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Base Spaces of Non-Isotrivial Families of Smooth Minimal Models

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Complex Geometry

Abstract

Given a polynomial h of degree n let M h be the moduli functor of canonically polarized complex manifolds with Hilbert polynomial h. By [23] there exist a quasi-projective scheme M h together with a natural transformation

$$ \Psi :\mathcal{M}_h \to Hom(\_,M_h ) $$

such that M h is a coarse moduli scheme for M h . For a complex quasi-projective manifold U we will say that a morphism ϕ U → M h factors through the moduli stack, or that ϕ is induced by a family, if ϕ lies in the image of Ψ(U), hence if ϕ = Ψ(ƒ: V → U).

This work has been supported by the “DFG-Forschergruppe Arithmetik und Geometrie” and the “DFG-Schwerpunktprogramm Globale Methoden in der Komplexen Geometrie”. The second named author is supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK 4239/01P).

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References

  1. Abramovich, D., Karu, K., Weak semi-stable reduction in characteristic 0, Invent. Math. 139 (2000), 241–273.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bedulev, E., Viehweg, E., On the Shafarevich conjecture for surfaces of general type over function fields. Invent. Math. 139 (2000), 603–615.

    Article  MathSciNet  MATH  Google Scholar 

  3. Brückmann, P., Rackwitz, H.-G,, T-symmetrical tensor forms on complete intersections. Math. Ann. 288 (1990), 627–635.

    Article  MathSciNet  MATH  Google Scholar 

  4. Deligne, P., Équations différentielles à points singuliers réguliers, Lecture Notes in Math. 163 (1970), Springer, Berlin Heidelberg New York.

    Google Scholar 

  5. Esnault, H., Viehweg, E., Revêtement cycliques, in Algebraic Threefolds, Proc. Varenna 1981, Springer Lect. Notes in Math.. 947 (1982), 241–250.

    MathSciNet  Google Scholar 

  6. Esnault, H., Viehweg, E., Lectures on vanishing theorems, DMV Seminar 20 (1992), Birkhäuser, Basel Boston.

    Book  MATH  Google Scholar 

  7. Griffiths, P. (Editor), Topics in transcendental algebraic geometry, Ann. of Math. Stud. 106 (1984), Princeton Univ. Press, Princeton, NJ.

    MATH  Google Scholar 

  8. Grothendieck, A., Techniques de construction et théorèmes d’existence en géométrie algébrique, IV: Les schémas de Hilbert, Sém. Bourbaki 221 (1960/61), in: Fondements de la Géométrie Algébrique Sém. Bourbaki, Secrétariat, Paris 1962.

    Google Scholar 

  9. Grauert, H., Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper, Publ. Math. IHES 25 (1965), 131–149.

    MathSciNet  Google Scholar 

  10. Kawamata, Y., Minimal models and the Kodaira dimension of algebraic fibre spaces, Journ. Reine Angew. Math. 363 (1985), 1–46.

    MathSciNet  MATH  Google Scholar 

  11. Kempf, G., Knudsen, F., Mumford, D. and Saint-Donat, B., Toroidal embeddings I, Lecture Notes in Math. 339 (1973), Springer, Berlin Heidelberg New York.

    Google Scholar 

  12. Kovács, S., Algebraic hyperbolicity of fine moduli spaces, J. Alg. Geom. 9 (2000), 165–174.

    MATH  Google Scholar 

  13. Kovács, S., Logarithmic vanishing theorems and Arakelov-Parshin boundedness for singular varieties, preprint (AG/0003019), to appear in Comp. Math.

    Google Scholar 

  14. Kovács, S., Families over a base with a birationally nef tangent bundle. Math. Ann. 308 (1997), 347–359.

    Article  MathSciNet  MATH  Google Scholar 

  15. Lu, S.S-Y., On meromorphic maps into varieties of log-general type, Proc. Symp. Amer. Math. Soc. 52 (1991), 305–333.

    Google Scholar 

  16. Migliorini, L., A smooth family of minimal surfaces of general type over a curve of genus at most one is trivial, J. Alg. Geom. 4 (1995), 353–361.

    MathSciNet  MATH  Google Scholar 

  17. Oguiso, K., Viehweg, E., On the isotriviality of families of elliptic surfaces, J. Alg. Geom. 10 (2001), 569–598.

    MathSciNet  MATH  Google Scholar 

  18. Schmid, W., Variation of Hodge structure: The singularities of the period mapping. Invent. Math. 22 (1973), 211–319.

    Article  MathSciNet  MATH  Google Scholar 

  19. Simpson, C, Higgs bundles and local systems, Publ. Math. I.H.E.S 75 (1992), 5–95.

    MathSciNet  MATH  Google Scholar 

  20. Viehweg, E., Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, in: Algebraic Varieties and Analytic Varieties, Advanced Studies in Pure Math. 1 (1983), 329–353.

    MathSciNet  Google Scholar 

  21. Viehweg, E., Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces II. The local Torelli map, in: Classification of Algebraic and Analytic Manifolds, Progress in Math. 39 (1983), 567–589.

    MathSciNet  Google Scholar 

  22. Viehweg, E., Weak positivity and the stability of certain Hilbert points. Invent. Math. 96 (1989), 639–667.

    Article  MathSciNet  MATH  Google Scholar 

  23. Viehweg, E., Quasi-projective Moduli for Polarized Manifolds, Ergebnisse der Mathematik, 3, Folge 30 (1995), Springer Verlag, Berlin-Heidelberg-New York.

    Google Scholar 

  24. Viehweg, E., Zuo K., On the isotriviality of families of projective manifolds over curves, J. Alg. Geom. 10 (2001), 781–799.

    MathSciNet  MATH  Google Scholar 

  25. Viehweg, E., Zuo K, On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds, preprint (AG/0101004).

    Google Scholar 

  26. Zuo, K., On the negativity of kernels of Kodaira-Spencer maps on Hodge bundles and applications, Asian J. of Math. 4 (2000), 279–302.

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. FB6 Mathematik, Universität Essen, 45117, Essen, Germany

    Eckart Viehweg

  2. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Kang Zuo

Authors
  1. Eckart Viehweg
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  2. Kang Zuo
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Editor information

Editors and Affiliations

  1. Faculty for Mathematics and Physics, University of Bayreuth, Universitätsstraße 30, 95447, Bayreuth, Germany

    Ingrid Bauer , Fabrizio Catanese  & Thomas Peternell ,  & 

  2. Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153, Japan

    Yujiro Kawamata

  3. Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA, 02138, USA

    Yum-Tong Siu

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© 2002 Springer-Verlag Berlin Heidelberg

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Viehweg, E., Zuo, K. (2002). Base Spaces of Non-Isotrivial Families of Smooth Minimal Models. In: Bauer, I., Catanese, F., Peternell, T., Kawamata, Y., Siu, YT. (eds) Complex Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56202-0_16

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  • DOI: https://doi.org/10.1007/978-3-642-56202-0_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62790-3

  • Online ISBN: 978-3-642-56202-0

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