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On existence of log minimal models

Part of: Birational geometry

Published online by Cambridge University Press:  05 February 2010

Caucher Birkar
Caucher Birkar*
Affiliation:
DPMMS, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge, CB3 0WB, UK (email: [email protected])
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Abstract

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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 minimal models for effective lc pairs in dimension four and of the Shokurov reduction theorem.

Keywords

minimal modelsMori fibre spacesLMMP with scaling

MSC classification

Secondary: 14E30: Minimal model program (Mori theory, extremal rays)
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 4 , July 2010 , pp. 919 - 928
Copyright
Copyright © Foundation Compositio Mathematica 2010

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