ar X iv : 0 80 6 . 07 35 v 1 [ m at h . D G ] 4 J un 2 00 8 ON ASTHENO - KÄHLER METRICS

A Hermitian metric on a complex manifold of complex dimension n is called astheno-Kähler if its fundamental 2-form F satisfies the condition ∂∂F n−2 = 0 and it is strong KT if F is ∂∂-closed. We prove that a conformally balanced astheno-Kähler metric on a compact manifod of complex dimension n ≥ 3, whose Bismut connection has (restricted) holonomy contained in SU(n), is necessarily Kähler. We provide compact examples of locally conformally balanced astheno-Kähler manifolds of complex dimension 3 for which the trace of R∧R vanishes, where R is the curvature of their Bismut connection. We study blow-ups of astheno-Kähler manifolds for which ∂∂F = 0 and ∂∂F 2 = 0 and we apply these results to orbifolds. Finally, we construct a family of astheno-Kähler 2-step nilmanifolds of complex dimension 4, showing that, in general, for n > 3, there is no relation between the astheno-Kähler and strong KT condition.