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 $K_X+B \geq 0$ or $n(K_X+B+L)=3$ and $\kappa(K_X+B )>0$.