Compact complex parallelisable nilmanifolds with unobstructed deformations

Compact quotients of non-abelian complex nilpotent Lie groups are non-Kähler manifolds with trivial canonical bundle, the most prominent example being the Iwasawa manifold. We give a precise characterisation of the (rare) cases where deformations of such manifolds are unobstructed in terms of so-called verbal ideals in free nilpotent Lie-algebras. On our way, we will have to review some basic ideas of deformation theory. (joint work with M. Paulsen and K. Wehler)