Fujiki Class $\mathcal C$ Varieties and a K\"ahler Criterion

Abstract

In this article, we show that flips and divisorial contractions preserve the K\"ahler condition (for strongly $\mathbb{Q}$-factorial compact K\"ahler generalized klt pairs with $B+\beta_X$ big), and we give a criterion for varieties in Fujiki's class $\mathcal C$ to be K\"ahler. We also prove the existence of small $\mathbb Q$-factorializations for generalized klt pairs and of dlt modifications for generalized pairs.