On astheno-Kaehler metrics

Abstract

A Hermitian metric on a complex manifold of complex dimension $n$ is called {\em astheno-Kähler} 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-Kähler manifolds of complex dimension $n > 3$. Moreover, we construct a family of astheno-Kähler (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-Kähler condition and (locally) conformally balanced one and we provide examples of locally conformally balanced astheno-Kähler metrics on $\T^2$-bundles over (non-Kähler) homogeneous complex surfaces.