Existence of minimal models for varieties of log general type
Authors:
Caucher Birkar, Paolo Cascini, Christopher D. Hacon and James McKernan
Journal:
J. Amer. Math. Soc. 23 (2010), 405-468
MSC (2010):
Primary 14E30
DOI:
https://doi.org/10.1090/S0894-0347-09-00649-3
Published electronically:
November 13, 2009
MathSciNet review:
2601039
Full-text PDF Free Access
View in AMS MathViewer
Abstract | References | Similar Articles | Additional Information
Abstract: We prove that the canonical ring of a smooth projective variety is finitely generated.
- 1. Valery Alexeev, Christopher Hacon, and Yujiro Kawamata, Termination of (many) 4-dimensional log flips, Invent. Math. 168 (2007), no. 2, 433–448. MR 2289869, https://doi.org/10.1007/s00222-007-0038-1
- 2. Caucher Birkar, Ascending chain condition for log canonical thresholds and termination of log flips, Duke Math. J. 136 (2007), no. 1, 173–180. MR 2271298, https://doi.org/10.1215/S0012-7094-07-13615-9
- 3. Daniel Bump, Lie groups, Graduate Texts in Mathematics, vol. 225, Springer-Verlag, New York, 2004. MR 2062813
- 4. Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi, Singular Kähler-Einstein metrics, J. Amer. Math. Soc. 22 (2009), no. 3, 607–639. MR 2505296, https://doi.org/10.1090/S0894-0347-09-00629-8
- 5. Osamu Fujino, Termination of 4-fold canonical flips, Publ. Res. Inst. Math. Sci. 40 (2004), no. 1, 231–237. MR 2030075
- 6. Osamu Fujino and Shigefumi Mori, A canonical bundle formula, J. Differential Geom. 56 (2000), no. 1, 167–188. MR 1863025
- 7. Angela Gibney, Sean Keel, and Ian Morrison, Towards the ample cone of \overline𝑀_{𝑔,𝑛}, J. Amer. Math. Soc. 15 (2002), no. 2, 273–294. MR 1887636, https://doi.org/10.1090/S0894-0347-01-00384-8
- 8. C. Hacon and J. McKernan, Existence of minimal models for varieties of log general type II, J. Amer. Math. Soc, posted on November 13, 2009, PII: S 0894-0347(09)00651-1.
- 9. 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
- 10. Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- 11. Brendan Hassett and Donghoon Hyeon, Log canonical models for the moduli space of curves: the first divisorial contraction, Trans. Amer. Math. Soc. 361 (2009), no. 8, 4471–4489. MR 2500894, https://doi.org/10.1090/S0002-9947-09-04819-3
- 12. Yi Hu and Sean Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786494, https://doi.org/10.1307/mmj/1030132722
- 13. Masayuki Kawakita, Inversion of adjunction on log canonicity, Invent. Math. 167 (2007), no. 1, 129–133. MR 2264806, https://doi.org/10.1007/s00222-006-0008-z
- 14. Yujiro Kawamata, The Zariski decomposition of log-canonical divisors, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 425–433. MR 927965
- 15. Yujiro Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2) 127 (1988), no. 1, 93–163. MR 924674, https://doi.org/10.2307/1971417
- 16. Yujiro Kawamata, On the length of an extremal rational curve, Invent. Math. 105 (1991), no. 3, 609–611. MR 1117153, https://doi.org/10.1007/BF01232281
- 17. Yujiro Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 419–423. MR 2426353, https://doi.org/10.2977/prims/1210167332
- 18. Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR 946243, https://doi.org/10.2969/aspm/01010283
- 19. János Kollár, Flips, flops, minimal models, etc, Surveys in differential geometry (Cambridge, MA, 1990) Lehigh Univ., Bethlehem, PA, 1991, pp. 113–199. MR 1144527
- 20. János Kollár, Effective base point freeness, Math. Ann. 296 (1993), no. 4, 595–605. MR 1233485, https://doi.org/10.1007/BF01445123
- 21. Flips and abundance for algebraic threefolds, Société Mathématique de France, Paris, 1992. Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991; Astérisque No. 211 (1992) (1992). MR 1225842
- 22. 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
- 23. Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471
- 24. Yuri Manin, Moduli stacks \overline𝐿_{𝑔,𝑆}, Mosc. Math. J. 4 (2004), no. 1, 181–198, 311 (English, with English and Russian summaries). MR 2074988, https://doi.org/10.17323/1609-4514-2004-4-1-181-198
- 25. Kenji Matsuki, Termination of flops for 4-folds, Amer. J. Math. 113 (1991), no. 5, 835–859. MR 1129294, https://doi.org/10.2307/2374787
