Finite generation of the log canonical ring in dimension four

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

2008-03-12

arxiv.org/abs/0803.1691v3

Mathematics

Algebraic Geometry

14 pages; v2: completely revised and expanded version, v3: Section 5 in v2 was removed because it contained a conceptual mista

Scientific paper

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 $\leq n$.

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