An inductive approach to generalized abundance using nef reduction

Abstract

We use the canonical bundle formula for parabolic fibrations to give an inductive approach to the generalized abundance conjecture using nef reduction. In particular, we observe that generalized abundance holds for a klt pair $(X,B)$ if the nef dimension $n(K_X+B+L)=2$ and $K_X+B \ge 0$ or $n(K_X+B+L)=3$ and $\kappa (K_X+B)>0$.