# Existence of log canonical flips and a special LMMP

@article{Birkar2011ExistenceOL,
title={Existence of log canonical flips and a special LMMP},
author={Caucher Birkar},
journal={Publications math{\'e}matiques de l'IH{\'E}S},
year={2011},
volume={115},
pages={325-368}
}
• C. Birkar
• Published 2011
• Mathematics
• Publications mathématiques de l'IHÉS
Let (X/Z,B+A) be a Q-factorial dlt pair where B,A≥0 are Q-divisors and KX+B+A∼Q0/Z. We prove that any LMMP/Z on KX+B with scaling of an ample/Z divisor terminates with a good log minimal model or a Mori fibre space. We show that a more general statement follows from the ACC for lc thresholds. An immediate corollary of these results is that log flips exist for log canonical pairs.
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Termination of (many) 4-dimensional log flips
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• 2007
We prove that any sequence of 4-dimensional log flips that begins with a klt pair (X,D) such that -(KX+D) is numerically equivalent to an effective divisor, terminates. This implies termination ofExpand
Letters of a Bi-rationalist. VII Ordered termination
To construct a resulting model in the LMMP, it is sufficient to prove the existence of log flips and their termination for some sequences. We prove that the LMMP in dimension d − 1 and theExpand
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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) AbundanceExpand
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• Compositio Mathematica
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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 canonicalExpand
On Finiteness of B-representation and Semi-log Canonical Abundance
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We give a new proof of the finiteness of B-representations. As a consequence of the finiteness of B-representations and Koll\'ar's gluing theory on lc centers, we prove that the (relative) abundanceExpand
On existence of log minimal models
• C. Birkar
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• 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) inExpand
On existence of log minimal models and weak Zariski decompositions
We first introduce a weak type of Zariski decomposition in higher dimensions: an $${\mathbb {R}}$$ -Cartier divisor has a weak Zariski decomposition if birationally and in a numerical sense it can beExpand