Vaisman's theorem and local reducibility

TL;DR

This paper broadens the understanding of Vaisman's theorem by identifying conditions under which it applies to compact K"ahler spaces that are locally reducible, extending previous results to more general settings.

Contribution

The paper provides a new sufficient condition ensuring Vaisman's theorem holds for compact K"ahler spaces that are locally reducible, beyond the previously known irreducible case.

Findings

Vaisman's theorem applies to a broader class of K"ahler spaces.

Counterexamples exist when local irreducibility is not assumed.

New conditions ensure the theorem's validity in reducible cases.

Abstract

As proven in a celebrated theorem due to Vaisman, pure locally conformally K\"ahler metrics do not exist on compact K\"ahler manifolds. In a previous paper, we extended this result to the singular setting, more precisely to K\"ahler spaces which are locally irreducible. Without the additional assumption of local irreducibility, there are counterexamples for which Vaisman's theorem does not hold. In this article, we give a much broader sufficient condition under which Vaisman's theorem still holds for compact K\"ahler spaces which are locally reducible.