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. The author shows that, up to equivalence of the complex structure, the strong Kähler structures \((J, g)\) with torsion on \(M\) are parametrized by the points in a subset of the Euclidean space.
A classification of six-dimensional nilmanifolds admitting balanced Hermitian structures \((J, g)\) is given. As an application, it classifies the nilmanifold \(\Gamma/(H^3\times H^3)\). The balanced condition is not stable under small deformations.
The author proves 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 Kähler metrics, and the complex structures underlying such metrics are all equivalent.