Skip to main content
SpringerLink
Log in

Arakelov–Parshin rigidity of towers of curve fibrations

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Arakelov–Parshin rigidity is concerned with varieties mapping rigidly to the moduli stack \(\mathfrak {M}_h\) of canonically polarized manifolds. Affirmative answer for any class of maps implies finiteness of the given class. This article studies Arakelov–Parshin rigidity on an open subspace of \(\mathfrak {M}_h\), on the locus \(\mathfrak {K}\mathfrak {F}_h\) of iterated Kodaira fibrations. First, we prove rigidity for all complete curves mapping finitely onto \(\mathfrak {K}\mathfrak {F}_h\). Then, for generic affine curves mapping into \(\mathfrak {K}\mathfrak {F}_h\), rigidity is shown when \(\deg h =2\). The method used in the latter part is showing that the iterated Kodaira–Spencer map is injective.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramovich, D., Olsson, M., Vistoli, A.: Tame stacks in positive characteristic. Ann. Inst. Fourier (Grenoble) 58(4), 1057–1091 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abramovich, D., Vistoli, A.: Complete moduli for fibered surfaces, Recent progress in intersection theory (Bologna, 1997), Trends Mathematics, pp. 1–31. Birkhäuser Boston, Boston (2000)

  3. Abramovich, D., Vistoli, A.: Compactifying the space of stable maps. J. Am. Math. Soc. 15(1), 27–75 (2002) (electronic)

  4. Arakelov, S.J.: Families of algebraic curves with fixed degeneracies. Izv. Akad. Nauk SSSR Ser. Mat. 35, 1269–1293 (1971)

    MathSciNet  MATH  Google Scholar 

  5. Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4. Springer, Berlin (1984)

    Google Scholar 

  6. Bedulev, E., Viehweg, E.: On the Shafarevich conjecture for surfaces of general type over function fields. Invent. Math. 139(3), 603–615 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Behrend, K., Noohi, B.: Uniformization of Deligne-Mumford curves. J. Reine Angew. Math. 599, 111–153 (2006)

    MathSciNet  MATH  Google Scholar 

  8. de Jong, A.J.: Families of curves and alterations. Ann. Inst. Fourier (Grenoble) 47(2), 599–621 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Esnault, H., Viehweg, E.: Lectures on Vanishing Theorems, DMV Seminar, vol. 20. Birkhäuser Verlag, Basel (1992)

    Book  Google Scholar 

  10. Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73(3), 349–366 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hacon, C.D., Kovács, S.J.: Classification of Higher Dimensional Algebraic Varieties, Oberwolfach Seminars, vol. 41. Birkhäuser Verlag, Basel (2010)

    Book  Google Scholar 

  12. Hacon, C.D., McKernan, J.: Boundedness of pluricanonical maps of varieties of general type. Invent. Math. 166(1), 1–25 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hartshorne, R.: Algebraic geometry, Springer, New York (1977). Graduate Texts in Mathematics, No. 52

  14. Hassett, B., Kovács, S.J.: Reflexive pull-backs and base extension. J. Algebr. Geom. 13(2), 233–247 (2004)

    Article  MATH  Google Scholar 

  15. Jabbusch, K., Kebekus, S.: Families over special base manifolds and a conjecture of Campana. Math. Z. 269(3–4), 847–878 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kebekus, S., Kovács, S.J.: Families of canonically polarized varieties over surfaces. Invent. Math. 172(3), 657–682 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kebekus, S., Kovács, S.J.: Families of varieties of general type over compact bases. Adv. Math. 218(3), 649–652 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kebekus, S., Kovács, S.J.: The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties. Duke Math. J. 155(1), 1–33 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kollár, J.: Moduli of varieties of general type. arXiv:1008.0621 (2010)

  20. Kollár, J.: Singularities of the Minimal Model Program, Cambridge Tracts in Mathematics, vol. 200 (2013)

  21. Kollár, J., Mori, S.: Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge (1998). With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original

  22. Kovács, S.J.: Smooth families over rational and elliptic curves. J. Algebr. Geom. 5(2), 369–385 (1996)

    MATH  Google Scholar 

  23. Kovács, S.J.: On the minimal number of singular fibres in a family of surfaces of general type. J. Reine Angew. Math. 487, 171–177 (1997)

