Families of Varieties of General Type: the Shafarevich Conjecture and Related Problems

  • Sándor J. Kovács
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 12)


Consider the curves on the (x, y)-plane parametrized by A and given by the equation, \({y^2} = {\rm{ }}{{\rm{x}}^{\rm{5}}} - {\rm{5}}\lambda {\rm{x }} + {\rm{ 4}}\lambda .\)


Modulus Space Line Bundle Hilbert Scheme Ample Line Bundle Smooth Family 
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  1. [1]
    U. Angehrn and Y.-T. Sin, Effective freeness and point separation for adjoint bundles, Invent Math., 122 (1995), 291–308.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    Y. Akizuki and S. Nakano, Note on Kodaira-Spencer’s proof of Lefschetz theorems, Proc. Jap. Acad., 30 (1954), 266–272.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    S. Arakelov, Families of algebraic curves with fixed degeneracies, Izv. A. N. SSSR, 35 (1971), 1269–1293.MathSciNetMATHGoogle Scholar
  4. [4]
    M. Artin, Versal deformations and algebraic stacks, Invent. Math., 27 (1974), 165–189.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 4., Springer (1984).CrossRefMATHGoogle Scholar
  6. [6]
    A. Beauville, Le nombre minimal de fibres singulières d’une courbe stable sur IP1, Astérisque, 86 (1981), 97–108.MATHGoogle Scholar
  7. [7]
    E. Bedulev and E. Vieh weg, On the Shafarevich conjecture for surfaces of general type over function fields, Invent. Math., 139 (2000), 603–615.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    F. Catanese and M. Schneider, Polynomial bounds for abelian groups of automorphisms, Compositio Math., 97 (1995), 1–15.MathSciNetMATHGoogle Scholar
  9. [9]
    B. Conrad, Grothendieck Duality and Base Change, LNM 1750, Springer (2000).Google Scholar
  10. [10]
    M. A. A. de Cataldo, Vanishing via lifting to second Witt vectors and a proof of an isotriviality result, J. Algebra, 219 (1999), 255–265.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    M. de Franchis, Un teorema sulle involuzioni irrazionali, Rend. Circ. Mat. Palermo, 36 (1913), 368. See also in “Collected works of Michele de Franchis”, Edited by C. Ciliberto and E. Sernesi., Rend. Circ. Mat. Palermo (2) Suppl., 27 (1991), 523.CrossRefMATHGoogle Scholar
  12. [12]
    A. J. de Jong and F. Oort, On extending families of curves, J. Algebraic Geom., 6 (1997), 545–562.MathSciNetMATHGoogle Scholar
  13. [13]
    P. Deligne and D. Mumford, The irreducibility of space of curves of given genus, Publ. Math. IHES, 36 (1969), 75–110.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    Ph. Du Bois, Complex de De Rham filtré d’une variété singulière, Bull. Soc. Math. France, 109 (1981), 41–81.MathSciNetMATHGoogle Scholar
  15. [15]
    L. Ein, Multiplier ideals, vanishing theorems and applications, in: Algebraic Geometry, Santa Cruz 1995 (S. Bloch et al., eds.), Proc. Symp. Pure Math., vol. 62 (1997), pp. 203–219.CrossRefGoogle Scholar
  16. [16]
    H. Esnault and E. Viehweg, Logarithmic de Rham complexes and vanishing theorems, Invent. Math., 86 (1986), 161–194.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    H. Esnault and E. Viehweg, Effective bounds for semipositive sheaves and for the height of points on curves over complex function fields, Compositio Math., 76 (1990), 69–85.MathSciNetMATHGoogle Scholar
  18. [18]
    H. Esnault and E. Viehweg, Lectures on vanishing theorems, DMV Seminar, vol. 20, Birkhäuser (1992).CrossRefMATHGoogle Scholar
  19. [19]
    G. Faltings, Arakelov’s Theorem for abelian varieties, Invent. Math., 73 (1983), 337–348.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73 (1983), 349–366.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    J.-M. Fontaine, Il n’y a pas de variété abélienne sur ℤ, Invent. Math., 81 (1985), 515–538.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    H. Grauert and O. Riemenschneider, Verschwindungssätze für analytische Koho-mologiegruppen auf komplexen Räumen, Invent. Math., 11 (1970), 263–292.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    F. Guillen, V. Navarro-Aznar, P. Pascual-Gainza and F. Puerta, Hyperrésolutions cubiques et descente cohomologique, LNM 1335, Springer (1988).MATHGoogle Scholar
