In this thesis, we are concerned with the classification of complex nilmanifolds.
To this end, we study two parameter spaces: the moduli space and the Teichmüller space for complex nilmanifolds. The moduli space parametrises the biholomorphism classes of complex nilmanifolds with a fixed underlying smooth nilmanifold, while the Teichmüller space parametrises complex nilmanifolds up to biholomorphisms smoothly isotopic to the identity.
Both of these spaces carry a natural topology and the main purpose of this work is to investigate when these spaces can also be endowed with a complex structure.
We first prove that, under a certain geometric hypothesis, every holomorphic map between complex nilmanifolds is, up to translations, induced by a group homomorphism on the universal cover. This allows for a much simpler description of the Teichmüller- and moduli space.
We will then present several existence and non-existence criteria for a complex structure on these spaces, which depend to a large extent on the behaviour of certain period maps.
Among other classes, we study the Teichmüller- and moduli spaces for almost abelian nilmanifolds. We prove that while the Teichmüller space always admits a complex structure in this class, the moduli space is often non-Hausdorff. Nevertheless, we construct a family of almost abelian nilmanifolds of unbounded dimension and nilpotency index where the moduli space is Hausdorff and admits a complex structure.
Finally, building on several classification results, we provide a fairly complete description of the Teichmüller- and moduli spaces for nilmanifolds of dimension six.
