¯ ξ

C C

Here we use the general fact that we can move the operator ‚‚z inside ¯

the integral sign, since the integration is performed over some compact

domain of C. So we have proved the Dolbeault Lemma in dimension 1.

We notice that if our function f we started with depends holomorphi-

cally upon some variables ·2 , ..., ·l then the same proof works and we

still have

¯

1 ‚ dξ § dξ

f (z, ·2 , ..., ·l ) = √ ( f (ξ + z, ·2 , ..., ·l ) ),

2π ’1 ‚ z

¯ ξ

C

so that we have f = ‚‚z g, where g depends holomorphically on (·2 , ..., ·l ).

¯

Now we treat the case n > 1. As before let us have two polydiscs

∆ ‚‚ ∆ and a smooth di¬erential form φ of type (p, q) in ∆ with

d φ = 0. Then we are to ¬nd a smooth di¬erential from ψ of type

(p, q ’ 1) in a neighbourhood of ∆ such that d ψ = φ. We assume that

our form φ does not involve d¯k+1 , ..., d¯n . Our purpose is to represent

z z

φ as φ = d · + φ in such a way that φ does not involve d¯k , ..., d¯n .

z z

First, we write φ = d¯k § ± + β, where β does not involve d¯k , ..., d¯n .

z z z

Next one can say that

±= fI,J dzJ § d¯I .

z

I‚{1,...,k’1};#I=q’1

J‚{1,...,n};#J=p

We think of the fI,J as functions of (zk , · · · , zn ), which are holomorphic

in the variables (zk+1 , · · · , zn ). This last fact follows from d β = d¯k §

z

d ±. For instance, to see that ‚fI,J /‚ zn = 0 one writes

¯

‚fI,J

d ± = d¯k §

z d¯n § dzJ § d¯I + · · ·

z z

‚ zn

¯

78 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

and one notices that d β does not have any term which involves d¯k §

z

d¯n .

z

Now, applying the result from dimension 1 we can say that fI,J =

‚gI,J

in the smaller polydisk ∆. Furhermore we can arrange that each

‚ zk

¯

fI,J is holomorphic in (zk+1 , ..., zn ).

Finally, one puts

·= gI,J dzJ § d¯I ,

z

I,J

so that

‚gI,J

d ·= d¯k § dzJ § d¯I + δ,

z z

‚ zk

¯

I,J

where δ involves only d¯1 , ..., d¯k’1 . It means that the form φ ’ d ·

z z

involves d¯1 , ..., d¯k’1 only. Thus we have managed to write down the

z z

form φ as φ = d · + φ as desired. Now if we apply our procedure

inductively we will prove Dolbeault Lemma by an explicit construction.

So far we have assumed that the forms in questions were de¬ned in

a bigger polydisk. Now we show that one can drop this assumption.

THEOREM 2.12.2 (Dolbeault, Grothendieck.) Let ∆ ‚ Cn be

a polydisc and let φ be a smooth (p, q) - form in ∆ such that d φ = 0.

Then there exists a smooth (p, q ’ 1) - form ψ in ∆ such that d ψ = φ.

Proof. We will ¬rst give a proof for q > 1. In the case q = 1 one has

to be a bit more cautious. We represent our polydisk as an increasing

union of smaller polydisks:

¯

∆ = ∪n ∆n , ∆n ‚ ∆n+1 .

We shall construct inductively a sequence of (p, q ’ 1) - forms ψn on

∆n such that d ψn = φ on ∆n and (ψn+1 )|∆n = ψn . If we are able to

do so, the theorem automatically follows.

We know already that the Dolbeault Lemma works in a smaller

polydisk, thus it provides us with the existence of ψ1 . Next we assume

that we have already constructed the sequence ψ1 , ..., ψn satisfying the

condition above. Applying Dolbeault Lemma for a smaller polydisk

once again, we see that we can ¬nd a (p, q ’1) - form ψ n+1 in ∆n+1 such

2.12. DOLBEAULT COHOMOLOGY 79

that d ψ n+1 = φ. It implies that on ∆n one has d (ψ n+1 ’ ψn ) = 0.

