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 $\le n$.

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

References in corpus (5)

• Introduction to the log minimal model program for log canonical pairs
• Existence of minimal models for varieties of log general type
• On existence of log minimal models
• The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension
• Fundamental theorems for the log minimal model program

Cited by in corpus (16)

• Introduction to the log minimal model program for log canonical pairs
• On existence of log minimal models
• Some remarks on the minimal model program for log canonical pairs
• Introduction to the theory of quasi-log varieties
• The GIT-stability of Polarised Varieties via discrepancy
• On the canonical line bundle and negative holomorphic sectional curvature
• On maximal Albanese dimensional varieties
• Minimal model theory for log surfaces
• Existence of log canonical flips and a special LMMP
• Strictly nef divisors on K-trivial fourfolds
• Minimal model theory for relatively trivial log canonical pairs
• Log Iitaka conjecture for abundant log canonical fibrations
• Minimal model theory for log surfaces in Fujiki's class $\mathcal{C}$
• Remarks on the abundance conjecture
• On termination of log flips in dimension four
• Some non-vanishing results on log canonical pairs of dimension 4