Small deformations and non-left-invariant complex structures on six-dimensional compact solvmanifolds

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. We also provide a complete classification of three-dimensional compact homogeneous complex solvmanifolds; and determine which of them admit pseudo-Kähler structures.