This paper classifies Hermitian structures on 6-dimensional nilmanifolds M=Γ G 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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Fino et al. (2004) studied this question.
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