Hermitian structures on six-dimensional almost nilpotent solvmanifolds

TL;DR

This paper classifies six-dimensional almost nilpotent Lie algebras with complex structures, explores invariant Hermitian metrics, and confirms conjectures about special metric types on solvmanifolds.

Contribution

It completes the classification of complex structures on six-dimensional strongly unimodular almost nilpotent Lie algebras and analyzes the existence of various Hermitian metrics.

Findings

Identified new balanced solvmanifolds

Confirmed conjecture on SKT and balanced structures

Proved non-existence of certain complex structures on almost abelian Lie algebras

Abstract

We complete the classification of six-dimensional strongly unimodular almost nilpotent Lie algebras admitting complex structures. For several cases we describe the space of complex structures up to isomorphism. As a consequence we determine the six-dimensional almost nilpotent solvmanifolds admitting an invariant complex structure and study the existence of special types of Hermitian metrics, including SKT, balanced, locally conformally K\"ahler, and strongly Gauduchon metrics. In particular, we determine new balanced solvmanifolds and confirm a conjecture by the first author and Vezzoni regarding SKT and balanced structures in the six-dimensional strongly unimodular almost nilpotent case. Moreover, we prove some negative results regarding complex structures tamed by symplectic forms, showing in particular that in every dimension such structures cannot exist on non-K\"ahler almost abelian Lie algebras.