Advertisement

Archiv der Mathematik

, Volume 103, Issue 1, pp 75–84 | Cite as

A sharp cusp count for complex hyperbolic surfaces and related results

  • Gabriele Di Cerbo
  • Luca F. Di Cerbo
Article

Abstract

We derive a sharp cusp count for finite volume complex hyperbolic surfaces which admit smooth toroidal compactifications. We use this result, and the techniques developed in Di Cerbo and Di Cerbo (see [5]), to study the geometry of cusped complex hyperbolic surfaces and their compactifications.

Mathematical Subject Classification

32Q45 57M50 14C20 

Keywords

Complex hyperbolic surfaces Toroidal compactifications Cusp count 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Ash et al. Smooth compactifications of locally symmetric varieties. Second edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2010.Google Scholar
  2. 2.
    Baily W.L., Borel A.: Compactifications of arithmetic quotients of bounded symmetric domains. Ann. of Math. 84, 442–528 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    W. Barth et al. Compact complex surfaces. Ergebnisse der Mathematik undihrer Grenzgebiete (3), 4. Springer-Verlag, Berlin 2004.Google Scholar
  4. 4.
    A. Borel and L. Ji, Compactifications of locally symmetric spaces. Mathematics: Theory & Applications, Birkäuser Boston, Inc. Boston, MA, 2006.Google Scholar
  5. 5.
    G. Di Cerbo and L. F. Di Cerbo, Effective results for complex hyperbolic manifolds, arXiv:1212.0501 [mathDG], 2012.
  6. 6.
    L. F. Di Cerbo, Classification of toroidal compactifications with \({3\overline{c}_{2}=\overline{c}^{2}_{1}}\) and \({\overline{c}_{2}=1}\), arXiv:1309.5516 [mathAG], 2013.
  7. 7.
    Di Cerbo L.F.: Finite-volume complex-hyperbolic surfaces, their toroidal compactifications, and geometric applications. Pacific J. Math. 255, 305–315 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Fujiki A.: An L 2 Dolbeault lemma and its applications. Publ. Res. Inst. Math. Sci. 28, 845–884 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Griffiths P., Harris J.: Principles of Algebraic Geometry. Pure and Applied Mathematics. Wiley-Interscience, New York (1978)Google Scholar
  10. 10.
    Gromov M.: Volume and bounded cohomology. Publ. Math. Inst. Hautes Et́udes Sci. 56, 5–99 (1982)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Hirzebruch F.: Chern numbers of algebraic surfaces: an example. Math. Ann. 266, 351–356 (1982)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Hwang J.-M.: On the number of complex hyperbolic manifolds of bounded volume. Internat. J. Math. 8, 863–873 (2005)CrossRefGoogle Scholar
  13. 13.
    J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 32. Springer-Verlag, Berlin, 1996.Google Scholar
  14. 14.
    Mumford D.: Hirzebruch’s proportionality theorem in the non-compact case. Invent. Math. 42, 239–272 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Parker J.R.: On the volume of cusped, complex hyperbolic manifolds and orbifolds. Duke Math. J. 94, 433–464 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Reider I.: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math. 127, 309–316 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Stover M.: On the number of ends of rank one locally symmetric spaces. Geom. Topol. 17, 905–924 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Friedman R.: Algebraic Surfaces and Holomorphic Vector Bundles, Univesitext. Springer-Verlag, New York (1998)CrossRefGoogle Scholar
  19. 19.
    Y. T. Siu and S. T. Yau, Compactification of negatively curved complete Kähler manifolds of finite volume, Seminars in Differential Geometry, pp. 363–380, Ann. of Math. Stud., Vol. 102, Princeton Univ. Press, Princeton, N. J., 1982.Google Scholar
  20. 20.
    H. C. Wang, Topics on totally discontinuous groups, Symmetric Spaces, 459–487, edited by W. B. Boothby and G. L. Weiss, Pure and Appl. Math. 8, Marcel Dekker, New York, 1972.Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Department of MathematicsDuke UniversityDurhamUSA

Personalised recommendations