Kyoto Journal of Mathematics

Finite generation of the log canonical ring in dimension four

Osamu Fujino

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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 n.

Article information

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

First available in Project Euclid: 29 November 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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.

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