Skip to main content
SpringerLink
Log in

Hyperbolicity of Moduli Spaces of Log-Canonically Polarized Manifolds

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

In recent years, there are quite a lot of interests and results related to hyperbolicity properties of the base spaces of various families of projective algebraic varieties. Not much is known for families of higher dimensional quasi-projective varieties. The goal of this paper is address the problem for the case of an effectively parametrized family of log-canonically polarized manifolds. We construct a Finsler metric on the base manifold of such a family with the property that its holomorphic sectional curvature is bounded from above by a negative constant, and as a consequence, we deduce the Kobayashi hyperbolicity of the base manifold. The method relies on developing analytic tools to investigate geometry of families of quasi-projective manifolds equipped with Kähler–Einstein metrics, which leads to an appropriate modification of the Weil–Petersson metric on the base manifold.

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. Ahlfors, L.: Some remarks on Teichmüller’s space of Riemann surfaces. Ann. Math. 74, 171–191 (1961)

    Article  MathSciNet  Google Scholar 

  2. Ahlfors, L.: Curvature properties of Teichmüller’s space. J. Anal. Math. 9, 161–176 (1961/1962)

  3. Aubin, T.: Some Non-linear Problems in Riemannian Geometry. Springer, Berlin (1998)

    Book  Google Scholar 

  4. Andreotti, A., Vesentini, E.: Carleman estimates for the Laplace-Beltrami equation on complex manifolds. Inst. Hautes Études Sci. Publ. Math. 25, 81–130 (1965)

    Article  MathSciNet  Google Scholar 

  5. Calderón, A.P., Zygmund, A.: On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952)

    Article  MathSciNet  Google Scholar 

  6. Cheeger, J.: On the Hodge theory of Riemannian pseudomanifolds. Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), pp. 91-146, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I. (1980)

  7. Cheng, S.Y., Yau, S.-T.: On the existence of complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation. Commun. Pure Appl. Math. 33, 507–544 (1980)

    Article  Google Scholar 

  8. DiBenedetto, E.: Degenerate Parabolic Equations. Springer, New York (1993)

    Book  Google Scholar 

  9. Gaffney, M.P.: A special Stokes’s theorem for complete Riemannian manifolds. Ann. Math. 60, 140–145 (1954)

    Article  MathSciNet  Google Scholar 

  10. Gaffney, M.P.: The heat equation method of Milgram and Rosenbloom for open Riemannian manifolds. Ann. Math. 60, 458–466 (1954)

    Article  MathSciNet  Google Scholar 

  11. Guenancia, H., Wu, D.: On the boundary behavior of Kähler-Einstein metrics on log canonical pairs. Math. Ann. 366(1–2), 101–120 (2016)

    Article  MathSciNet  Google Scholar 

  12. Kawamata, Y.: On deformations of compactifiable complex manifolds. Math. Ann. 235, 247–265 (1978)

    Article  MathSciNet  Google Scholar 

  13. Kobayashi, R.: Kähler-Einstein metric on an open algebraic manifolds. Osaka J. Math. 21, 399–418 (1984)

    MathSciNet  MATH  Google Scholar 

  14. Milgram, A., Rosenbloom, P.: Harmonic forms and heat conduction, I: Closed Riemannian manifolds. Proc. Natl. Acad. Sci. U.S.A. 37, 180–184 (1951)

    Article  MathSciNet  Google Scholar 

  15. Mok, N., Zhong, J.-Q.: Compactifying complete Kähler-Einstein manifolds of finite topological type and bounded curvature. Ann. Math. 129, 417–470 (1989)

    MATH  Google Scholar 

  16. Royden, H.L.: Intrinsic metrics on Teichmüller space. In: Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2, pp. 217–221. Canad. Math. Congress, Montreal, QC (1975)

  17. Schumacher, G.: Moduli of framed manifolds. Invent. Math. 134(2), 229–249 (1998)

    Article  MathSciNet  Google Scholar 

  18. Siu, Y.-T.: Curvature of the Weil-Petersson metric in the moduli space of compact Kähler-Einstein manifolds of negative first Chern class. Contributions to several complex variables, pp. 261–298, Aspects Math., E9, Vieweg, Braunschweig (1986)

  19. Tian, G., Yau, S.-T.: Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry. Proc. Conf. San Diego. Adv. Ser. Math. Phys. 1, 574–629 (1987)

    MATH  Google Scholar 

  20. To, W.-K., Yeung, S.-K.: Kähler metrics of negative holomorphic bisectional curvature on Kodaira surfaces. Bull. Lond. Math. Soc. 43, 507–512 (2011)

    Article  MathSciNet  Google Scholar 

  21. To, W.-K., Yeung, S.-K.: Finsler metrics and Kobayashi hyperbolicity of the moduli spaces of canonically polarized manifolds. Ann. Math. 181(2), 547–586 (2015)

    Article  MathSciNet  Google Scholar 

  22. To, W.-K., Yeung, S.-K.: Augmented Weil-Petersson metrics on moduli spaces of polarized Ricci-flat Kähler manifolds and orbifolds. Asian J. Math. 22, 705–728 (2018)

    Article  MathSciNet  Google Scholar 

  23. Tsuji, H.: A characterization of ball quotients with smooth boundary. Duke Math. J. 57, 537–553 (1988)

    Article  MathSciNet  Google Scholar 

  24. Viehweg, E.: Quasi-projective moduli for polarized manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 Folge, Band 30. Springer, Berlin (1995)

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

  26. Wolpert, S.A.: Chern forms and the Riemann tensor for the moduli space of curves. Invent. Math. 85, 119–145 (1986)

    Article  MathSciNet  Google Scholar 

  27. Wu, D.: Kähler-Einstein metrics of negative Ricci curvature on general quasi-projective manifolds. Commun. Anal. Geom. 16(2), 395–435 (2008)

    Article  Google Scholar 

  28. Yau, S.-T.: A general Schwarz lemma for Kähler manifolds. Am. J. Math. 100, 197–203 (1978)

    Article  Google Scholar 

  29. Yeung, S.-K.: Compactification of complete Kähler manifolds with negative Ricci curvature. Invent. Math. 106, 13–26 (1991)

    Article  MathSciNet  Google Scholar 

  30. Zucker, S.: Hodge theory with degenerating coefficients: \(L^2\)-cohomology in the Poincaré metric. Ann. Math. 109, 415–476 (1979)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore, 119076, Singapore

    Wing-Keung To

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Sai-Kee Yeung

Authors
  1. Wing-Keung To
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sai-Kee Yeung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sai-Kee Yeung.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author was partially supported by the Singapore Ministry of Education Academic Research Fund Tier 1 Grant R-146-000-254-114. The second author was partially supported by a grant from the National Science Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

To, WK., Yeung, SK. Hyperbolicity of Moduli Spaces of Log-Canonically Polarized Manifolds. J Geom Anal 31, 2941–2969 (2021). https://doi.org/10.1007/s12220-020-00380-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-020-00380-8

Keywords

Search

Navigation