Hermitian structures on six-dimensional nilmanifolds

Abstract

Let (J,g) be a Hermitian structure on a six-dimensional compact nilmanifold M with invariant complex structure J and compatible metric g, which is not required to be invariant. We show that, up to equivalence of the complex structure, the strong Kahler with torsion structures (J,g) on M are parametrized by the points in a subset of the Euclidean space, in particular, the region inside a certain ovaloid corresponds to such structures on the Iwasawa manifold and the region outside to strong Kahler with torsion structures with nonabelian J on the nilmanifold $\Gamma\backslash (H^3\times H^3),$ where H³ is the Heisenberg group. A classification of six-dimensional nilmanifolds admitting balanced Hermitian structures (J,g) is given, and as an application we classify the nilmanifolds having invariant complex structures which do not admit Hermitian structure with restricted holonomy of the Bismut connection contained in SU(3). It is also shown that on the nilmanifold $\Gamma\backslash (H^3\times H^3)$ the balanced condition is not stable under small deformations. Finally, we prove that a compact quotient of $H(2,1)\times \mathbb{R},$ where H(2,1) is the five-dimensional generalized Heisenberg group, is the only six-dimensional nilmanifold having locally conformal Kahler metrics, and the complex structures underlying such metrics are all equivalent. Moreover, this nilmanifold is a Vaisman manifold for any invariant locally conformal Kahler metric.