ˇ ˇ

Cech cohomology groups H j (X, V ) are ¬nite-dimensional vector spaces

and they vanish for j > dimC X.

ˇ

As we stated before, to compute Cech cohomology groups we need

an open covering (Ui ) such that each Ui is Stein. For CPn we will

use the standard covering by n + 1 open sets U0 , ..., Un , where Ui is

de¬ned in homogeneous coordinates [z0 : · · · : zn ] by the inequality

zi = 0. Each set Ui is naturally identi¬ed with Cn . We also notice that

computations for CP1 which we have done before show that it is enough

to consider only those sections of L—m which are meromorphic on CP1 .

The foundation for this assumption is provided by

THEOREM 2.3.2 (Serre). For any algebraic vector bundle V over

ˇ

CPn we get the same Cech cohomology using either all holomorphic

sections of V over Ui0 ...ip or all algebraic sections of V over Ui0 ...ip (that

are meromorphic over the whole CPn ).

So, what this theorem really means is that essensial singularities do not

contribute to cohomology. And now we state the main result of this

section.

42 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

PROPOSITION 2.3.3 (I)

{ homogeneous polynomials in (z0 , ..., zn ) of deg = m} if m > 0

ˇ

H 0 (CPn , L—m ) = .

0 if m < 0

(II)

{ homogeneous polynomials in (z0 , ..., zn ) of deg = d} if d > 0

ˇ

H n (CPn , L—m ) = ,

0 if d < 0

where d = ’n ’ 1 ’ m.

ˇ

(III) H j (CPn , L—m ) = 0, j = 0, n.

ˇ

Proof. During the proof we shall use skew-symmetric Cech cochains;

the degree p component of those cochains is given by

“(Ui0 ...ip , L—m ) = ( “(Ui0 ...ip , L—m ))skew part

.

i0 <···<ip i0 ,...,ip

Consider our basic ¬bration

C— ’ Cn+1 \ {0}

“q .

CPn

We also recall that a section of L—m over an open set W ‚ CPn is a

holomorphic function over q ’1 (W ) which is homogeneous of degree m.

Due to the fact that the cover (Ui ) is choosen in such a way that Ui0 ...ip

is the locus where zi0 = 0, ..., zip = 0, all the sections we are interested

in are given by meromorphic functions

g(z0 , ..., zn )

f= ,

zik0 · · · zikp

where g(z0 , ..., zn ) is a homogeneous polynomial in Cn+1 of degree m +

ˇ ˇ

k(p + 1). We have then a natural subspaces of Cech cochains C p (k) ‚

ˇ

C p consisting of those having poles of order at most k in CPn . The

ˇ

di¬erential ‚ preserves this ¬ltration, so we have a subcomplex C • (k):

ˇ ˇ ˇ

‚ ‚

C p’1 ’ Cp ’ C p+1

··· ’ ’ ···

∪ ∪ ∪

ˇ ˇ ˇ

‚ ‚

· · · ’ C p’1 (k) ’ C p (k) ’ C p+1 (k) ’ · · ·

2.3. COHOMOLOGY OF PROJECTIVE SPACE 43

ˇ

Our strategy is to compute the cohomology of subcomplexes C • (k)

ˇ

and then the cohomology of C • (U, L—m ) will be the direct limit of the

cohomologies of subcomplexes. In fact, the cohomology will stabilize

ˇ

as k increases. As we see, we can identify the space C p (k) with the

set of skew cochains (i0 , ..., ip ) ’ gi0 ...ip - a homogeneous polynomial of

degree m + k(p + 1) and the expression of the di¬erential ‚ in terms of

homogeneous polynmials is given by

p+1

(’1)j zikj gi0 ...iˆ ...ip+1 .

(‚g)i0 ...ip+1 = j

j=0

We have a homogeneous piece of the so-called Koszul complex which

¬rst was invented for Lie algebra homology purposes. Let us denote by

A = C[z0 , ..., zn ] the polynomial algebra in n + 1 variables and then the

Koszul complex or exterior algebra complex is

d d

K : A ’ §1 (Cn+1 )—A ’ §2 (Cn+1 )—A ’ · · · ’ §n+1 (Cn+1 )—A ’ 0.

