Deformations of Dolbeault cohomology classes

TL;DR

This paper develops a deformation theory for Dolbeault cohomology classes in holomorphic tensor bundles, constructing a canonical family of deformations and analyzing their properties, including unobstructedness and cohomology dimension jumps.

Contribution

It introduces an extension equation analogous to Maurer-Cartan, constructs a complete deformation family, and explains cohomology dimension jumps on complex manifolds.

Findings

Deformation of Dolbeault classes can be unobstructed under mild conditions.

The theory explains the jumping phenomenon of Dolbeault cohomology.

Constructs a canonical complete family of deformations using power series.

Abstract

In this paper, we establish a deformation theory for Dolbeault cohomology classes valued in holomorphic tensor bundles. We prove the extension equation which will play the role of Maurer-Cartan equation. Following the classical theory of Kodaira-Spencer-Kuranishi, we construct a canonical complete family of deformations by using the power series method. We also prove a simple relation between the existence of deformations and the varying of the dimensions of Dolbeault cohomology. The deformations of $(p,q)$-forms is shown to be unobstructed under some mild conditions. By analyzing Nakamura's example of complex parallelizable manifolds, we will see that the deformation theory developed in this work provides precise explanations to the jumping phenomenon of Dolbeault cohomology.