Generalized K\"ahler almost abelian Lie groups

TL;DR

This paper classifies six-dimensional almost abelian Lie groups with generalized K"ahler structures, identifying conditions for complex, Hermitian, and Poisson structures, and explores their geometric flows and compact quotients.

Contribution

It provides a comprehensive classification of six-dimensional generalized K"ahler almost abelian Lie groups and analyzes their geometric and complex structures.

Findings

Classification of six-dimensional almost abelian Lie groups with complex structures.

Identification of those with $ ext{∂∂}$-closed Hermitian forms.

Results on holomorphic Poisson structures and pluriclosed flow.

Abstract

We study left-invariant generalized K\"ahler structures on almost abelian Lie groups, i.e., on solvable Lie groups with a codimension-one abelian normal subgroup. In particular, we classify six-dimensional almost abelian Lie groups which admit a left-invariant complex structure and establish which of those have a left-invariant Hermitian structure whose fundamental 2-form is $\partial \bar \partial$-closed. We obtain a classification of six-dimensional generalized K\"ahler almost abelian Lie groups and determine the 6-dimensional compact almost abelian solvmanifolds admitting an invariant generalized K\"ahler structure. Moreover, we prove some results in relation to the existence of holomorphic Poisson structures and to the pluriclosed flow.