Minimal model program for algebraically integrable foliations on klt varieties

TL;DR

This paper establishes foundational results for the minimal model program in the context of algebraically integrable foliations on klt varieties, including base-point-freeness, contractions, flips, and existence of minimal models.

Contribution

It proves key theorems like base-point-freeness and flips for these foliations, removing previous assumptions and establishing the MMP in this setting.

Findings

Proved the base-point-freeness theorem for lc algebraically integrable foliations.

Established the existence of flips and contractions without assuming termination.

Showed that certain klt varieties with Fano foliations are Mori dream spaces.

Abstract

For lc algebraically integrable foliations on klt varieties, we prove the base-point-freeness theorem, the contraction theorem, and the existence of flips. The first result resolves a conjecture of Cascini and Spicer, while the latter two results strengthen a result of Cascini and Spicer by removing their assumption on the termination of flips. Moreover, we prove the existence of the minimal model program for lc algebraically integrable foliations on klt varieties and the existence of good minimal models or Mori fiber spaces for lc algebraically integrable foliations polarized by ample divisors on klt varieties. As a consequence, we show that $\mathbb{Q}$-factorial klt varieties with lc algebraically integrable Fano foliation structures are Mori dream spaces. We also show the existence of a Shokurov-type polytope for lc algebraically integrable foliations.