Complex structures on two-step nilpotent Lie groups

TL;DR

This paper characterizes 2-step nilpotent Lie algebras that admit invariant complex structures, studies Hermitian metrics on these groups, and explores conditions for pluriclosed metrics and hyper-Kähler with torsion structures.

Contribution

It provides a comprehensive characterization of complex structures on 2-step nilpotent Lie groups and analyzes conditions for pluriclosed Hermitian metrics, including special cases and related geometric structures.

Findings

Characterization of complex structures on 2-step nilpotent Lie groups.

Conditions for pluriclosed Hermitian metrics on nilpotent Lie groups.

Existence of pluriclosed metrics on certain nilmanifolds and hyper-Kähler with torsion metrics.

Abstract

We give a characterization of the $2$-step nilpotent Lie algebras whose corresponding Lie groups admit a left invariant complex structure. This is done by considering separately the cases when the complex structure is 2-step or 3-step nilpotent in the sense of [Cordero-Fern\'andez-Gray-Ugarte, 2000]. We also study the Hermitian geometry 2-step nilpotent Lie groups. We show that if a left invariant Hermitian metric on such Lie group is pluriclosed, then the corresponding complex structure is 2-step nilpotent. Moreover, we obtain a necessary and sufficient condition for such a metric to be pluriclosed in case the complex structure is abelian. This allows us to show that pluriclosed metrics on nilpotent Lie algebras with one dimensional commutator ideal can only occur on trivial central extensions of the $3$-dimensional Heisenberg Lie algebra. We show that certain Hermitian nilmanifolds constructed from compact semisimple irreducible Hermitian symmetric pairs are pluriclosed with respect to a canonically defined abelian complex structure. Finally, we give necessary and sufficient conditions for a naturally reductive Riemannian metric on a nilmanifold to be Hermitian with respect to an abelian complex structure. We prove the analogue of this result in the hypercomplex case, thereby obtaining a distinguished family of hyper-K\"ahler with torsion metrics.