nLab
Dolbeault cohomology

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Idea

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

Properties

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

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