On astheno-Kähler metrics

A Hermitian metric on a complex manifold of complex dimension n is called astheno-Kaehler if its fundamental 2-form F satisfies the condition ∂∂ ̄F^{ n−2} = 0. If n = 3, then the metric is strong KT, that is, F is ∂∂ ̄-closed. By using blow-ups and the twist construction, we construct simply connected astheno-Kaehler manifolds of complex dimension n > 3. Moreover, we construct a family of astheno-Kaehler (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 the astheno-Kaehler condition and the (locally) conformally balanced condition and we provide examples of locally conformally balanced astheno-Kaehler metrics on T^2-bundles over (non-Kaehler) homogeneous complex surfaces.