nLab
Dolbeault theorem

Contents

Context

Complex geometry

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

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

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

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

References

Review includes

  • Zachary Maddock, Dolbeault cohomology (pdf)

Last revised on July 1, 2016 at 03:28:56. See the history of this page for a list of all contributions to it.