Do products of compact complex manifolds admit LCK metrics?

TL;DR

This paper investigates whether products of compact complex manifolds can admit LCK metrics, proving that such products do not support LCK structures if the factors belong to certain classified types.

Contribution

It establishes that the product of an LCK manifold with another LCK manifold of specific classes cannot admit an LCK metric, advancing understanding of LCK geometric structures.

Findings

Products of LCK manifolds of certain classes do not admit LCK metrics.

Classified all known compact LCK manifolds into three categories.

Proved non-existence of LCK structures on these product manifolds.

Abstract

An LCK (locally conformally Kahler) manifold is a Hermitian manifold which admits a Kahler cover with deck group acting by holomorphic homotheties with respect to the Kahler 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.