Consequently, we can ¬nd a (p, q ’ 2) - form ± in ∆n+1 such that

d ± = ψ n+1 ’ ψn on ∆n . Put ψn+1 = ψ n+1 ’ d ±; the form ψn+1 is

the desired continuation of the sequence ψ1 , ..., ψn .

In the case q = 1 we proceed di¬erently. There is no real loss of gen-

erality in assuming p = 0. For each n we construct a smooth function

fn on ∆ such that d fn = φ on ∆n . Furthermore we can choose the fn

so that |fn (x) ’ fn’1 (x)| ¤ 2’n for all x ∈ ∆n’2 . To construct such fn

inductively, ¬rst use the Dolbeault lemma to construct some smooth

function fn on ∆n such that d fn = φ on ∆n . Then notice that fn ’fn’1

is holomorphic on ∆n’1 and let ± c± z ± be its Taylor expansion at the

origin. This Taylor expansion is uniformly convergent in ∆n’2 , which

is relatively compact in ∆n’1 . Then let g(x) = |±|≥m c± z ± , where

m is such that |fn (x) ’ fn’1 (x) ’ g(x)| ¤ 2’n on ∆n’2 . We then put

fn (x) = fn (x) ’ g(x). Then the sequence (fn ) converges locally uni-

formly on ∆. The limit function f satis¬es d f = φ on each ∆m ; indeed

over ∆m , the sequence fn ’ fm of holomorphic functions converges

locally uniformly to f ’ fm , which therefore is holomorphic. Hence

d f = d fn = φ over ∆n . This proves the theorem for q = 1.

Let us consider now a holomorphic vector bundle E of rank r over

X. In this case the Dolbeault complex is de¬ned to be

d d

“sm (X, E) ’ “sm (X, T — X — E) ’ “sm (X, §2 T — X — E) ’ · · · ,

where the subscript sm is to indicate that we consider the smooth

sections and the bundle T — X is the complex conjugate bundle to the

holomorphic cotangent bundle T — X. The space of sections of this bun-

dle “sm (T — X) = A0,1 (X) is the space of 1 forms of type (0, 1). The way

in which the di¬erential d works can be seen if we pick a local basis

(e1 , ..., er ) of E consisting of holomorphic sections. Then

d( fi ei ) = (d fi ) — ei .

The cohomology of the above complex is what is called Dolbeault co-

homology of X with coe¬cients in E. The corresponding cohomology

q

groups are denoted by HDolb (X, E). If we take E = §p T — X then we

¬nd the same complex as before:

d d d

Ap,0 (X) ’ Ap,1 (X) ’ · · · ’ Ap,n .

80 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

For the convenience of the reader we notice that often in the liter-

¯

ature the notation ‚ is used instead of d and ‚ instead of d .

ˇ

The following theorem shows that Cech and Dolbeault cohomology

have a lot in common.

THEOREM 2.12.3 If E be a holomorphic vector bundle over a com-

plex manifold X, then

ˇ j

H j (X, E) = HDolb (X, E).

We will construct concrete maps in both directions. Let (Ui ) be an

open locally ¬nite covering of X by Stein open sets and let (fi ) be

a partition of unity subordinate to this covering: fi = 1. For a

holomorphic section φi0 ...ip ∈ “(Ui0 ...ip , E) we would like to construct a

d -closed p-form of type (0, p) with values in E. Over an open set Ui

we introduce the form

·i = d fi1 § · · · § d fip — φii1 ...ip ,

i1 ,...,ip

so that the equality d ·i = 0 is automatically satis¬ed. What is left to

do is to show that on Uij one has ·i = ·j . We take p = 2 for simplicity

(for general p it works analogously):

·j ’·i = d fi1 §d fi2 —(φji1 i2 ’φii1 i2 ) = d fi1 §d fi2 —(φiji1 ’φiji2 ) =

i1 ,i2 i1 ,i2

=( d fi1 — φiji1 ) § ( d fi2 ) ’ ( d fi1 ) § ( d fi2 — φiji2 ) = 0,

i1 i2 i1 i2

because for instance d fi2 = d fi2 = d 1 = 0, since (fi ) is a

ˇ

partition of unity. We also used the fact that φ is a Cech cocycle and

whence

φji1 i2 ’ φii1 i2 + φiji2 ’ φiji1 = 0.

