Blow-ups and modifications of lcK spaces

TL;DR

This paper investigates the behavior of locally conformally Kähler (lcK) spaces under blow-ups and modifications, establishing conditions for when these spaces retain lcK properties and introducing quasi-lcK metrics for modified spaces.

Contribution

It generalizes a known theorem to singular spaces and shows that modifications of lcK spaces always admit quasi-lcK metrics, expanding understanding of lcK geometry.

Findings

Blow-up of lcK spaces under certain conditions remains lcK.

Modifications of lcK spaces are not always lcK but admit quasi-lcK metrics.

Generalization of Ornea-Verbitsky-Vuletescu theorem to singular spaces.

Abstract

In this article, we prove that the blow-up of a locally irreducible lcK space $X$ along a subspace $Z$ which verifies certain conditions is lcK if and only if $X$ is induced gcK, generalizing a theorem of Ornea-Verbitsky-Vuletescu to singular locally irreducible spaces. We also show that even if modifications of lcK spaces are not always of lcK type, they always admit quasi-lcK metrics.