Advertisement

On a connectedness principle of Shokurov-Kollár type

  • Christopher D. Hacon
  • Jingjun Han
Articles
  • 11 Downloads

Abstract

Let (X, Δ) be a log pair over S, such that-(KX + Δ) is nef over S. It is conjectured that the intersection of the non-klt (non Kawamata log terminal) locus of (X, Δ) with any fiber Xs has at most two connected components. We prove this conjecture in dimension no larger than 4 and in arbitrary dimension assuming the termination of klt flips.

Keywords

minimal model program non-klt locus connectedness 

MSC(2010)

14E30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The first author was supported by National Science Foundation of USA (Grant Nos. DMS-1300750 and DMS-1265285) and by a grant from the Simons Foundation (Grant No. 256202). The first author thanks the Department of Mathematics and the Research Institute for Mathematical Sciences located in Kyoto University. Much of this work was done when the second author visited the first author at the University of Utah, the second author thanks the University of Utah for its hospitality. The authors thank Chen Jiang for providing useful comments on previous drafts. The second author thanks his advisors Gang Tian and Chenyang Xu for constant support and encouragement.

References

  1. 1.
    Alexeev V, Hacon C, Kawamata Y. Termination of (many) 4-dimensional log flips. Invent Math, 2007, 168: 433–448MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Birkar C. On existence of log minimal models. Compos Math, 2010, 146: 919–928MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Fujino O. Abundance theorem for semi log canonical threefolds. Duke Math J, 2000, 102: 513–532MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gongyo Y. On weak Fano varieties with log canonical singularities. J Reine Angew Math, 2012, 665: 237–252MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hacon C D, McKernan J, Xu C. ACC for log canonical thresholds. Ann of Math (2), 2014, 180: 523–571MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kollár J. Singularities of the Minimal Model Program. Cambridge Tracts in Mathematics, vol. 200. Cambridge: Cambridge University Press, 2013Google Scholar
  7. 7.
    Kollár J, Kovács S J. Log canonical singularities are Du Bois. J Amer Math Soc, 2010, 23: 791–813MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kollár J, Mori S. Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134. Cambridge: Cambridge University Press, 1998Google Scholar
  9. 9.
    Prokhorov Y G. Lectures on Complements on Log Surfaces. MSJ Memoirs, vol. 10. Tokyo: Mathematical Society of Japan, 2001Google Scholar
  10. 10.
    Shokurov V V. Three-dimensional log perestroikas. Izv Ross Akad Nauk Ser Mat, 1992, 56: 105–203Google Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  2. 2.Beijing International Center for Mathematical ResearchPeking UniversityBeijingChina
  3. 3.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA

Personalised recommendations