# On existence of log minimal models

@article{Birkar2010OnEO,
title={On existence of log minimal models},
author={C. Birkar},
journal={Compositio Mathematica},
year={2010},
volume={146},
pages={919 - 928}
}
• C. Birkar
• Published 2010
• Mathematics
• Compositio Mathematica
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 dimension d. In fact, we prove that the same conclusion follows from a weaker assumption, namely, the log minimal model program with scaling in dimension d−1. This enables us to prove that effective lc pairs in dimension five have log minimal models. We also give new proofs of the existence of log… Expand
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We prove that one can run the log minimal model program for log canonical $3$-fold pairs in characteristic $p>5$. In particular we prove the Cone Theorem, Contraction Theorem, the existence of flipsExpand
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• Mathematics
• 2008
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Supplement to the paper "On existence of log minimal models II"
We prove a stronger version of a termination theorem appeared in the paper "On existence of log minimal models II". We essentially just get rid of the redundant assumptions so the proof is almost theExpand
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• 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 theExpand

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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 inExpand
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) AbundanceExpand
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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Following Shokurov’s ideas, we give a short proof of the following klt version of his result: termination of terminal log flips in dimension d implies that any klt pair of dimension d has a logExpand
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In this paper, we discuss a proof of existence of log minimal models or Mori fibre spaces for klt pairs $(X/Z,B)$ with $B$ big$/Z$. This then implies existence of klt log flips, finite generation ofExpand
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In this paper, we discuss a proof of existence of log minimal models or Mori fibre spaces for klt pairs (X/Z, B) with B big/Z. This then implies existence of klt log flips, finite generation of kltExpand
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Abstract Minimal log discrepancies (mld's) are related not only to termination of log flips [Shokurov, Algebr. Geom. Metody 246: 328–351, (2004)] but also to the ascending chain condition (ACC) ofExpand
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We prove that 3-fold log flips exist. We deduce the existence of log canonical and -factorial log terminal models, as well as a positive answer to the inversion problem for log canonical and logExpand
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Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.