Frölicher spectral sequence of compact complex manifolds with special Hermitian metrics

In this paper we focus on the interplay between the behaviour of the Fr & ouml;licher spectral sequence and the existence of special Hermitian metrics on the manifold, such as balanced, SKT or generalized Gauduchon. The study of balanced metrics on nilmanifolds endowed with strongly non-nilpotent complex structures allows us to provide infinite families of compact balanced manifolds with Fr & ouml;licher spectral sequence not degenerating at the second page. Moreover, this result is extended to non-degeneration at any arbitrary page. Similar results are obtained for the Fr & ouml;licher spectral sequence of compact generalized Gauduchon manifolds. We also find a compact SKT manifold whose Fr & ouml;licher spectral sequence does not degenerate at the second page, thus providing a counterexample to a conjecture by Popovici.