An abundance theorem for generalised pairs

TL;DR

This paper proves finiteness results for B-representations of generalized pairs and establishes an abundance theorem for generalized dlt pairs, advancing the minimal model program in algebraic geometry.

Contribution

It introduces new finiteness and abundance results for generalized pairs, linking semi-log canonical pairs to their normalizations and supporting the minimal model program.

Findings

Finiteness of B-representations for generalized log canonical pairs

Abundance for generalized semi-log canonical pairs follows from their normalizations

An abundance theorem for generalized dlt pairs with abundant data

Abstract

We prove the finiteness of $B$-representations of generalised log canonical pairs. As a consequence, we prove that, the (relative) abundance for a generalised semi-log canonical pair is implied by the abundance for its normalisation. Furthermore, we obtain an abundance theorem for generalised dlt pairs with abundant data, which is a natural inductive step of generalised minimal model program.