Nef and abundant divisors, semiampleness and canonical bundle formula

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.