Letters of a Bi-rationalist. VII Ordered termination

Article

Abstract

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 the termination of terminal log flips in dimension d imply, for any log pair of dimension d, the existence of a resulting model: a strictly log minimal model or a strictly log terminal Mori log fibration, and imply the existence of log flips in dimension d + 1. As a consequence, we prove the existence of a resulting model of 4-fold log pairs, the existence of log flips in dimension 5, and Geography of log models in dimension 4.

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References

  1. 1.
    V. Alexeev, Ch. Hacon, and Yu. Kawamata, “Termination of (Many) 4-Dimensional Log Flips,” Invent. Math. 168, 433–448 (2007).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    F. Ambro, “Quasi-log Varieties,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 240, 220–239 (2003) [Proc. Steklov Inst. Math. 240, 214–233 (2003)].MathSciNetGoogle Scholar
  3. 3.
    Ch. D. Hacon and J. McKernan, “On the Existence of Flips,” arXiv:math.AG/0507597.Google Scholar
  4. 4.
    V. A. Iskovskikh and V. V. Shokurov, “Birational Models and Flips,” Usp. Mat. Nauk 60(1), 29–98 (2005) [Russ. Math. Surv. 60, 27–94 (2005)].MathSciNetGoogle Scholar
  5. 5.
    Y. Kawamata, K. Matsuda, and K. Matsuki, “Introduction to the Minimal Model Problem,” in Algebraic Geometry, Sendai, 1985 (Kinokuniya, Tokyo, 1987), Adv. Stud. Pure Math. 10, pp. 283–360.Google Scholar
  6. 6.
    Yu. G. Prokhorov and V. V. Shokurov, “Toward the Second Main Theorem on Complements,” J. Algebr. Geom. 18, 151–199 (2009).MATHMathSciNetGoogle Scholar
  7. 7.
    V. V. Shokurov, “The Nonvanishing Theorem,” Izv. Akad. Nauk SSSR, Ser. Mat. 49(3), 635–651 (1985) [Math. USSR, Izv. 26, 591–604 (1986)].MathSciNetGoogle Scholar
  8. 8.
    V. V. Shokurov, “3-fold Log Flips,” Izv. Ross. Akad. Nauk, Ser. Mat. 56(1), 105–203 (1992) [Russ. Acad. Sci., Izv. Math. 40, 95–202 (1993)].MathSciNetGoogle Scholar
  9. 9.
    V. V. Shokurov, “Anticanonical Boundedness for Curves,” Appendix to V. V. Nikulin, “The Diagram Method for 3-folds and Its Application to the Kähler Cone and Picard Number of Calabi-Yau 3-folds. I,” in Higher Dimensional Complex Varieties, Trento, June 1994, Ed. by M. Andreatta and T. Peternell (de Gruyter, Berlin, 1996), pp. 321–328.Google Scholar
  10. 10.
    V. V. Shokurov, “Letters of a Bi-rationalist. I: A Projectivity Criterion,” in Birational Algebraic Geometry (Am. Math. Soc., Providence, RI, 1997), Contemp. Math. 207, pp. 143–152.Google Scholar
  11. 11.
    V. V. Shokurov, “3-fold Log Models,” J. Math. Sci. 81, 2667–2699 (1996).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    V. V. Shokurov, “Prelimiting Flips,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 240, 82–219 (2003) [Proc. Steklov Inst. Math. 240, 75–213 (2003)].MathSciNetGoogle Scholar
  13. 13.
    V. V. Shokurov, “Letters of a Bi-rationalist. V: Mld’s and Termination of Log Flips,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 246, 328–351 (2004) [Proc. Steklov Inst. Math. 246, 315–336 (2004)].MathSciNetGoogle Scholar

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© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA
  2. 2.Steklov Institute of MathematicsRussian Academy of SciencesMoscowRussia

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