On the relative Minimal Model Program for fourfolds in positive and mixed characteristic

TL;DR

This paper proves specific cases of the four-dimensional Minimal Model Program in characteristic p>5, including contractions and semi-stable families, and explores their implications for liftability and Calabi-Yau varieties.

Contribution

It establishes the validity of certain minimal model program cases in positive and mixed characteristic for fourfolds, extending known results and implications for liftability.

Findings

Validity of contractions to Q-factorial fourfolds in characteristic p>5

Semi-stable minimal model program over curves in positive characteristic

Liftability of threefolds and Calabi-Yau varieties as birational invariants

Abstract

We show the validity of two special cases of the four-dimensional Minimal Model Program in characteristic $p>5$: for contractions to $\mathbb{Q}$-factorial fourfolds and in families over curves ("semi-stable mmp"). We also provide their mixed characteristic analogues. As a corollary, we show that liftability of positive characteristic threefolds is stable under the minimal model program, and that liftability of three-dimensional Calabi-Yau varieties is a birational invariant. Our results are partially contingent upon the existence of log resolutions.