Local and global applications of the Minimal Model Program for co-rank one foliations on threefolds

TL;DR

This paper applies the minimal model program to study co-rank one foliations on threefolds, providing new local and global results including singularity classification, termination of flips, and hyperbolicity properties.

Contribution

It introduces novel applications of the minimal model program to co-rank one foliations, including classification of singularities and global geometric properties.

Findings

Proved a singular variant of Malgrange's theorem.

Classified terminal foliation singularities.

Established termination of flips and hyperbolicity properties.

Abstract

We provide several applications of the minimal model program to the local and global study of co-rank one foliations on threefolds. Locally, we prove a singular variant of Malgrange's theorem, a classification of terminal foliation singularities and the existence of separatrices for log canonical singularities. Globally, we prove termination of flips, a connectedness theorem on lc centres, a non-vanshing theorem and some hyperbolicity properties of foliations.