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.

20 pages. To be published in J. Lond. Math. Soc

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