A basepoint free theorem for algebraically integrable foliations

TL;DR

This paper proves a semi-ampleness result for the canonical divisor of algebraically integrable foliations under certain positivity conditions, extending the basepoint free theorem to foliated varieties.

Contribution

It establishes a basepoint free theorem for algebraically integrable foliations on $Q$-factorial varieties, including applications to contraction theorems and the b-semiampleness conjecture.

Findings

Proves semi-ampleness of $K_{F}+A+B$ under specified conditions.

Provides a contraction theorem for F-dlt pairs.

Offers a case of the b-semiampleness conjecture.

Abstract

We show that if $\mathcal{F}$ is an algebraically integrable foliation on a $\mathbb{Q}$-factorial normal projective variety $X$, $ A, B \geq 0$ are $\mathbb{Q}$-divisors on $X$ with $A$ ample such that $(\mathcal{F}, B)$ is foliated dlt and $K_{\mathcal{F}}+ A+B$ is nef, then $K_{\mathcal{F}}+A+B$ is semiample. We also provide some applications of this and related results such as contraction theorem for F-dlt pairs and a special case of the b-semiampleness conjecture.