Advertisement

Geometriae Dedicata

, Volume 137, Issue 1, pp 85–111 | Cite as

Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type

  • Vincent Koziarz
  • Julien MaubonEmail author
Original Paper

Abstract

Let Г be a torsion-free uniform lattice of SU(m, 1), m > 1. Let G be either SU(p, 2) with p ≥ 2, \({{\rm Sp}(2,\mathbb {R})}\) or SO(p, 2) with p ≥ 3. The symmetric spaces associated to these G’s are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of Kähler manifolds and Higgs bundles we study representations of the lattice Г into G. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily G = SU(p, 2) with p ≥ 2m and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(p, 2)/S(U(p) × U(2)), on which it acts cocompactly.

Keywords

Complex hyperbolic space Lattice Hermitian symmetric space Toledo invariant Higgs bundles Rigidity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bradlow S.B., Garcia-Prada O., Gothen P.B.: Surface group representations and U(p,q)-Higgs bundles. J. Diff. Geom. 64, 111–170 (2003)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bradlow S.B., Garcia-Prada O., Gothen P.B.: Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces. Geom. Dedicata. 122, 185–213 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Burger M., Iozzi A.: Bounded differential forms, generalized Milnor-Wood inequality and an application to deformation rigidity. Geom. Dedicata. 125, 1–23 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Burger M., Iozzi A., Wienhard A.: Surface group representations with maximal Toledo invariant. C.R. Acad. Sci. Paris 336, 387–390 (2003)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Burger, M., Iozzi, A., Wienhard, A.: Surface group representations with maximal Toledo invariant. arXiv:math.DG/0605656 v2 (2006)Google Scholar
  6. 6.
    Clerc J.-L., Ørsted B.: The Maslov index revisited. Tranform. Groups. 6, 303–320 (2001)zbMATHCrossRefGoogle Scholar
  7. 7.
    Corlette K.: Flat G-bundles with canonical metrics. J. Diff. Geom. 28, 361–382 (1988)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Domic A., Toledo D.: The Gromov norm of the Kähler class of symmetric domains. Math. Ann. 276, 425–432 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Eberlein, P.: Geometry of nonpositively curved manifolds. In: Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1996)Google Scholar
  10. 10.
    Goldman, W.M.: Discontinuous groups and the Euler class. Thesis, University of California at Berkeley (1980)Google Scholar
  11. 11.
    Goldman W.M.: Topological components of spaces of representations. Invent. Math. 93(3), 557–607 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Goldman W.M., Millson J.J.: Local rigidity of discrete groups acting on complex hyperbolic space. Invent. Math. 88, 495–520 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Corrected reprint of the 1978 original, Graduate Studies in Mathematics, 34. American Mathematical Society, Providence (2001)Google Scholar
  14. 14.
    Hitchin N.J.: The self-duality equations on a Riemann surface. Proc. Lon. Math. Soc. 55(3), 59–126 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hitchin N.J.: Lie groups and Teichmüller space. Topology 31, 449–473 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves. In: Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig (1997)Google Scholar
  17. 17.
    Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Princeton University Press (1987)Google Scholar
  18. 18.
    Koziarz V., Maubon J.: Harmonic maps and representations of non-uniform lattices of PU(m,1). Ann. Inst. Fourier (Grenoble) 58(2), 507–558 (2008)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Labourie F.: Existence d’applications harmoniques tordues à valeurs dans les variétés à courbure négative. Proc. Am. Math. Soc. 111(3), 877–882 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Royden H.L.: The Ahlfors-Schwarz lemma in several complex variables. Comment. Math. Helvetici. 55, 547–558 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sampson J.H.: Applications of harmonic maps to Kähler geometry. Contemp. Math. 49, 125–133 (1986)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Satake, I.: Algebraic structures of symmetric domains. In: Kanô Memorial Lectures, 4, Iwanami Shoten, Tokyo. Princeton University Press, Princeton (1980)Google Scholar
  23. 23.
    Simpson C.T.: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Am. Math. Soc. 1, 867–918 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Simpson C.T.: Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 75, 5–95 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Simpson C.T.: Moduli of representations of the fundamental group of a smooth projective variety I. Inst. Hautes Études Sci. Publ. Math. 79, 47–129 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Simpson C.T.: Moduli of representations of the fundamental group of a smooth projective variety II. Inst. Hautes Études Sci. Publ. Math. 80, 5–79 (1994)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Toledo D.: Harmonic mappings of surfaces to certain Kähler manifolds. Math. Scand. 45, 13–26 (1979)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Toledo D.: Representations of surface groups in complex hyperbolic space. J. Diff. Geom. 29, 125–133 (1989)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Viehweg E., Zuo K.: Arakelov inequalities and the uniformization of certain rigid Shimura varieties. J. Diff. Geom. 77(2), 291–352 (2007)zbMATHMathSciNetGoogle Scholar
  30. 30.
    Xia E.Z.: The moduli of flat PU(2,1) structures over Riemann surfaces. Pacific J. Math. 193, 231–256 (2000)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Institut Elie CartanUniversité Henri PoincaréVandoeuvre-les-Nancy CedexFrance

Personalised recommendations