Simpson ’ s Theory and Superrigidity of Complex Hyperbolic Lattices
نویسنده
چکیده
We attack a conjecture of J. Rogawski: any cocompact lattice in SU(2, 1) for which the ball quotient X = B/Γ satisfies b1(X) = 0 and H (X)∩H(X,Q) ≈ Q is arithmetic. We prove the Archimedian suprerigidity for representation of Γ is SL(3,C). Théorie de Simpson et superrigidité des réseaux hyperboliques complexes Résumé Soit Γ ⊂ SU(2, 1) un reseau cocompact et soit X = B/Γ. Nous preuvons: si b1(X) = 0 et H (X) ∩ H(X,Q) ≈ Q allors tous les representations ρ de Γ dans SL(3,C) sont conjugué à le représentation naturelle ou la fermeture de Zariski de l’image p(Γ) est compacte. Version française abrégéé Le théorème classique de Margulis dit que tous les réseaux dans les groupes de Lie semi-simples sont superrigides. Ceci a eté generalisé par Corlette [C] à la superrigidite des réseaux quaternioniques et de Cayley. D’autre part, Johnson et Millson ont montré qú il existait des deformations des réseaux cocompact dans SO(n, 1) si on regarde SO(n, 1) comme plongé dans SO(n+ 1, 1). C’est une question d’un intérêt fondamental de savoir si les réseaux hyperboliques complexes sont superrigides. Dans cet article, nons considerons la question suivante de J. Rogawski. Hypothése Soit X = B/Γ,Γ ⊂ SU(2, 1) une surface hyperbolique complexe compacte. Supposons b1(X) = 0 et H (X) ∩ H(X,Q) = Q. Allors Γ est arithmetique et provient d’une algébre avec division E|Q de rang 3 avec une involution. Observons que pour tous les réseux provenant d’algébres avec division, on a effectivement b1(X) = 0 et H (X) ∩H(X,Q) = Q [Rog]. Soit l un fibré linéaire tautologiue sur X [Re]. La condition H(X)∩H(X,Q) = Q dit que [l] = k· générateur dans Pic(X)/tors ≈ Z.
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