Bounding singular surfaces via Chern numbers

  • Joaquín MoragaEmail author


We prove the existence of a bound on the number of steps of the minimal model program for singular surfaces in terms of discrepancies and Chern numbers. As an application, we prove that given \(R\in {\mathbb {R}}_{>0}\) and \(\epsilon \in (0,1)\), the class \({\mathcal {F}}(R,\epsilon )\) of 2-dimensional pairs (XD) of general type with \(\epsilon \)-klt singularities, D with standard coefficients, and \(4c_2(X,D)-c_1^2(X,D)\le R\), forms a bounded family.

Mathematics Subject Classification

Primary 14E30 Secondary 14J17 



The author would like to thank Christopher Hacon and Valery Alexeev for many useful comments. He would like to express his gratitude to the anonymous referees for their careful report and the many suggestions.


  1. 1.
    Alexeev, V.: Two two-dimensional terminations. Duke Math. J. 69(3), 527–545 (1993). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alexeev, V.: Boundedness and \(K^2\) for log surfaces. Internat. J. Math. 5(6), 779–810 (1994). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alexeev, V., Mori, S.: Bounding singular surfaces of general type. Algebra, Arithmetic and Geometry with Applications, vol. 2004, pp. 143–174. Springer, Berlin (2004)CrossRefGoogle Scholar
  4. 4.
    Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bogomolov, F.A.: Holomorphic tensors and vector bundles on projective manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 42(6):1227–1287 (1978) (in Russian)Google Scholar
  6. 6.
    Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics, vol. 52. Springer Verlag, New York (1977)CrossRefGoogle Scholar
  7. 7.
    Hacon, C., McKernan, J., Xu, C.: Boundedness of moduli of varieties of general type. arXiv:1412.1186 (2014)
  8. 8.
    Kollár, J., Mori, S.: Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original (1998)Google Scholar
  9. 9.
    Kollar, J., et al.: 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)Google Scholar
  10. 10.
    Kollár, J.: Singularities of the minimal model program. Cambridge Tracts in Mathematics, vol. 200. Cambridge University Press, Cambridge. With a collaboration of Sándor Kovács (2013)Google Scholar
  11. 11.
    Kobayashi, R.: Uniformization of complex surfaces. In: Kähler Metric and Moduli Spaces. Advanced Studies in Pure Mathematics, vol. 18, Academic Press, Boston, pp. 313–394 (1990)Google Scholar
  12. 12.
    Langer, A.: Chern classes of reflexive sheaves on normal surfaces. Math. Z. 235(3), 591–614 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Langer, A.: The Bogomolov–Miyaoka–Yau inequality for log canonical surfaces. J. Lond. Math. Soc. (2) 64(2), 327–343 (2001). MathSciNetCrossRefGoogle Scholar
  14. 14.
    Langer, A.: The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic. Duke Math. J. 165(14), 2737–2769 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Megyesi, G.: Generalisation of the Bogomolov–Miyaoka–Yau inequality to singular surfaces. Proc London Math Soc (3) 78(2), 241–282 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Miyaoka, Y.: On the Chern numbers of surfaces of general type. Invent. Math. 42, 225–237 (1977). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Miyaoka, Y.: The maximal number of quotient singularities on surfaces with given numerical invariants. Math. Ann. 268(2), 159–171 (1984). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sakai, F.: Semistable curves on algebraic surfaces and logarithmic pluricanonical maps. Math. Ann. 254(2), 89–120 (1980). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tsunoda, S.: Structure of open algebraic surfaces. I. J. Math. Kyoto Univ. 23(1), 95–125 (1983)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wahl, J.: Second Chern class and Riemann–Roch for vector bundles on resolutions of surface singularities. Math. Ann. 295(1), 81–110 (1993). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Yau, S.T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. USA 74(5), 1798–1799 (1977)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

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