## Kyoto Journal of Mathematics

### Finite generation of the log canonical ring in dimension four

Osamu Fujino

#### 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 ${}\leq n$.

#### Article information

Source
Kyoto J. Math., Volume 50, Number 4 (2010), 671-684.

Dates
First available in Project Euclid: 29 November 2010

https://projecteuclid.org/euclid.kjm/1291041214

Digital Object Identifier
doi:10.1215/0023608X-2010-010

Mathematical Reviews number (MathSciNet)
MR2740690

Zentralblatt MATH identifier
1210.14020

Subjects
Primary: 14J35: $4$-folds
Secondary: 14E30: Minimal model program (Mori theory, extremal rays)

#### Citation

Fujino, Osamu. Finite generation of the log canonical ring in dimension four. Kyoto J. Math. 50 (2010), no. 4, 671--684. doi:10.1215/0023608X-2010-010. https://projecteuclid.org/euclid.kjm/1291041214

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