Small deformations and non-left-invariant complex structures on a compact solvmanifold

TL;DR

This paper presents a six-dimensional compact solvmanifold with a continuous family of non-left-invariant complex structures and classifies three-dimensional homogeneous complex solvmanifolds, including their pseudo-Kaehler structures.

Contribution

It provides the first example of a higher-dimensional solvmanifold with non-left-invariant complex structures and offers a complete classification of three-dimensional cases.

Findings

Existence of non-left-invariant complex structures in six dimensions

Complete classification of three-dimensional homogeneous complex solvmanifolds

Identification of pseudo-Kaehler structures in these manifolds

Abstract

We observed in our previous paper that all the complex structures on four-dimensional compact solvmanifolds, including tori, are left-invariant. In this paper we will give an example of a six-dimensional compact solvmanifold which admits a continuous family of non-left-invariant complex structures. Furthermore, we will make a complete classification of three-dimensional compact homogeneous complex solvmanifolds; and determine which of them admit pseudo-Kaehler structures.