Do products of compact complex manifolds admit LCK metrics?
Differential Geometry
2024-07-08 v2 Algebraic Geometry
Complex Variables
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.
Keywords
Cite
@article{arxiv.2211.08111,
title = {Do products of compact complex manifolds admit LCK metrics?},
author = {Liviu Ornea and Misha Verbitsky and Victor Vuletescu},
journal= {arXiv preprint arXiv:2211.08111},
year = {2024}
}
Comments
15 pages, version 2.0, final version, many small changes