Rank one foliations on toroidal varieties

TL;DR

This paper studies rank one foliations on toroidal varieties, establishing a relation between their canonical divisors and log canonical pairs, with applications to birational geometry.

Contribution

It proves the existence of a divisor relating the canonical class of the foliation to a log canonical pair, advancing understanding of foliations on toroidal varieties.

Findings

Existence of a divisor $Gamma$ such that $(X, Gamma)$ is log canonical and relates to the foliation's canonical class.

Application of the main result to birational geometry of rank 1 log canonical foliations.

Insights into the structure of foliations on log homogeneous varieties.

Abstract

Consider a log canonical pair $(X,B)$ such that there is a Cartier divisor $D$ for which $T_X(-\log B) \otimes \mathcal O(D)$ is locally free and globally generated. Let $\mathcal F$ be a log canonical foliation of rank 1 on $X$. We prove that there exists a divisor $\Gamma$ such that $(X, \Gamma)$ is log canonical and $K_X + \Gamma \sim K_{\mathcal F} + D$. We then apply this result to prove several statements on the birational geometry of rank 1 log canonical foliations on log homogeneous varieties.