Non-vanishing theorem for log canonical pairs
Author:
Osamu Fujino
Journal:
J. Algebraic Geom. 20 (2011), 771-783
DOI:
https://doi.org/10.1090/S1056-3911-2010-00558-9
Published electronically:
January 3, 2011
MathSciNet review:
2819675
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Abstract |
References |
Additional Information
Abstract: We obtain a correct generalization of Shokurov’s non-vanishing theorem for log canonical pairs. It implies the base point free theorem for log canonical pairs. We also prove the rationality theorem for log canonical pairs. As a corollary, we obtain the cone theorem for log canonical pairs. We do not need Ambro’s theory of quasi-log varieties.
References
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- Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. MR 2601039, DOI https://doi.org/10.1090/S0894-0347-09-00649-3
- Alessio Corti (ed.), Flips for 3-folds and 4-folds, Oxford Lecture Series in Mathematics and its Applications, vol. 35, Oxford University Press, Oxford, 2007. MR 2352762
- O. Fujino, Introduction to the log minimal model program for log canonical pairs, preprint (2009).
- Osamu Fujino, On injectivity, vanishing and torsion-free theorems for algebraic varieties, Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 8, 95–100. MR 2561896, DOI https://doi.org/10.3792/pjaa.85.95
- O. Fujino, Introduction to the theory of quasi-log varieties, to appear in the proceeding of the “Classification of Algebraic Varieties” conference, Schiermonnikoog, Netherlands, May 10–15, 2009.
- O. Fujino, Fundamental theorems for the log minimal model program, to appear in Publ. Res. Inst. Math. Sci.
- Yujiro Kawamata, On the length of an extremal rational curve, Invent. Math. 105 (1991), no. 3, 609–611. MR 1117153, DOI https://doi.org/10.1007/BF01232281
- Steven L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293–344. MR 206009, DOI https://doi.org/10.2307/1970447
- J. Kollár, S. Kovács, Log canonical singularities are Du Bois, preprint (2009).
- János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959
- V. V. Shokurov, A nonvanishing theorem, Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), no. 3, 635–651 (Russian). MR 794958
References
- F. Ambro, Quasi-log varieties, Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 220–239; translation in Proc. Steklov Inst. Math. 2003, no. 1 (240), 214–233. MR 1993751 (2004f:14027)
- C. Birkar, P. Cascini, C. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. MR 2601039
- O. Fujino, What is log terminal?, in Flips for $3$-folds and $4$-folds (Alessio Corti, ed.), 29–62, Oxford University Press, 2007. MR 2352762 (2008j:14031)
- O. Fujino, Introduction to the log minimal model program for log canonical pairs, preprint (2009).
- O. Fujino, On injectivity, vanishing and torsion-free theorems for algebraic varieties, Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 8, 95–100. MR 2561896
- O. Fujino, Introduction to the theory of quasi-log varieties, to appear in the proceeding of the “Classification of Algebraic Varieties” conference, Schiermonnikoog, Netherlands, May 10–15, 2009.
- O. Fujino, Fundamental theorems for the log minimal model program, to appear in Publ. Res. Inst. Math. Sci.
- Y. Kawamata, On the length of an extremal rational curve, Invent. Math. 105 (1991), no. 3, 609–611. MR 1117153 (92m:14026)
- S. L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293–344. MR 0206009 (34:5834)
- J. Kollár, S. Kovács, Log canonical singularities are Du Bois, preprint (2009).
- J. Kollár, S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, Vol. 134, 1998. MR 1658959 (2000b:14018)
- V. V. Shokurov, The nonvanishing theorem, Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), no. 3, 635–651. MR 794958 (87j:14016)
Additional Information
Osamu Fujino
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502 Japan
MR Author ID:
652921
Email:
fujino@math.kyoto-u.ac.jp
Received by editor(s):
March 23, 2009
Received by editor(s) in revised form:
December 1, 2009
Published electronically:
January 3, 2011
Additional Notes:
The author was partially supported by The Inamori Foundation and by the Grant-in-Aid for Young Scientists (A) $\sharp$20684001 from JSPS
