Nef and abundant divisors, semiampleness and canonical bundle formula

TL;DR

This paper generalizes Kawamata's theorem by using canonical bundle formulas to show that nef and abundant log canonical divisors are semiample under certain conditions, extending the concept of abundance.

Contribution

It introduces a new approach to prove semiampleness for nef and abundant divisors using canonical bundle formulas, broadening the scope of Kawamata's theorem.

Findings

Nef and abundant divisors become semiample under specified conditions

Generalization of Kawamata's theorem to the setting of generalized abundance

Application of canonical bundle formulas to log canonical pairs

Abstract

In this paper, we use canonical bundle formulas to prove some generalizations of an old theorem of Kawamata on the semiampleness of nef and abundant log canonical divisors. In particular, we show that for klt pairs $(X,B)$ with $K_X+B$ effective, $L \in Pic (X)$ nef, nefness and abundance of $K_X+B+L$ implies semiampleness. This essentially generalizes Kawamata's theorem to the setting of generalized abundance.