Astheno-K\"ahler and balanced structures on fibrations

We study the existence of three classes of Hermitian metrics on certain types of compact complex manifolds. More precisely, we consider balanced, SKT and astheno-K\"ahler metrics. We prove that the twistor spaces of compact hyperk\"ahler and negative quaternionic-K\"ahler manifolds do not admit astheno-K\"ahler metrics. Then we provide examples of astheno-K\"ahler structures on toric bundles over K\"ahler manifolds. In particular, we find examples of compact complex non-K\"ahler manifolds which admit a balanced and an astheno-K\"ahler metrics, thus answering to a question in [52] (see also [24]). One of these examples is simply connected. We also show that the Lie groups $SU(3)$ and $G_2$ admit SKT and astheno-K\"ahler metrics, which are different. Furthermore, we investigate the existence of balanced metrics on compact complex homogeneous spaces with an invariant volume form, showing in particular that if a compact complex homogeneous space $M$ with invariant volume admits a balanced metric, then its first Chern class $c_1(M)$ does not vanish. Finally we characterize Wang C-spaces admitting SKT metrics.