Complex geometry and Hermitian metrics on the product of two Sasakian manifolds

Abstract

A Sasakian manifold is a Riemannian manifold whose metric cone admits a certain Kähler structure which behaves well under homotheties. We show that the product of two compact Sasakian manifolds admits a family of complex structures indexed by a complex nonreal parameter, none of whose members admits either compatible Kähler metrics, locally conformally Kähler metrics or balanced metrics, if both Sasakian manifolds are of dimension greater than 3. We compare this family with another family of complex structures which has been studied in the literature. We compute the Dolbeault cohomology groups of these products of compact Sasakian manifolds.