Geometry of nilmanifolds with left-invariant complex structure and deformations in the large

TL;DR

This paper explores the geometric structure of nilmanifolds with left-invariant complex structures, providing criteria for their deformations and offering a comprehensive view in complex dimension three.

Contribution

It clarifies the relationship between nilmanifolds and holomorphic torus bundles and establishes conditions for large deformations to preserve this structure.

Findings

Nilmanifolds with left-invariant complex structures relate to iterated principal holomorphic torus bundles.

Criteria are provided for deformations in the large to remain of similar type.

A comprehensive classification is achieved in complex dimension three.

Abstract

The relation between nilmanifolds with left-invariant complex structure and iterated principal holomorphic torus bundles is clarified and we give criteria under which deformations in the large are again of such type. As an application we obtain a fairly complete picture in complex dimension three.