Diophantine approximations and foliations

  • Michael McQuillan


In this paper we indicate the proof of an effective version of the Green-Griffiths conjecture for surfaces of general type and positive second Segre class (i.e.c12>c2). Naturally this effective version is stronger than the Green-Griffiths conjecture itself.


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  1. [A-S-S]
    E. Arrondo, I. Sols, B. Speiser,Global Moduli For Contacts, MSRI preprint (1992).Google Scholar
  2. [B-B]
    P. Baum, R. Bott, Singularities Of Holomorphic Foliations,J. Differential Geometry7 (1972), 279–342.MATHMathSciNetGoogle Scholar
  3. [B1]
    F. Bogomolov, Families Of Curves On Surfaces Of General Type,Soviet Math. Dokl.18 (1977), 1294–1297.MATHGoogle Scholar
  4. [B2]
    F. Bogomolov, Holomorphic tensors and vector bundles on projective varieties,Math. USSR Izv.13 (1978), 499–555.MATHCrossRefGoogle Scholar
  5. [Bo]
    E. Bombieri, The Mordell Conjecture Revisited,Ann. Scuola Norm. Sup. Pisa Cl. Sci (4)17 (1990), no 4, 615–640. Errata-corrigé: The Mordell conjecture revisited,Ann. Scuola Norm. Sup. Pisa Cl. Sci (4)18 (1991), no 3, 473.MATHMathSciNetGoogle Scholar
  6. [C]
    A. Connes,Non Commutative Geometry, Academic Press, 1994.Google Scholar
  7. [C-M]
    D. Cerveau, J.-F. Mattei, Formes intégrables holomorphes singulières,Astérisque vol. 97 (1982).Google Scholar
  8. [D]
    M. Deschamps, Courbes de genres géométrique borné sur une surface de type général (d’après F. A. Bogomolov),Séminaire Bourbaki519 (1977/1978).Google Scholar
  9. [De1]
    J. P. Demailly, Monge-Ampère operators, Lelong numbers, and intersection theory,Complex Analysis and Geometry, V. Ancona and A. Silva Eds, Plenum, New York, London (1993).Google Scholar
  10. [De2]
    J. P. Demailly,Algebraic Criteria for Kobayashi Hyperbolic Varieties and Jet Differentials, Lecture notes of a course given at AMS summer research Institute, Santa Cruz (1995).Google Scholar
  11. [De3]
    J. P. Demailly, Regularization of closed positive currents and intersection theory,JAG1 (1992), 361–409.MATHMathSciNetGoogle Scholar
  12. [E]
    T. Ekedahl, Foliations and Inseparable Morphisms, Algebraic Geometry, Bowdoin,Proc. Symp. Pure Math.46, S. Bloch ed., AMS, Providence (1987), 139–149.Google Scholar
  13. [F]
    G. Faltings, Diophantine Approximation on abelian Varieties,Ann. Math.133 (1991), 549–576.CrossRefMathSciNetGoogle Scholar
  14. [Fu]
    W. Fulton,Intersection Theory, New York, Springer Verlag, Berlin, Heidelberg (1984).MATHGoogle Scholar
  15. [Fuj]
    T. Fujita, On the Zariski problem,Proc. Japan Acad.55 (1979), 106–110.MATHCrossRefGoogle Scholar
  16. [G-G]
    M. Green, P. Griffiths, Two applications of algebraic geometry to entire holomorphic mappings,The Chern Symposium, New York, Springer Verlag, Berlin, Heidelberg, (1979), 41–74.Google Scholar
  17. [H]
    R. Hartshorne,Ample subvarieties of algebraic varieties, LNM156 (1970).Google Scholar
  18. [J]
    J. P. Jouanolou, Hypersurface solutions d’une équation de Pfaff analytique,Math. Ann.232 (1978), 239–245.MATHCrossRefMathSciNetGoogle Scholar
  19. [L1]
    S. Lang,Number Theory III, Encyclopedia of Mathematical Sciences, volume 60, New York, Springer Verlag, Berlin, Heidelberg, 1991.MATHGoogle Scholar
