This paper classifies Hermitian structures on 6-dimensional nilmanifolds for which the fundamental 2-form is -closed, a condition that is shown to depend only on the underlying complex structure J of M. The space of such J is described when G is the complex Heisenberg group, and explicit solutions are obtained from a limaçon-shaped curve in the complex plane. Related theory is used to provide examples of various types of Ricci-flat structures.
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Anna Fino, Simon Salamon, Maurizio Parton, Families of strong KT structures in six dimensions. Comment. Math. Helv. 79 (2004), no. 2, pp. 317–340DOI 10.1007/S00014-004-0803-3