Inventiones mathematicae

, Volume 153, Issue 1, pp 105–143 | Cite as

Sur la rigidité de certains groupes fondamentaux, l’arithméticité des réseaux hyperboliques complexes, et les “faux plans projectifs”

Article
First Online:
Received:
Accepted:

Abstract

The motivation of this work comes from the study of lattices in real simple Lie groups. The famous Margulis’s superrigidity theorem claims that finite dimensional reductive representations of any lattice of a real simple Lie group of real rank ≥2 are superrigid. As a corollary such a lattice is arithmetic. These results extend to the real rank one case for lattices in Sp(n,1) and F4(-20) by the work of Corlette and Gromov-Schoen. On the other hand Mostow and Deligne-Mostow exhibited arithmetic lattices with non-superrigid representations as well as non-arithmetic lattices in the unitary group PU(2,1). A natural question is then to find simple sufficient conditions for superrigidity or arithmeticity of lattices in PU(2,1). Rogawski conjectured the following: let Γ be a torsion-free cocompact lattice in PU(2,1) such that the hyperbolic quotient M=Γ\B2ℂ verifies the cohomogical conditions b1(M)=0 and H1,1(M,ℂ)∩H2(M,ℚ)≃ℚ. Then Γ is arithmetic. In this paper we consider a smooth complex projective surface M verifying the above cohomological assumptions and study Zariski-dense representations of the fundamental group π1(M) in a simple k-group H of k-rank ≤2 (where k denotes a local field). Our main result states that there are strong restrictions on such representations, especially when k is non-archimedean (Theorem 5). We then consider some cocompact lattices in PU(2,1) of special geometric interest: recall that a “fake P2ℂ” is a smooth complex surface (distinct from P2ℂ) having the same Betti numbers as P2ℂ. “Fake P2ℂ” exist by a result of Mumford and are complex hyperbolic quotients Γ\H2ℂ by Yau’s proof of the Calabi conjecture. They obviously verify the hypotheses of Rogawski’s conjecture. In this case we prove that every Zariski-dense representation of Γ in PGL(3) is superrigid in the sense of Margulis (Theorem 3). As a corollary every “fake P2ℂ” is an arithmetic quotient of the ball B2ℂ.

