Blow-ups and modifications of lcK 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.