If a basis of Cn+1 is given by elements e0 , ..., en , then the corresponding

basis of §p Cn+1 consists of all elements of the form ei0 § · · · § eip , where

i0 < · · · < ip and the di¬erential d works as

k

d([ei0 § · · · § eip ] — g) = [ej § ei0 § · · · § eip ] — (zj g).

j

The next important observation we make is that the complex

ˇ ‚ˇ ‚ˇ

· · · ’ C p’1 (k) ’ C p (k) ’ C p+1 (k) ’ · · ·

can be embedded into the Koszul complex as follows. We will forget

about the degree 0 part of the Koszul complex (which is A) and put

§1 (Cn+1 ) — A in the degree 0 of the new complex. Besides, we shall

impose homogeneity degrees to get the embedding

§1 (Cn+1 ) — Am+k ’ §2 (Cn+1 ) — Am+2k ’ · · ·

© © ,

§1 (Cn+1 ) — A §2 (Cn+1 ) — A

’ ’ ···

where in the lower line we have the chopped o¬ the Koszul complex.

We also notice that A is a graded algebra and the grading is by the

44 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

degree of homogeneity: A = •s≥0 As and As consists of homogeneous

polynomials of degree s.

A very useful remark is that the original Koszul complex K • does

not have cohomology except for the degree n + 1. The only non-zero

cohomology group is H n+1 (K • ) = A/(z0 , z1 , ..., zn ). The basis of this

kk k

± ±n

vector space is given by monomials z0 0 · · · zn where 0 ¤ ±i ¤ k ’ 1.

To see this we shall ¬x k and vary n, so that we have a Koszul complex

K • (p) corresponding to the algebra C[z1 , ..., zp ]. It is easy to understand

that K • (p) — K • (q) K • (p + q). At the ¬rst level we have the complex

zk

•

K (1) : C[z] ’ C[z] which shows that the statement is true for p = 1.

Now we apply the K¨nneth theorem and induction on p. In particular

u

j •

H (K (p)) = 0 for j < p.

Before we chop o¬ K • it had cohomology only in its maximal degree.

The same happens with the complex in which the degree of homogeneity

is ¬xed. After we chop the Koszul complex o¬, we can only change the

degree zero cohomology. So we can summarize our achievements in the

following statement. One has

0 if j = 0, n

ˇ

H j (C • (k)) = Am if j = 0, m ≥ 0 .

k k

[A/(z0 , ..., zn )]m+k(n+1) if j=n

What also should be emphasized is that the vector space in the last

line is nonzero only if m ¤ ’n ’ 1.

The next step is to show that for k ≥ ’m ’ n one has an isomor-

ˇˇ ˇˇ

phism H n (C • (k)) H n (C • (k + 1)). It is enough to prove that the

natural map

φ

kk k k+1 k+1 k+1

[A/(z0 , z1 , ..., zn )]m+k(n+1) ’ [A/(z0 , z1 , ..., zn )]m+(k+1)(n+1) :

φ(P ) = z0 z1 · · · zn P

kk k

is an isomorphism. We take a basis S of [A/(z0 , z1 , ..., zn )]m+k(n+1)

± ±

consisting of all monomials z0 0 · · · zn n with 0 ¤ ±i ¤ k ’ 1 and such

that n ±j = m + k(n + 1).

j=0

Let us consider a map Sk ’ Sk+1 : (±0 , ..., ±n ) ’ (±0 + 1, ..., ±n + 1).

We claim that this map is a bijection. The only thing we have to verify

2.3. COHOMOLOGY OF PROJECTIVE SPACE 45

is that if (β0 , ..., βn ) ∈ Sk+1 then each βj ≥ 1. Assume otherwise that

for instance β0 = 0. Then one has

n n

βj = βj ¤ kn,

j=0 j=1

so m + (k + 1)(n + 1) ¤ kn which gives us the contradiction k ¤

ˇˇ ˇˇ

’m ’ n ’ 1. We conclude that H n (C • (U, L—m )) = H n (C • (k)) for

k ≥ ’m ’ n.

