Generalized Kähler almost abelian Lie groups

Abstract

We study left-invariant generalized Kähler 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 Kähler almost abelian Lie groups and determine the 6-dimensional compact almost abelian solvmanifolds admitting an invariant generalized Kähler structure. Moreover, we prove some results in relation to the existence of holomorphic Poisson structures and to the pluriclosed flow.

arXiv comment28 pages, to appear in Annali di Matematica Pura ed Applicata

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Cited by in corpus (5)

• Hypercomplex almost abelian solvmanifolds
• The structure of almost Abelian Lie algebras
• Applications of the quaternionic Jordan form to hypercomplex geometry
• Almost Abelian Lie groups, subgroups and quotients
• A note on $p$-Kähler structures on compact quotients of Lie groups