Deformations of Dolbeault cohomology classes for Lie algebra with complex structures

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 $B$.