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 4. Berlin, New York: Springer 1984 Google Scholar
  2. 2.
    Blasius, D., Rogawski, J.: Cohomology of congruence subgroups and Hodge cycles. In: Regulators in Analysis, Geometry and Number Theory. Prog. Math. 171 (2000) Google Scholar
  3. 3.
    Bourbaki: Groupes et Algèbres de Lie. Chap. 4–6. Masson 1981 Google Scholar
  4. 4.
    Carlson, J., Toledo, D.: Harmonic mappings of Kähler manifolds to locally symmetric spaces. Publ. Math., Inst. Hautes Étud. Sci. 69, 173–201 (1989) Google Scholar
  5. 5.
    Corlette, K.: Archimedean superrigidity and hyperbolic geometry. Ann. Math. 135, 165–182 (1992) MathSciNetMATHGoogle Scholar
  6. 6.
    Deligne, P., Mostow, G.D: Monodromy of hypergeometric functions and non-lattice integral monodromy. Publ. Math., Inst. Hautes Étud. Sci. 63, 5–89 (1986) Google Scholar
  7. 7.
    Fujiki, A.: Finite automorphism groups of complex tori of dimension two. Publ. Res. Inst. Math. Sci. 24, 1–98 (1988) MathSciNetMATHGoogle Scholar
  8. 8.
    Griffiths, P.: Periods of integrals on algebraic manifolds III. Publ. Math., Inst. Hautes Étud. Sci. 38, 125–180 (1970) Google Scholar
  9. 9.
    Griffiths, P., Schmid, W.: Locally homogeneous complex manifolds. Acta Math. 123, 253–302 (1969) MATHGoogle Scholar
  10. 10.
    Gromov, M., Piatetski-Shapiro, I.: Non-arithmetic groups in Lobachevsky spaces. Publ. Math., Inst. Hautes Étud. Sci. 65, 93–103 (1988) Google Scholar
  11. 11.
    Gromov, M., Schoen, R.: Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Publ. Math., Inst. Hautes Étud. Sci. 76, 165–246 (1992) Google Scholar
  12. 12.
    Grothendieck, A.: Cohomologie globale des faisceaux cohérents et Théorèmes de Lefschetz locaux et globaux (SGA2) (1962). Advanced Studies in Pure Mathematics, Vol. 2. Amsterdam: North-Holland Publishing Co.; Paris: Masson et Cie, Éditeur 1968 Google Scholar
  13. 13.
    Hartman, P.: On homotopic harmonic maps. Can. J. Math. 19, 673–687 (1967) MATHGoogle Scholar
  14. 14.
    Jost, J., Zuo, K.; Harmonic maps and SL(r,ℂ)-representations of π1 of quasi-projective manifolds. J. Algebr. Geom. 5, 77–106 (1996) Google Scholar
  15. 15.
    Jost, J., Zuo, K.: Harmonic maps into Bruhat-Tits buildings and factorizations of p-adically unbounded representations of π1 of algebraic varieties. J. Algebr. Geom. 9, 1–42 (2000) MathSciNetMATHGoogle Scholar
  16. 16.
    Kato, F.: On p-adic uniformization of fake projective planes. Preprint Google Scholar
  17. 17.
    Katzarkov, L.: On the Shafarevich maps. Algebraic geometry, Santa Cruz 1995, pp. 173–216. Proc. Sympos. Pure Math. 62, Part 2. Providence, RI: Am. Math. Soc. 1997 Google Scholar
  18. 18.
    Katzarkov, L., Pantev, T.: Representations of fundamental groups whose Higgs bundles are pullbacks. J. Differ. Geom. 39, 103–121 (1994) MathSciNetMATHGoogle Scholar
  19. 19.
    Makarov, V.S.: On a certain class of discrete subgroups of Lobachevsky space having an infinite fundamental region of finite measure. Sov. Math. Dokl. 7, 328–331 (1966) Google Scholar
  20. 20.
    Margulis, G.A.: Discrete subgroups of semisimple Lie groups. Ergebnisse der Mathematik und ihrer Grengebiete 17. Springer-Verlag 1991 Google Scholar
  21. 21.
    Mok, N.: Metric rigidity theorem on Hermitian Symmetric Manifolds. Singapore: World Science Publishing 1989 Google Scholar
  22. 22.
    Mostow, G.D.: On a remarkable class of polyhedra in complex hyperbolic space. Pac. J. Math. 86, 171–276 (1980) MathSciNetMATHGoogle Scholar
  23. 23.
    Mumford, D.: An algebraic surface with K ample, (K 2)=9, p g=q=0. Am. J. Math. 101, 233–244 (1979) MathSciNetMATHGoogle Scholar
  24. 24.
    Reznikov, A.: Simpson’s theory and superrigidity of complex hyperbolic lattices. C. R. Acad. Sci., Paris, Sér. I, Math. 320, 1061–1064 (1995) Google Scholar
  25. 25.
    Schoen, R., Yau, S.T.: Lectures on harmonic maps. Conference Proceedings and Lecture Notes in Geometry and Topology. International Press 1997 Google Scholar
  26. 26.
    Serre, J.P.: Revêtements ramifiés du plan projectif. Sémin. Bourbaki 204 (1960) Google Scholar
  27. 27.
    Simpson, C.: The ubiquity of variations of Hodge structure. Proc. Symp. Pure Math. 53 (1991) Google Scholar
  28. 28.
    Simpson, C.: Lefschetz theorems for the integral leaves of a holomorphic one-form. Compos. Math. 87, 99–113 (1993) MathSciNetMATHGoogle Scholar
  29. 29.
    Simpson, C.: Higgs bundles and local systems. Publ. Math., Inst. Hautes Étud. Sci. 75, 5–95 (1992) Google Scholar
  30. 30.
    Simpson, C.: Moduli of representations of the fundamental group of a smooth projective variety I. Publ. Math., Inst. Hautes Étud. Sci. 79, 47–129 (1994) Google Scholar
  31. 31.
    Vinberg, E.B.: Discrete groups generated by reflections in Lobachevsky spaces. Math. USSR SB 1, 429–444 (1967) Google Scholar
  32. 32.
    Wang, M.: On representations of complex hyperbolic lattices. Math. Res. Lett. 6, 99–105 (1999) MathSciNetGoogle Scholar
  33. 33.
    Zimmer, R.: Ergodic theory and semisimple groups. Monogr. Math. 81. Birkhäuser Verlag 1984 Google Scholar
  34. 34.
    Zuo, K.: Factorization of non-rigid Zariski-dense representation of π1 of projective manifolds. Invent. Math. 118, 37–46 (1994)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Institut Post-Doctoral EuropéenIHESBures-sur-YvetteFrance

Personalised recommendations