On astheno-Kähler metrics

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