Finite generation of the log canonical ring in dimension four

TL;DR

This paper proves the finite generation of the log canonical ring and the abundance theorem for irregular fourfolds, advancing the minimal model program in four dimensions using new and traditional methods.

Contribution

It establishes the finite generation of the log canonical ring in four dimensions and proves the abundance conjecture for irregular fourfolds under certain assumptions.

Findings

Finite generation of the log canonical ring in dimension four.

Proof of the abundance conjecture for irregular fourfolds.

Utilization of Fukuda's theorem and traditional arguments.

Abstract

We treat two different topics on the log minimal model program, especially for four-dimensional log canonical pairs. (a) Finite generation of the log canonical ring in dimension four. (b) Abundance theorem for irregular fourfolds. We obtain (a) as a direct consequence of the existence of four-dimensional log minimal models by using Fukuda's theorem on the four-dimensional log abundance conjecture. We can prove (b) only by using traditional arguments. More precisely, we prove the abundance conjecture for irregular $(n+1)$-folds on the assumption that the minimal model conjecture and the abundance conjecture hold in dimension $\leq n$.