Finite generation of the log canonical ring in dimension four

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 $\le n$.