# Dolbeault cohomology

In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups Hp,q(M,C) depend on a pair of integers p and q and are realized as a subquotient of the space of complex differential forms of degree (p,q).

Contents

1 Construction of the cohomology groups
2 Dolbeault cohomology of vector bundles
3 Dolbeault’s theorem

3.1 Proof

4 Footnotes
5 References

Construction of the cohomology groups
Let Ωp,q be the vector bundle of complex differential forms of degree (p,q). In the article on complex forms, the Dolbeault operator is defined as a differential operator on smooth sections

¯

:
Γ
(

Ω

p
,
q

)

Γ
(

Ω

p
,
q
+
1

)

{\displaystyle {\bar {\partial }}:\Gamma (\Omega ^{p,q})\rightarrow \Gamma (\Omega ^{p,q+1})}

Since

¯

2

=
0

{\displaystyle {\bar {\partial }}^{2}=0}

this operator has some associated cohomology. Specifically, define the cohomology to be the quotient space

H

p
,
q

(
M
,

C

)
=

ker

(

¯

:
Γ
(

Ω

p
,
q

,