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

TL;DR

This paper investigates the minimal model program on the generic fiber of a fibration with log Calabi-Yau type fibers, establishing conditions under which the generic fiber is pklt and enabling MMP runs.

Contribution

It proves that the geometric generic fiber of a fibration with epsilon-lc log Calabi-Yau fibers is pklt, allowing MMP procedures on the generic fiber.

Findings

The geometric generic fiber is pklt under certain conditions.

MMP can be run on the generic fiber with big divisors.

Results apply to fibrations with epsilon-lc log Calabi-Yau fibers.

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 $X_{\overline{\eta}}$.