On astheno-Kaehler metrics

TL;DR

This paper introduces and constructs new examples of astheno-K"ahler metrics on complex manifolds, explores their properties, and investigates their relation to other geometric structures like strong KT and conformally balanced metrics.

Contribution

It constructs simply-connected and nilmanifold examples of astheno-K"ahler manifolds, and analyzes their deformation and relation to conformally balanced metrics.

Findings

Constructed simply-connected astheno-K"ahler manifolds for complex dimension > 3.

Developed examples of non strong KT astheno-K"ahler nilmanifolds of complex dimension 4.

Explored the relation between astheno-K"ahler and conformally balanced conditions.

Abstract

A Hermitian metric on a complex manifold of complex dimension $n$ is called {\em astheno-K\"ahler} if its fundamental $2$-form $F$ satisfies the condition $\partial \overline \partial F^{n - 2} =0$. If $n =3$, then the metric is {\em strong KT}, i.e. $F$ is $\partial \overline \partial$-closed. By using blow-ups and the twist construction, we construct simply-connected astheno-K\"ahler manifolds of complex dimension $n > 3$. Moreover, we construct a family of astheno-K\"ahler (non strong KT) $2$-step nilmanifolds of complex dimension $4$ and we study deformations of strong KT structures on nilmanifolds of complex dimension $3$. Finally, we study the relation between astheno-K\"ahler condition and (locally) conformally balanced one and we provide examples of locally conformally balanced astheno-K\"ahler metrics on $\T^2$-bundles over (non-K\"ahler) homogeneous complex surfaces.