nLab Dolbeault theorem

Idea

The complex geometry-analog of the de Rham theorem is the Dolbeault theorem:

For $X$ a complex manifold then its Dolbeault cohomology in bi-degree $(p,q)$ is naturally isomorphic to the abelian sheaf cohomology in degree $q$ of the abelian sheaf $\Omega^p \coloneqq \Omega^{p,0}$ of holomorphic p-forms

$$H^{p,q}(X)\simeq H^q(X,\Omega^p) \,.$$

Related concepts

• de Rham theorem

• Hodge theorem

References

Review includes

• Zachary Maddock, Dolbeault cohomology (pdf)