Deformations of Dolbeault cohomology classes for Lie algebra with complex structures

TL;DR

This paper investigates how complex structures on Lie algebras deform and how these deformations affect Dolbeault cohomology classes, establishing a Kuranishi-like family and analyzing invariant cohomology in deformed manifolds.

Contribution

It constructs a complete deformation framework for complex structures on Lie algebras and demonstrates the stability of invariant Dolbeault cohomology under such deformations.

Findings

Constructed a Kuranishi-like deformation family for complex structures.

Proved the extension isomorphism holds in this deformation setting.

Identified an analytic open subset where Dolbeault cohomology remains invariant under deformation.

Abstract

In this paper, we study deformations of complex structures on Lie algebras and its associated deformations of Dolbeault cohomology classes. A complete deformation of complex structures is constructed in a way similar to the Kuranishi family. The extension isomorphism is shown to be valid in this case. As an application, we prove that given a family of left invariant deformations $\{M_t\}_{t\in B}$ of a compact complex manifold $M=(\Gamma\setminus G, J)$ where $G$ is a Lie group, $\Gamma$ a sublattice and $J$ a left invariant complex structure, the set of all $t\in B$ such that the Dolbeault cohomology on $M_t$ may be computed by left invariant tensor fields is an analytic open subset of $B$.