An inductive approach to generalized abundance using nef reduction

TL;DR

This paper introduces an inductive method leveraging nef reduction and the canonical bundle formula to advance the generalized abundance conjecture for certain klt pairs with specific nef dimensions and positivity conditions.

Contribution

It provides a new inductive approach to the generalized abundance conjecture using nef reduction and the canonical bundle formula, establishing results for particular nef dimensions.

Findings

Generalized abundance holds for n(K_X+B+L)=2 with K_X+B ≥ 0.

It also holds for n(K_X+B+L)=3 when κ(K_X+B)>0.

The approach applies to klt pairs with specified nef dimensions.

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 $\kappa(K_X+B )>0$.