Rank one foliations on toroidal 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 $\hat{I}“$ such that $(X,\hat{I}“)$ is log canonical and $K_X+\hat{I}“\sim K_{\mathcal{F}}+D$. We then apply this result to prove several statements on the birational geometry of rank 1 log canonical foliations on log homogeneous varieties.

20 pages