  24. [24]
    J. Harris, Algebraic Geometry: A First Course, Graduate Texts in Mathematics, vol. 133, Springer (1992).CrossRefMATHGoogle Scholar
  25. [25]
    J. Harris and I. Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer (New York, 1998).MATHGoogle Scholar
  26. [26]
    R. Hartshorne, Residues and Duality, LNM 20, Springer (1966).MATHGoogle Scholar
  27. [27]
    R. Hartshorne, On the De Rham cohomology of algebraic varieties, Publ. Math. IHES, 45 (1975), 5–99.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer (1977).CrossRefMATHGoogle Scholar
  29. [29]
    Y. Kawamata, A generalisation of Kodaira-Ramanujam’s vanishing theorem, Math. Ann., 261 (1982), 43–46.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    Y. Kawamata, Kodaira dimension of algebraic fiber spaces over curves, Invent. Math., 66 (1982). 57–71.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the Minimal Model Problem, in: Algebraic Geometry (Sendai, 1985) (T. Oda, ed.), Adv. Stud. Pure Math., vol. 10. Kinokuniya — North-Holland (1987), pp. 283–360.Google Scholar
  32. [32]
    F. F. Knudsen and D. Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand., 39 (1976), 19–55.MathSciNetMATHGoogle Scholar
  33. [33]
    K. Kodaira, On a differential geometric method in the theory of analytic stacks, Proc. Nat. Acad. USA, 39 (1953), 1268–1273.MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    J. Kollár, Vanishing theorems for cohomology groups, in: Algebraic Geometry Bowdoin 1985, Proc. Symp. Pure Math., vol. 46 (1987), pp. 233–243.CrossRefGoogle Scholar
  35. [35]
    J. Kollár, Subadditivity of the Kodaira dimension: Fibers of general type, in: Algebraic Geometry (Sendai, 1985), (T. Oda, ed.), Advanced Studies in Pure Math., vol. 10, Kinokuniya-North Holland (1987), pp. 361–398.Google Scholar
  36. [36]
    J. Kollár, Projectivity of Complete Moduli, J. Diff. Geom., 32 (1990), 235–268.MATHGoogle Scholar
  37. [37]
    J. Kollár, Rational Curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 32. Springer (1996).CrossRefGoogle Scholar
  38. [38]
    J. Kollár, Singularities of pairs, in: Algebraic Geometry Santa Cruz 1995, Proc. Symp. Pure Math., vol. 62 (1997), pp. 221–287.Google Scholar
  39. [39]
    S. J. Kovács, Smooth families over rational and elliptic curves, J. Algebraic Geom., 5 (1996), 369–385.MathSciNetMATHGoogle Scholar
  40. [40]
    S.J. Kovács, On the minimal number of singular fibers in a family of surfaces of general type, J. reine angew. Math., 487 (1997), 171–177.MathSciNetMATHGoogle Scholar
  41. [41]
    S. J. Kovács, Families over a base with a birationally nef tangent bundle, Math. Ann., 308 (1997), 347–359.MathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    S. J. Kovács, Relative De Rham complexes for non-smooth morphisms, in: Birational Algebraic Geometry (Y. Kawamata, V. Shokurov eds.), Contemp. Math., vol. 207 (1997), pp. 89–100.CrossRefGoogle Scholar
  43. [43]
    S. J. Kovács, Algebraic hyperbolicky offine moduli spaces, J. Algebraic Geom., 9 (2000), 165–174.MathSciNetMATHGoogle Scholar
  44. [44]
    S. J. Kovács, A characterization of rational singularities, Duke Math. J.f 102 (2000), 187–191.CrossRefMATHGoogle Scholar
  45. [45]
    S. J. Kovács, Rational, log canonical, Du Bois singularities II: Kodaira vanishing and small deformations, Compositio Math., 121 (2000), 297–304.MathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    S. J. Kovács, Logarithmic vanishing theorems and Arakelov-Parshin boundedness for singular varieties, Compositio Math., 131 (2002), 291–317.MathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    S. J. Kovács, Vanishing theorems, boundedness and hyperbolicity over higher dimensional bases, Proc. AMS (to appear).Google Scholar
  48. [48]
    G. Laumon and L. Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 39, Springer (2000).MATHGoogle Scholar
