On astheno-Kähler nilmanifolds with balanced metrics

In this paper we study the structure of complex nilmanifolds $X$ admitting some special classes of Hermitian metrics, namely, astheno-K\"ahler, strongly Gauduchon and balanced metrics. We prove that, in complex dimension 4, the existence of a (non necessarily invariant) astheno-K\"ahler metric on $X$ implies that the nilmanifold is at most $2$-step and it has first Betti number $\geq 6$. Moreover, the complex structure has a very specific form, sometimes called "of special type" in the literature. We also study the interplay between the existence of astheno-K\"ahler metrics and that of strongly Gauduchon or balanced metrics. A key result is the use of some obstructions that are preserved by what we call $\mathfrak{b}$-extensions. This allows us to study the existence of these metrics on important classes of complex nilmanifolds, such as almost abelian, those having maximal nilpotent complex structures, and 8-dimensional nilmanifolds with non-nilpotent complex structures. We also construct, in every complex dimension $n\geq 4$, complex nilmanifolds admitting both an astheno-K\"ahler metric (possibly also being strongly Gauduchon) and another metric that is balanced. As an application, astheno-K\"ahler nilmanifolds with balanced metrics and with Fr\"olicher spectral sequence not degenerating at the second or third pages are found. To our knowledge, these are the first compact astheno-K\"ahler manifolds with such properties.