • (S) ’ C • (E) ’ C • (Q) ’ 0

which gives rise to the long exact sequence in cohomology by basic

arguments of homological algebra:

ˇ ˇ ˇ ˇ

0 ’ H 0 (U, S) ’ H 0 (U, E) ’ H 0 (U, Q) ’ H 1 (U, S) ’ · · · .

Example. We have made a commitment each time to see how all our

constructions work in simple cases, so let us take X = CP1 = U0 ∪ U1

- our usual covering with local coordinate z on U0 and w on U1 . As

before, on their intersection U01 one has zw = 1. We trivialize the line

bundle L—n on U0 and U1 in such a way that the transition function

ˇ

on U01 will be z n . Here the Cech complex is consisting of only two

non-trivial terms:

‚0

0 ’ “(U0 , L—n ) • “(U1 , L—n ) ’ “(U01 , L—n ) ’ 0.

Using the trivializations we can write “(Ui , L—n ) H(Ui ) and “(U01 , L—n )

ˇ

H(U01 ) so that the Cech complex is

‚0

0 ’ H(C) • H(C) ’ H(C— ),

34 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

where ‚ 0 works as follows:

‚ 0 (f (z), g(w)) = z n g(1/z) ’ f (z).

ˇ

So, H 0 (U, L—n ) = Ker(‚ 0 ) = {f, g : ‚ 0 (f, g) = 0}. Then one easily

observes that if n < 0 then there is no global holomorphic sections

and for a non-negative n the dimension of this space is n + 1 which

ˇ

con¬rms our earlier calculations. Now, H 1 (U, L—n ) = Coker(‚ 0 ) =

H(C— )/ f (z), z n g(1/z) . Here for n < 0 the basis of Coker(‚ 0 ) is given

ˇ

by z ’1 , z ’2 , ..., z n+1 and hence H 1 (U, L—n ) has dimension ’(n + 1). For

n ≥ ’1, Coker(‚ 0 ) = 0. Thus we have proved

ˇ

PROPOSITION 2.1.1 (I). If n ¤ ’2 then dim H 1 (U, L—n ) = ’n ’

1.

ˇ

(II)If n ≥ ’1 then dim(H 1 (U, L—n )) = 0.

Remark. If we were working with polynomials rather than with holo-

morphic functions, we would get the same answers, but we would have

a smaller complex:

‚0

C[z] • C[w] ’ C[z, z ’1 ],

which gives the same cohomology. Of course, in the case X = CP1 and

the cover we have choosen there are no cohomology groups of degree

more than 1 for any vector bundle as a coe¬cient system. That is why

a short exact sequence 0 ’ S ’ E ’ Q ’ 0 gives rise to the following

”long” exact sequence in cohomology:

0 ’ “(CP1 , S) ’ “(CP1 , E) ’ “(CP1 , Q) ’

ˇ ˇ ˇ

H 1 (U, S) ’ H 1 (U, E) ’ H 1 (U, Q) ’ 0.

ˇ

Now we show several applications of our computation of H j (U, L—n ) for

the open covering U = (U0 , U1 ) of CP1 . We will also use the notation

ˇ

H j (CP1 , L—n ) since it is a basic theorem which tells us that the Cech

cohomology groups are independent on the open covering by open sets

which are Stein. A corollary of this theorem is that if X itself is Stein

then for the obviuos reasons H j (X, V ) = 0 for j > 0.

THEOREM 2.1.2 (Birkho¬, Grothendieck.) For any holomor-

phic vector bundle V over CPn we have a unique non-increasing se-

quence of natural numbers (a1 , ..., ar ) such that V = L—a1 • · · · • L—ar .

ˇ

2.1. CECH COHOMOLOGY FOR VECTOR BUNDLES 35

We need the following

LEMMA 2.1.3 (True for any compact Riemann surface.) Given the

vector bundle V on CP1 , there exists an exact sequence of holomorphic