ˇ

p

Now we want to go in the opposite direction: HDolb (X, E) ’ H p (X, E).

Let (Ui ) an open covering as before and also without loss of generality

we take p = 2. We start with a 2-form · of type (0, 2) with val-

ues in E such that d · = 0. Over Ui by Dolbeault Lemma we have

·|Ui = d ±i , where ±i ∈ “sm (Ui , T — X — E). Then over Uij we have

2.12. DOLBEAULT COHOMOLOGY 81

d (±j ’ ±i ) = 0 and again by Dolbeault Lemma (±j ’ ±i ) = d sij

for some sij ∈ “sm (Uij , E). We de¬ne φijk = sjk ’ sik + sij and it is

holomorphic because

d φijk = ±k ’ ±j ’ ±k + ±i + ±j ’ ±i = 0.

ˇ

Thus we get a Cech 2 - cocycle φijk ∈ “hol (Uijk , E) which is a smooth

(not necessarily holomorphic) coboundary. Our procedure can be de-

scribed using the following staircase diagram:

φijk

‘δ

d

sij ’ ±j ’ ±i ,

‘δ

d

±i ’ ·|Ui

ˇ

where δ is the Cech coboundary map which acts in the vertical direction

and the Dolbeault di¬erential d acts horizontally.

Let us consider an example where the Dolbeault cohomology groups

ˇ

are much easier to compute then the Cech cohomology groups. Let X

be a compact complex manifold of complex dimension n. The group

ˇ

H n (X, §n T — X) is equal to

H n,n (X) = An,n (X)/d An,n’1 (X) C.

LEMMA 2.12.4 For a compact connected complex manifold X we

have

H n,n (X) C.

The map to C works as follows. For ω ∈ An,n (X) we take X ω ∈ C.

This map is well de¬ned on the factor because if ± ∈ An,n’1 (X) then

d ± = d± and hence

d ±= d± = 0

X X

by Stokes™ theorem.

Another advantage of Dolbeault cohomology is that the ring struc-

ture given by cup product is transparent here. Let E and F be two

82 CHAPTER 2. COHOMOLOGY OF VECTOR BUNDLES

holomorphic vector bundles over X. We would like to have an explicit

construction of cup product

∪ a+b

a b

HDolb (X, E) — HDolb (X, F ) ’ HDolb (X, E — F )

on the level of di¬erential forms. Let us have

± ∈ §a T — X — E, d ± = 0,

β ∈ §b T — X — F, d β = 0,

then the class [±] ∪ [β] is represented by the form

± § β ∈ §a+b T — X — E — F.

The form ± § β is d -closed since we have the usual rule of di¬erential

calculus

d (± § β) = (d ±) § β + (’1)deg(±) ± § d β.

Applying this result one can deduce the Serre duality theorem which

involves the non-degenerate pairing

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

(this uses the natural pairing E — E — ’ C).

Next we consider a holomorphic vector bundle E over X endowed

with a hermitian metric and the unique connection compatible with

both holomorphic and hermitian structures. Let R be the curvature of

this connection, which is a 2-form of type (1, 1) with values in End(E).

We have the Chern-Weil 2p-form „¦p = P (R) of type (p, p). The form

„¦p is closed, hence it is d -closed too. Thus we arrive at

PROPOSITION 2.12.5 One has a well-de¬ned class

p

cDolb ∈ H p,p (X) = HDolb (X, §p T — X).

p

Sometimes the class cDolb turns out to be more useful then the usual

p

Chern class cp (E) ∈ H 2p (X, C).

The Dolbeault cohomology groups come from an elliptic complex