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.
Categories: math.AG