Geometriae Dedicata

, Volume 150, Issue 1, pp 377–389 | Cite as

Forgetful maps between Deligne–Mostow ball quotients

Original Paper


We study forgetful maps between Deligne–Mostow moduli spaces of weighted points on \({\mathbb{P}^1}\) , and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livné fibrations and the Mostow/Toledo maps between complex hyperbolic surfaces. They also include a retraction of a 3-dimensional ball quotient onto one of its 1-dimensional totally geodesic complex submanifolds.


Ball quotients Holomorphic maps Deligne–Mostow moduli spaces 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bergeron, N., Haglund, F., Wise, D.: Hyperplane sections in arithmetic hyperbolic manifolds, Preprint (2008)Google Scholar
  2. 2.
    Deligne P., Mostow G.D.: Monodromy of hypergeometric functions and non-lattice integral monodromy. Inst. Hautes Études Sci. Publ. Math. 63, 5–89 (1986)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Deligne P., Mostow G.D.: Commensurabilities Among Lattices in PU(1,n), Ann. Math. Stud., vol. 132. Princeton University Press, Princeton (1993)Google Scholar
  4. 4.
  5. 5.
    Deraux, M.: Complex Surfaces of Negative Curvature. Ph.D. thesis, University of Utah (2001)Google Scholar
  6. 6.
    Deraux M.: On the universal cover of certain exotic Kähler surfaces of negative curvature. Math. Ann. 329(4), 653–683 (2004)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Deraux M.: A negatively curved Kähler threefold not covered by the ball. Inv. Math. 160(3), 501–525 (2005)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Kirwan F.C., Lee R., Weintraub S.H.: Quotients of the complex ball by discrete groups. Pac. J. Math. 130, 115–141 (1987)MathSciNetMATHGoogle Scholar
  9. 9.
    Koziarz, V., Mok, N.: Nonexistence of holomorphic submersions between complex unit balls equivariant with respect to a lattice and their generalizations. Preprint, arXiv:0804.2122 (2008)Google Scholar
  10. 10.
    Livné, R.A.: On certain covers of the universal elliptic curve. Ph.D. thesis, Harvard University (1981)Google Scholar
  11. 11.
    Looijenga E.: Uniformization by Lauricella functions—an overview of the theory of Deligne–Mostow, Arithmetic and geometry around hypergeometric functions. In: Holzapfel, R.-P., Uludăg, A.M., Yoshida, M. (eds) Progress in Mathematics, vol. 260, pp. 207–244. Birkhäuser, Basel (2007)Google Scholar
  12. 12.
    Mok N.: Uniqueness theorems of hermitian metrics of seminegative curvature on quotients of bounded symmetric domains. Ann. Math. 125(2), 105–152 (1987)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Mostow G.D.: Generalized Picard lattices arising from half-integral conditions. Inst. Hautes Études Sci. Publ. Math. 63, 91–106 (1986)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Mostow G.D.: On discontinuous action of monodromy groups on the complex n-ball. J. Amer. Math. Soc. 1, 555–586 (1988)MathSciNetMATHGoogle Scholar
  15. 15.
    Mostow G.D., Siu Y.T.: A compact Kähler surface of negative curvature not covered by the ball. Ann. Math. 112, 321–360 (1980)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Siu, Y.-T.: Some recent results in complex manifold theory related to vanishing theorems for the semipos- itive case, Arbeitstagung Bonn 1984. In: Hirzebruch, F., Schwermer, J., Suter, S. (eds.). Lecture Notes in Mathematics, vol. 1111, pp. 169–192. Springer (1985)Google Scholar
  17. 17.
    Thurston W.P.: Shapes of polyhedra and triangulations of the sphere. Geom. Topol. Monogr. 1, 511–549 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Toledo D.: Maps between complex hyperbolic surfaces. Geom. Dedicata 97, 115–128 (2003)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Institut FourierUniversité de Grenoble I, UMR 5582Saint-Martin d’HèresFrance

Personalised recommendations