Vaisman theorem for lcK spaces

TL;DR

This paper extends Vaisman's theorem, originally for smooth lcK manifolds, to include compact complex spaces with singularities, broadening the understanding of lcK metrics in complex geometry.

Contribution

The paper generalizes Vaisman's theorem to singular compact complex spaces, providing new insights into lcK metrics beyond smooth manifolds.

Findings

Vaisman's theorem holds for certain singular complex spaces.

Extension of lcK metric properties to singular settings.

Broader applicability of conformal K"ahler geometry.

Abstract

Vaisman's theorem for locally conformally K\"ahler (lcK) compact manifolds states that any lcK metric on a compact complex manifold which admits a K\"ahler metric is, in fact, globally conformally K\"ahler (gcK). In this paper, we extend this theorem to compact complex spaces with singularities.