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[edit]
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

,