Lie-algebra Dolbeault cohomology and small deformations of nilmanifolds

TL;DR

This paper studies how small deformations of complex structures on nilmanifolds remain invariant, using Lie-algebra Dolbeault cohomology, and shows this is generally applicable.

Contribution

It establishes that small deformations of left-invariant complex structures on nilmanifolds are also left-invariant under certain cohomological conditions, extending previous results.

Findings

Small deformations preserve left-invariance in nilmanifolds.

Lie-algebra Dolbeault cohomology effectively computes deformations.

Generically, Dolbeault cohomology can be calculated using left-invariant forms.

Abstract

We consider nilmanifolds with left-invariant complex structure and prove that small deformations of such structures are again left invariant if the Dolbeault-cohomology of the nilmanifold can be calculated using left-invariant forms. By a result of Console and Fino this is generically the case. Our main tool is an analog of Dolbeault-cohomology for Lie-algebras with complex structure.