Locally conformal SKT structures

TL;DR

This paper introduces and studies locally conformal SKT structures on Hermitian manifolds, classifying their existence on Lie groups, especially in 6 dimensions, and exploring their relation to balanced and Kähler structures.

Contribution

It defines the new LCSKT condition, classifies 6-dimensional nilpotent Lie algebras with such structures, and characterizes almost abelian Lie algebras admitting LCSKT structures.

Findings

Existence of non-trivial LCSKT structures on certain 6-dimensional nilpotent Lie algebras.

Construction of explicit examples of 6-dimensional unimodular almost abelian Lie algebras with LCSKT structures.

LCSKT and balanced conditions are incompatible unless the structure is Kähler.

Abstract

A Hermitian metric on a complex manifold is called SKT (strong K\"ahler with torsion) if the Bismut torsion $3$-form $H$ is closed. As the conformal generalization of the SKT condition, we introduce a new type of Hermitian structure, called \emph{locally conformal SKT} (or shortly LCSKT). More precisely, a Hermitian structure $(J,g)$ is said to be LCSKT if there exists a closed non-zero $1$-form $\alpha$ such that $d H = \alpha \wedge H$. In the paper we consider non-trivial LCSKT structures, i.e. we assume that $d H \neq 0$ and we study their existence on Lie groups and their compact quotients by lattices. In particular, we classify 6-dimensional nilpotent Lie algebras admitting a LCSKT structure and we show that, in contrast to the SKT case, there exists a $6$-dimensional $3$-step nilpotent Lie algebra admitting a non-trivial LCSKT structure. Moreover, we show a characterization of even dimensional almost abelian Lie algebras admitting a non-trivial LCSKT structure, which allows us to construct explicit examples of $6$-dimensional unimodular almost abelian Lie algebras admitting a non-trivial LCSKT structure. The compatibility between the LCSKT and the balanced condition is also discussed, showing that a Hermitian structure on a 6-dimensional nilpotent or a $2n$-dimensional almost abelian Lie algebra cannot be simultaneously LCSKT and balanced, unless it is K\"{a}hler.