  20. [L2]
    S. Lang,Introduction to complex hyperbolic spaces, New York, Springer Verlag, Berlin, Heidelberg (1987).MATHGoogle Scholar
  21. [L-C]
    S. Lang, W. Cherry,Topics In Nevanlinna Theory, LNM1433 (1990).Google Scholar
  22. [L-Y]
    S. Lu, S. T. Yau, Holomorphic curves in surfaces of general type,Proc. Nat. Acad. Sci. USA87 (1990), 80–82.MATHCrossRefMathSciNetGoogle Scholar
  23. [M1]
    M. McQuillan, A new proof of the Bloch conjecture,JAG5 (1996), 107–117.MATHMathSciNetGoogle Scholar
  24. [M2]
    M. McQuillan,A Dynamical Counterpart To Faltings’ Diophantine Approximation On Abelian Varieties, IHES preprint, 1996.Google Scholar
  25. [M3]
    M. McQuillan,La mappa di Faltings e la construzione grafica di Macpherson, lectures given at the Università di Napoli, “Federico II”, to appear.Google Scholar
  26. [Mi1]
    Y. Miyaoka, Algebraic Surfaces With Positive Index,Classification of algebraic and analytic manifolds (Kata Symposium Proc. 1982),Progress in Math.39, Birkhauser, Boston Basel Stuttgart (1983), 281–301.Google Scholar
  27. [Mi2]
    Y. Miyaoka, Deformations of Morphisms Along a Foliation,Algebraic Geometry, Bowdoin, Proc. Symp. Pure Math.46, S. Bloch ed., AMS, Providence (1987), 245–268.Google Scholar
  28. [N]
    M. Nakamaye, conversation at tea, MSRI, 1992.Google Scholar
  29. [No]
    J. Noguchi, On the value distribution of meromorphic mappings of covering spaces over Cm into algebraic varieties,J. Math. Soc. Japan37 (1985), 295–313.MATHMathSciNetCrossRefGoogle Scholar
  30. [R]
    M. Reid, Bogomolov’s Theoremc12⩽ 4c2,Proc. Intern. Colloq. Algebraic Geometry (Kyoto, 1977), 623–642.Google Scholar
  31. [S]
    A. Seidenberg, Reduction of singularities of the differential equation Ady=Bdx, American Journ. Math.89 (1967), 248–269.MathSciNetGoogle Scholar
  32. [Sa]
    F. Sakai, Weil Divisors on normal surfaces,Duke Math. J.51 (1984), 877–887.MATHCrossRefMathSciNetGoogle Scholar
  33. [S-B]
    N. Shepherd-Baron, Miyaoka’s Theorems, Flips And Abundance For Algebraic Threefolds (János Kollár ed.),Asterisque vol.211 (1992), 103–114.Google Scholar
  34. [Si]
    Y. T. Siu, Analyticity of sets associated to Lelong numbers and the extension of closed positive currents,Invent. Math. (1974), 53–156.Google Scholar
  35. [So]
    C. Soulé,Lectures On Arakelov Geometry, Cambridge Studies in Advanced Mathematics33, Cambridge, 1992.Google Scholar
  36. [U]
    E. Ullmo,Positivité et discrétion des points algébriques des courbes, to appear.Google Scholar
  37. [V1]
    P. Vojta,Diophantine Approximations and Value Distribution Theory, LNM1239 (1987).Google Scholar
  38. [V2]
    P. Vojta, On Algebraic Points On Curves,Comp. Math.78 (1991), 29–36.MATHMathSciNetGoogle Scholar
  39. [V3]
    P. Vojta, Siegel’s Theorem In The Compact Case,Ann. Math.133 (1991), 509–548.CrossRefMathSciNetGoogle Scholar
  40. [V4]
    P. Vojta, Integral points on subvarieties of semi-abelian varieties I, to appear.Google Scholar

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© Publications Mathematiques de L’I.H.E.S 1998

Authors and Affiliations

  • Michael McQuillan
    • 1
  1. 1.All Souls CollegeOxfordAngleterre

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