Minimal model program on the generic fiber of log Calabi-Yau type fibration

Abstract

We study the minimal model program on the geometric generic fiber of a fibration $f:X\to S$ such that for a Zariski dense subset $S'\subseteq S$, $X_s$ is an $\varepsilon$-lc log Calabi--Yau type for every $s\in S'$. We prove that for a fibration $f:X\to S$ of varieties, if the fibers are of $\varepsilon$-lc log Calabi--Yau type, then the geometric generic fiber $X_{\overline{\eta}}$ is pklt. In particular, for any big divisor $D$ on $X_{\overline{\eta}}$, we can run the anticanonical MMP and $D$-MMP with scaling of an ample divisor on $X_{\overline{\eta}}$.