Invariant complex structures on 6-nilmanifolds: classification, Fr\"olicher spectral sequence and special Hermitian metrics

We classify invariant complex structures on 6-dimensional nilmanifolds up to equivalence. As an application, the behaviour of the associated Fr\"olicher sequence is studied as well as its relation to the existence of strongly Gauduchon metrics. We also show that the strongly Gauduchon property and the balanced property are not closed under holomorphic deformation.