- 26. Shigefumi Mori, Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1 (1988), no. 1, 117–253. MR 924704, https://doi.org/10.1090/S0894-0347-1988-0924704-X
- 27. David Mumford, Stability of projective varieties, Enseign. Math. (2) 23 (1977), no. 1-2, 39–110. MR 450272
- 28. Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR 2104208
- 29. Viacheslav V. Nikulin, The diagram method for 3-folds and its application to the Kähler cone and Picard number of Calabi-Yau 3-folds. I, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 261–328. With an appendix by Vyacheslav V. Shokurov. MR 1463184
- 30. Thomas Peternell, Towards a Mori theory on compact Kähler threefolds. II, Math. Ann. 311 (1998), no. 4, 729–764. MR 1637984, https://doi.org/10.1007/s002080050207
- 31. V. V. Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 1, 105–203 (Russian); English transl., Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95–202. MR 1162635, https://doi.org/10.1070/IM1993v040n01ABEH001862
- 32. V. V. Shokurov, 3-fold log models, J. Math. Sci. 81 (1996), no. 3, 2667–2699. Algebraic geometry, 4. MR 1420223, https://doi.org/10.1007/BF02362335
- 33. V. V. Shokurov, Letters of a bi-rationalist. I. A projectivity criterion, Birational algebraic geometry (Baltimore, MD, 1996) Contemp. Math., vol. 207, Amer. Math. Soc., Providence, RI, 1997, pp. 143–152. MR 1462930, https://doi.org/10.1090/conm/207/02725
- 34. V. V. Shokurov, Prelimiting flips, Tr. Mat. Inst. Steklova 240 (2003), no. Biratsion. Geom. Lineĭn. Sist. Konechno Porozhdennye Algebry, 82–219; English transl., Proc. Steklov Inst. Math. 1(240) (2003), 75–213. MR 1993750
- 35. V. V. Shokurov, Letters of a bi-rationalist. V. Minimal log discrepancies and termination of log flips, Tr. Mat. Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 328–351 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 3(246) (2004), 315–336. MR 2101303
- 36. Y-T. Siu, A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring. arXiv:math.AG/0610740
- 37. Kenji Ueno, Bimeromorphic geometry of algebraic and analytic threefolds, Algebraic threefolds (Varenna, 1981) Lecture Notes in Math., vol. 947, Springer, Berlin-New York, 1982, pp. 1–34. MR 672613
Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 14E30
Retrieve articles in all journals with MSC (2010): 14E30
Additional Information
Caucher Birkar
Affiliation:
DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
Email:
c.birkar@dpmms.cam.ac.uk
Paolo Cascini
Affiliation:
Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106 and Imperial College London, 180 Queens Gate, London SW7 2A2, United Kingdom
Email:
cascini@math.ucsb.edu, p.cascini@imperial.ac.uk
Christopher D. Hacon
Affiliation:
Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, Utah 84112
Email:
hacon@math.utah.edu
James McKernan
Affiliation:
Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106 and Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email:
mckernan@math.ucsb.edu, mckernan@math.mit.edu
DOI:
https://doi.org/10.1090/S0894-0347-09-00649-3
Received by editor(s):
August 13, 2008
Published electronically:
November 13, 2009
Additional Notes:
The first author was partially supported by EPSRC grant GR/S92854/02
The second author was partially supported by NSF research grant no: 0801258
The third author was partially supported by NSF research grant no: 0456363 and an AMS Centennial fellowship
The fourth author was partially supported by NSA grant no: H98230-06-1-0059 and NSF grant no: 0701101 and would like to thank Sogang University and Professor Yongnam Lee for their generous hospitality, where some of the work for this paper was completed
All authors would like to thank Dan Abramovich, Valery Alexeev, Florin Ambro, Tommaso de Fernex, Stephane Dreul, Seán Keel, Kalle Karu, János Kollár, Sándor Kovács, Michael McQuillan, Shigefumi Mori, Martin Olsson, Genia Tevelev, Burt Totaro, Angelo Vistoli and Chengyang Xu for answering many of our questions and pointing out some errors in an earlier version of this paper. They would also like to thank the referee for some useful comments.
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.