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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: c.birkar@dpmms.cam.ac.uk)
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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)

Information

Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 4 , July 2010 , pp. 919 - 928
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Alexeev, V., Hacon, C. and Kawamata, Y., Termination of (many) four-dimensional log flips, Invent. Math. 168 (2007), 433–448.CrossRefGoogle Scholar
[2]Birkar, C., On existence of log minimal models II, arXiv:0907.4170v1.Google Scholar
[3]Birkar, C., Ascending chain condition for lc thresholds and termination of log flips, Duke Math. J. 136 (2007), 173–180.CrossRefGoogle Scholar
[4]Birkar, C., Log minimal models according to Shokurov, J. Algebra and Number Theory 3 (2009), 951–958.CrossRefGoogle Scholar
[5]Birkar, C., Cascini, P., Hacon, C. and McKernan, J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405–468.CrossRefGoogle Scholar
[6]Birkar, C. and Păun, M., Minimal models, flips and finite generation: a tribute to V. V. Shokurov and Y.-T. Siu, arXiv:0904.2936v1.Google Scholar
[7]Birkar, C. and Shokurov, V. V., Mld’s vs thresholds and flips, arXiv:math/0609539v1.Google Scholar
[8]Fujino, O., Finite generation of the log canonical ring in dimension four, arXiv:0803.1691v2.Google Scholar
[9]Fujino, O., Special termination and reduction to pl flips, in Flips for 3-folds and 4-folds (Oxford University Press, Oxford, 2007).Google Scholar
[10]Hacon, C. and McKernan, J., Extension theorems and the existence of flips, in Flips for 3-folds and 4-folds (Oxford University Press, Oxford, 2007).Google Scholar
[11]Kawamata, Y., A generalization of Kodaira–Ramanujam’s vanishing theorem, Math. Ann. 261 (1982), 43–46.CrossRefGoogle Scholar
[12]Kawamata, Y., Termination of log flips for algebraic 3-folds, Internat. J. Math. 3 (1992), 653–659.CrossRefGoogle Scholar
[13]Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model problem, in Algebraic geometry (Sendai, 1985), Advanced Studies in Pure Mathematics, vol. 10 (North-Holland, Amsterdam, 1987), 283–360.CrossRefGoogle Scholar
[14]Kollár, J. and Mori, S., Birational geometry of algebraic varieties (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
[15]Mori, S., Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), 133–176.CrossRefGoogle Scholar
[16]Mori, S., Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1 (1988), 117–253.CrossRefGoogle Scholar
[17]Shokurov, V. V., A nonvanishing theorem, Math. USSR Izv. 26 (1986), 591–604.CrossRefGoogle Scholar
[18]Shokurov, V. V., Three-dimensional log flips, Russ. Acad. Sci. Izv. Math. 40 (1993), 95–202, with an appendix in English by Yujiro Kawamata.Google Scholar
[19]Shokurov, V. V., 3-fold log models, J. Math. Sci. 81 (1996), 2667–2699.CrossRefGoogle Scholar
[20]Shokurov, V. V., Prelimiting flips, Proc. Steklov Inst. Math. 240 (2003), 75–213.Google Scholar
[21]Shokurov, V. V., Letters of a bi-rationalist V. Mld’s and termination of log flips, Proc. Steklov Inst. Math. 246 (2004), 315–336.Google Scholar
[22]Shokurov, V. V., Letters of a bi-rationalist VII. Ordered termination, arXiv:math/0607822v2.Google Scholar
[23]Viehweg, E., Vanishing theorems, J. Reine Angew. Math. 335 (1982), 1–8.Google Scholar