Balanced Hermitian geometry on 6-dimensional nilmanifolds

Abstract

The invariant balanced Hermitian geometry of nilmanifolds of dimension 6 is described. We prove that the holonomy group of the associated Bismut connection reduces to a proper subgroup of SU(3) if and only if the complex structure is abelian. As an application we show that if J is abelian then any invariant balanced J-Hermitian structure provides solutions of the Strominger system.