Do products of compact complex manifolds admit LCK metrics?

Abstract

An locally conformally Kähler (LCK) manifold is a Hermitian manifold which admits a Kähler cover with deck group acting by holomorphic homotheties with respect to the Kähler metric. The product of two LCK manifolds does not have a natural product LCK structure. It is conjectured that a product of two compact complex manifolds is never LCK. We classify all known examples of compact LCK manifolds onto three not exclusive classes: LCK with potential, a class of manifolds we call of Inoue type, and those containing a rational curve. In the present paper, we prove that a product of an LCK manifold and an LCK manifold belonging to one of these three classes does not admit an LCK structure.