  49. [49]
    S. Lang, Introduction to complex hyperbolic spaces, Springer (1987).CrossRefMATHGoogle Scholar
  50. [50]
    S. Lang, Survey of Diophantine Geometry, Springer (1997).MATHGoogle Scholar
  51. [51]
    Ju. I. Manin, Rational points on algebraic curves over function fields, Izv. Akad. Nauk SSSR Ser. Mat, 27 (1963), 1395–1440.MathSciNetMATHGoogle Scholar
  52. [52]
    L. Migliorini, A smooth family of minimal surfaces of general type over a curve of genus at most one is trivial, J. Algebraic Geom., 4 (1995), 353–361.MathSciNetMATHGoogle Scholar
  53. [53]
    L. Moret-Bailly, Un théorème de pureté pour les families de courbes lisses, C.R. Acad. Sci. Paris Sér. I Math., 300 (1985), 489–492.MathSciNetMATHGoogle Scholar
  54. [54]
    D. Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34, Springer (1965).CrossRefMATHGoogle Scholar
  55. [55]
    V. Navarro-Aznar, Théorèmes dannulation, in: Hyperrésolutions cubiques et de-scente cohornologique, LNM 1335, Springer (1988), pp. 133–160.CrossRefGoogle Scholar
  56. [56]
    K. Oguiso and E. Viehweg, On the isotriviality of families of elliptic surfaces, J. Algebraic Geom., 10 (2001), 569–598.MathSciNetMATHGoogle Scholar
  57. [57]
    A. Parshin, Algebraic curves over function fields, Izv. A.N. SSSR, 32 (1968), 1145–1170.CrossRefGoogle Scholar
  58. [58]
    C. P. Ramanujam, Remarks on the Kodaira vanishing theorem, J. Indian Math. Soc., 36 (1972), 41–51.MathSciNetMATHGoogle Scholar
  59. [59]
    V. V. Shokurov, Letters of a bi-rationalist I. A projectivity criterion, in: Birational Algebraic Geometry, (Y. Kawamata, V. Shokurov, eds.), Contemp. Math., vol. 207 (1997), pp. 143–152.CrossRefGoogle Scholar
  60. [60]
    K. E. Smith, Vanishing, singularities and effective bounds via prime characteristic local algebra, in: Algebraic Geometry Santa Cruz 1995, Proc. Symp. Pure Math., vol. 62 (1997), pp. 289–325.CrossRefGoogle Scholar
  61. [61]
    J. H. M. Steenbrink, Vanishing theorems on singular spaces, Astérisque, 130 (1985), 330–341.MathSciNetGoogle Scholar
  62. [62]
    S. L. Tan, The minimal number of singular fibres of a semistable curve over ∙1, J. Algebraic Geom., 4 (1995), 591–596.MathSciNetMATHGoogle Scholar
  63. [63]
    E. Viehweg, Vanishing theorems, J. reine angew. Math., 335 (1982), 1–8.MathSciNetMATHGoogle Scholar
  64. [64]
    E. Viehweg, Weak positivity and the additivity of Kodaira dimension for certain fibre spaces, in: Algebraic Varieties and Analytic Varieties, Advanced Studies in Pure Math., vol. 1, North-Holland (1983), pp. 329–353.Google Scholar
  65. [65]
    E. Viehweg, Weak positivity and the additivity of the Kodaira dimension IL, in: Classification of algebraic and analytic manifolds, Progress in Math., vol. 39, Birkhäuser (1983), pp. 567–590.Google Scholar
  66. [66]
    E. Viehweg, Quasi-Projective Moduli of Polarized Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 30, Springer (1995).CrossRefGoogle Scholar
  67. [67]
    E. Viehweg, Positivity of direct image sheaves and applications to families of higher dimensional manifolds (2000) Lecture notes, “School on Vanishing Theorems and Effective Results in Algebraic Geometry”, ICTP, Trieste.Google Scholar
  68. [68]
    E. Viehweg and K. Zuo, On the isotriviality of families of projective manifolds over curves, J. Algebraic Geom., 10 (2001), 781–799.MathSciNetMATHGoogle Scholar
  69. [69]
    E. Viehweg and K. Zuo, Base spaces of non-isotrivial families of smooth minimal models, Complex geometry (Göttingen, 2000), 279–328, Springer (Berlin, 2002).Google Scholar
  70. [70]
    E. Viehweg and K. Zuo, On the Brody hyperbolicity of moduli spaces for canoni-cally polarized manifolds, preprint.Google Scholar
  71. [71]
    Q. Zhang, Global holomorphic one-forms on projective manifolds with ample canonical bundles, J. Alg. Geom., 6 (1997), 777–787.MATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Sándor J. Kovács
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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