In fact we also proved that this group is ¬nite dimensional and its

dimension coincides with the cardinality of the set Sk for k ≥ ’m ’ n.

Let us take d = ’n ’ m ’ 1 and establish a duality result

ˇ

dim H n (CPn , L—m ) = dim H 0 (CPn , L—d ).

The l.h.s. is just #(Sk ) and the r.h.s. is

n

#T = #{(β0 , ..., βn ), βj ≥ 0, βj = d}.

j=0

f

We would like to construct a bijection between those two sets Sk ’ T

as follows. Let f (±0 , ..., ±n ) = (k ’ 1 ’ ±0 , ..., k ’ 1 ’ ±n ), then one

has n {(k ’ 1) ’ ±i } = d. If one considers now the inverse map

i=0

’1

f (β0 , ..., βn ) = (k ’ 1 ’ β0 , ..., k ’ 1 ’ βn ) then one notices that each

βj ¤ d ¤ k ’ 1 by assumption and hence k ’ 1 ’ βj ≥ 0 and this

concludes the proof of the theorem.

A more general duality result is due to Serre

THEOREM 2.3.4 (Serre duality.) Let X be a compact complex

ˇ

manifold of dimension n. Then H n (X, §n T — X) C. For any holo-

morphic vector bundle E over X we have a non-degenerate pairing

ˇ ˇ ˇ

H j (X, E) — H n’j (X, E — — §n T — X) ’ H n (X, §n T — X) = C

ˇ ˇ

that allows one to identify cohomology groups H j (X, E) and H n’j (X, E — —

§n T — X).

46 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

In the case X = CPn we have §n T — X = L—’n’1 and L—m = §n T — X —

(L—d )— and thus our computations con¬rm Serre theorem in this par-

ticular case.

Let us again have a holomorphic vector bundle E over a compact

complex manifold X of dimension n. The integer χ(X, E) de¬ned by

n

ˇ

dim H j (X, E)

χ(X, E) =

j=0

is called the Euler characteristic of the bundle E. It represents a sort

of a robust invariant of the bundle. As we saw before a single coho-

mology group can not serve as an invariant due to the phenomenon of

degeneration. The Riemann-Roch formula by Hirzebruch allows one to

write down χ(X, E) in terms of Chern classes of E and the Todd class

of X as we shall see later.

Claim.

(m + 1)(m + 2) · · · (m + n)

χ(CPn , L—m ) = P (m) =

n!

Proof. First we notice that for m ≥ 0 as we have seen earlier

dim H 0 (CPn , L—m ) = P (m)

is the same polynomial in m that occurs in the statement of the claim.

Also one sees that P (’n ’ 1 ’ m) = (’1)n P (m) for plain arithmetical

reasons. The formula for m ¤ ’n’1 then follows from this observation

and from the duality:

ˇ

χ(CPn , L—m ) = (’1)n dim H n (L—m ) =

(’1)n dim H 0 (L—’n’m’ ) = (’1)n P (’n ’ m ’ 1) = P (m).

For all intermediate values of m, i.e. when m lies in the ¬nite interval

{’n, ’n + 1, ..., ’1} one has the vanishing of all cohomology groups

(hence χ(CPn , L—m ) = 0 too) as well as the vanishing of the polynomial

P for obvious reasons.

2.4. CHERN CLASSES OF COMPLEX VECTOR BUNDLES 47

In fact for any vector bundle E over CPn the Euler characteristic

χ(CPn , E — L—m ) is always a polynomial in m. Another important

theorem of Serre (Corollary 3.5 corresponds to the case n = 1) is the

theorem about vanishing of cohomology:

THEOREM 2.3.5 (Serre vanishing). In the present notation and

assumptions there exists an integer N such that for any m > N and

ˇ