    MathSciNet  MATH  Google Scholar 

  24. Kovács, S.J.: Algebraic hyperbolicity of fine moduli spaces. J. Algebr. Geom. 9(1), 165–174 (2000)

    MATH  Google Scholar 

  25. Kovács, S.J.: Logarithmic vanishing theorems and Arakelov–Parshin boundedness for singular varieties. Compos. Math. 131(3), 291–317 (2002)

    Article  MATH  Google Scholar 

  26. Kovács, S. J.: Vanishing theorems, boundedness and hyperbolicity over higher-dimensional bases. Proc. Am. Math. Soc. 131(11), 3353–3364 (2003) (electronic)

  27. Kovács, S. J.: Viehweg’s conjecture for families over \(\mathbb{P}^n\). Commun. Algebra 31(8), 3983–3991 (2003). Special issue in honor of Steven L. Kleiman

  28. Kovács, S. J.: Strong non-isotriviality and rigidity, Recent progress in arithmetic and algebraic geometry. Contemp. Math. vol. 386, pp. 47–55. Am. Math. Soc., Providence, RI (2005)

  29. Kovács, S.J., Lieblich, M.: Boundedness of families of canonically polarized manifolds: a higher dimensional analogue of shafarevich’s conjecture, vol. 172 (2010)

  30. Migliorini, L.: A smooth family of minimal surfaces of general type over a curve of genus at most one is trivial. J. Algebr. Geom. 4(2), 353–361 (1995)

    MathSciNet  MATH  Google Scholar 

  31. Olsson, M.C.: Deformation theory of representable morphisms of algebraic stacks. Math. Z. 253(1), 25–62 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Olsson, M.C.: \(\underline{\text{ Hom }}\)-stacks and restriction of scalars. Duke Math. J. 134(1), 139–164 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Paršin, A.N.: Algebraic curves over function fields. Dokl. Akad. Nauk SSSR 183, 524–526 (1968)

    MathSciNet  Google Scholar 

  34. Patakfalvi, Zs.: Fibered stable varieties. Trans Am Math Soc. arXiv:1208.1787 (2012) (to appear)

  35. Schneider, M.: Complex surfaces with negative tangent bundle, Complex analysis and algebraic geometry (Göttingen, 1985), Lecture Notes in Mathematics, vol. 1194, pp. 150–157. Springer, Berlin (1986)

  36. Viehweg, E.: Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Algebraic varieties and analytic varieties (Tokyo, 1981), Advanced Studies in Pure Mathematics, vol. 1, pp. 329–353. North-Holland, Amsterdam (1983)

  37. Viehweg, E.: Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 30. Springer, Berlin (1995)

    Google Scholar 

  38. Viehweg, E., Zuo, K.: On the isotriviality of families of projective manifolds over curves. J. Algebr. Geom. 10(4), 781–799 (2001)

    MathSciNet  MATH  Google Scholar 

  39. Viehweg, E., Zuo, K.: Base spaces of non-isotrivial families of smooth minimal models, Complex geometry (Göttingen, 2000), pp. 279–328. Springer. Berlin (2002)

  40. Viehweg, E., Zuo, K.: Discreteness of minimal models of Kodaira dimension zero and subvarieties of moduli stacks, Surveys in differential geometry, Vol. VIII, pp. 337–356 (Boston, MA, 2002). Subveys in Differential Geometry, VIII. International Press, Somerville (2003)

  41. Viehweg, E., Zuo, K.: On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds. Duke Math. J. 118(1), 103–150 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  42. Viehweg, E., Zuo, K.: Complex multiplication, Griffiths-Yukawa couplings, and rigidity for families of hypersurfaces. J. Algebraic Geom. 14(3), 481–528 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Vistoli, A.: Intersection theory on algebraic stacks and on their moduli spaces. Invent. Math. 97(3), 613–670 (1989)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

I would like to thank my advisor, Sándor Kovács, for the fantastic guidance throughout my PhD studies. I would also like to thank János Kollár for the useful remarks concerning a preprint version of the paper and Max Lieblich for discussing Sect. 7 with me.

Author information

Authors and Affiliations

  1. Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, NJ , 08544-1000, USA

    Zsolt Patakfalvi

Authors
  1. Zsolt Patakfalvi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zsolt Patakfalvi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Patakfalvi, Z. Arakelov–Parshin rigidity of towers of curve fibrations. Math. Z. 278, 859–892 (2014). https://doi.org/10.1007/s00209-014-1336-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-014-1336-0

Keywords

Search

Navigation