Nef and abundant divisors, semiampleness and canonical bundle formula

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In this paper, we use canonical bundle formulas to prove various 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 [Formula: see text] with [Formula: see text] effective, [Formula: see text] nef, nefness and abundance of [Formula: see text] implies semiampleness. This essentially extends Kawamata’s theorem to the setting of generalized abundance.