Fujiki Class $\mathcal{C}$ Varieties and a Kähler Criterion

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