On compact astheno-Kähler manifolds (Q2759164)

The authors obtain results on the class of Hermitian manifolds, called astheno-Kähler. These are complex Hermitian \(m\)-dimensional manifolds \( \left( M,g,J\right) \) for which the Kähler form \(\Omega \) satisfies \( \partial \overline{\partial }\Omega ^{m-2}=0,\) where \(\partial \) and \( \overline{\partial }\) are the complex exterior differentials and \(\Omega ^{m-2}=\Omega \wedge ...\wedge \Omega .\) If a compact astheno-Kähler manifold is balanced, that is, the torsion 1-form of the Hermitian connection vanishes everywhere, the authors prove that such a manifold is necessarily Kähler. Consequently, compact Hermitian-flat astheno-Kähler manifolds are Kähler. The authors also give examples of astheno-Kähler manifolds as products of 3-dimensional compact Sasakian manifolds, and as products of a 3-dimensional compact Sasakian manifold with an \(n\) -dimensional compact cosymplectic manifold for \(n\geq 3.\)