Astheno-K\"ahler and balanced structures on fibrations

TL;DR

This paper investigates the existence of balanced, SKT, and astheno-K"ahler Hermitian metrics on various compact complex manifolds, providing new examples and non-existence results, especially on fibrations and homogeneous spaces.

Contribution

It offers new examples of astheno-K"ahler and balanced metrics on non-K"ahler manifolds, and characterizes conditions for their existence on homogeneous spaces and specific Lie groups.

Findings

Twistor spaces of certain hyperk"ahler manifolds do not admit astheno-K"ahler metrics.

Examples of astheno-K"ahler structures on toric bundles over K"ahler manifolds.

Lie groups SU(3) and G2 admit distinct SKT and astheno-K"ahler metrics.

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] (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.