A note on the abundance conjecture

TL;DR

This paper shows that proving the abundance conjecture for non-uniruled klt pairs in a given dimension can imply its validity for uniruled klt pairs in the same dimension, assuming the MMP in lower dimensions.

Contribution

It establishes a logical link between the abundance conjecture for non-uniruled and uniruled klt pairs under the MMP framework.

Findings

Proves the implication from non-uniruled to uniruled cases in the abundance conjecture.

Assumes the validity of the Minimal Model Program in lower dimensions.

Provides a new approach to tackling the abundance conjecture for uniruled pairs.

Abstract

We prove that the abundance conjecture for non-uniruled klt pairs in dimension $n$ implies the abundance conjecture for uniruled klt pairs in dimension $n$, assuming the Minimal Model Program in lower dimensions.