# nLab Dolbeault cohomology

cohomology

### Theorems

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Idea

Dolbeault cohomology of a complex manifold $X$ is the abelian sheaf cohomology $H^q(X;\Omega_X^p)$, of the abelian sheaf $\Omega_X^p$ is Dolbeault complex of holomorphic $p$-forms.

## Properties

Given any holomorphic vector bundle $E$, one can form the Dolbeault resolution $E \otimes \Omega^{0,q}$, where $\Omega^{0,q}$ is the sheaf of $C^\infty$ $(0,q)$-forms. This is an acyclic resolution of $E$ and hence computes its sheaf cohomology.

Revised on July 12, 2013 16:35:46 by Urs Schreiber (82.113.121.27)