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‚z¯ ξ ‚z
¯ 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

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