Moduli spaces for complex nilmanifolds

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Nilmanifolds have been established as one of the go-to classes of manifolds if one is looking for an accessible example of an exotic complex or hermitian structure. The main feature is that the properties of left-invariant structures (often) can be studied with methods from Lie theory, i.e., only depend on the corresponding Lie algebra.Classification of such structures on nilpotent Lie algebras has received a lot of attention, but the additional difficulties that arise in the geometric setting have mostly been neglected so far. In this project we aim to classify isomorphism classes of left-invariant complex structures for certain classes of compact nilmanifolds. We want to identify cases where the moduli space exists as a complex space, possibly after fixing additional data, and study its properties.