# On log canonical rings

@article{Fujino2013OnLC,
title={On log canonical rings},
author={Osamu Fujino and Yoshinori Gongyo},
journal={arXiv: Algebraic Geometry},
year={2013}
}
• Published 21 February 2013
• Mathematics
• arXiv: Algebraic Geometry
We discuss the relationship among various conjectures in the minimal model theory including the finite generation conjecture of the log canonical rings and the abundance conjecture. In particular, we show that the finite generation conjecture of the log canonical rings for log canonical pairs can be reduced to that of the log canonical rings for purely log terminal pairs of log general type.
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## References

SHOWING 1-10 OF 31 REFERENCES
Remarks on the non-vanishing conjecture
We discuss a difference between the rational and the real non-vanishing conjecture for pseudo-effective log canonical divisors of log canonical pairs. We also show the log non-vanishing theorem for
Finite generation of the log canonical ring in dimension four
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
Log pluricanonical representations and the abundance conjecture
• Mathematics
Compositio Mathematica
• 2014
Abstract We prove the finiteness of log pluricanonical representations for projective log canonical pairs with semi-ample log canonical divisor. As a corollary, we obtain that the log canonical
On existence of log minimal models II
Abstract We prove that the existence of log minimal models in dimension d essentially implies the LMMP with scaling in dimension d. As a consequence we prove that a weak nonvanishing conjecture in
Fundamental Theorems for the Log Minimal Model Program
In this paper, we prove the cone theorem and the contraction theorem for pairs (X;B), where X is a normal variety and B is an effective R-divisor on X such that KX +B is R-Cartier.
On existence of log minimal models
• C. Birkar
• Mathematics
Compositio Mathematica
• 2010
Abstract In this paper, we prove that the log minimal model program in dimension d−1 implies the existence of log minimal models for effective lc pairs (e.g. of non-negative Kodaira dimension) in
ON NUMERICALLY EFFECTIVE LOG CANONICAL DIVISORS
Let (X,Δ) be a 4-dimensional log variety which is proper over the field of complex numbers and with only divisorial log terminal singularities. The log canonical divisor KX
Higher direct images of log canonical divisors
In this paper, we investigate higher direct images of log canonical divisors. After we reformulate Kollár’s torsion-free theorem, we treat the relationship between higher direct images of log
New outlook on the Minimal Model Program, II
• Mathematics
• 2013
We prove that the finite generation of adjoint rings implies all the foundational results of the Minimal Model Program: the Rationality, Cone and Contraction theorems, the existence of flips, and
Extension theorems, non-vanishing and the existence of good minimal models
• Mathematics
• 2010
We prove an extension theorem for effective purely log-terminal pairs (X, S + B) of non-negative Kodaira dimension $${\kappa (K_X+S+B)\ge 0}$$ . The main new ingredient is a refinement of the