Locally conformal SKT structures

A Hermitian metric on a complex manifold is called SKT (strong Kähler 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 locally conformal SKT (or shortly LCSKT). More precisely, a Hermitian structure (J, g) is said to be LCSKT if there exists a closed nonzero 1-form α such that dH = α ∧ H. In this paper, we consider nontrivial LCSKT structures, i.e. we assume that dH ≠ 0 and we study their existence on Lie groups and their compact quotients by lattices. In particular, we classify six-dimensional nilpotent Lie algebras admitting a LCSKT structure and we show that, in contrast to the SKT case, there exists a six-dimensional 3-step nilpotent Lie algebra admitting a nontrivial LCSKT structure. Moreover, we show a characterization of even dimensional almost abelian Lie algebras admitting a nontrivial LCSKT structure, which allows us to construct explicit examples of six-dimensional unimodular almost abelian Lie algebras admitting a nontrivial LCSKT structure. The compatibility between the LCSKT and the balanced condition is also discussed, showing that a Hermitian structure on a six-dimensional nilpotent or a 2n-dimensional almost abelian Lie algebra cannot be simultaneously LCSKT and balanced, unless it is Kähler.