Astheno-Kähler and balanced structures on fibrations

Abstract: 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. 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.