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Spectrum and homology of congruence complex hyperbolic manifolds. (Spectre et homologie des variétés hyperboliques complexes de congruence.) (French) Zbl 1011.32020
Summary: In the first part of this note, we show that the first non-zero eigenvalue of the Laplace operator on 1-forms of a standard congruence arithmetic complex hyperbolic \(n\)-manifold is always \(\geq{10n-11\over 25}\). The following parts of this note concern homological applications of this result. We prove, in particular, that if \(\text{Sh}^0 H\subset\text{Sh}^0G\) are two Shimura varieties of type \(U(n-1,1)\) and \(U(n,1)\), the natural map \(H_{2n-3}( \text{Sh}^0H)\to H_{2n-3} (\text{Sh}^0G)\) is injective, first step of a “Lefschetz theorem” for Shimura varieties.

MSC:
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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[1] Arthur, J., Unipotent automorphic representations: conjectures, Astérisque, 171-172, 13-172, (1989) · Zbl 0728.22014
[2] N. Bergeron, Asymptotique de la norme L^2 d’un cycle géodésique dans les revêtements de congruence d’une variété hyperbolique arithmétique, 2002, à paraı̂tre
[3] Borel, A.; Wallach, N., Continuous cohomology, discrete subgroups, and representations of reductive groups, (1980), Princeton University Press · Zbl 0443.22010
[4] Burger, M.; Sarnak, P., Ramanujan duals II, Invent. math., 106, 1-11, (1991) · Zbl 0774.11021
[5] Clozel, L.; Venkataramana, T.N., Restriction of the holomorphic cohomology of a Shimura variety to a smaller Shimura variety, Duke math. J., 95, 51-106, (1998) · Zbl 1037.11504
[6] Donnelly, H.; Fefferman, C., L^2-cohomology and index theorem for the Bergman metric, Ann. of math. (2), 118, 593-618, (1983) · Zbl 0532.58027
[7] N. Bergeron, L. Clozel, Article en préparation
[8] Harris, M.; Li, J.S., A Lefschetz property for subvarieties of Shimura varieties, J. algebraic geometry, 7, 77-122, (1998) · Zbl 0954.14016
[9] Matsushima, Y., A formula for the Betti numbers of compact locally symmetric Riemannian manifolds, J. differential geom., 1, 99-109, (1967) · Zbl 0164.22101
[10] Rogawski, J., Automorphic representations of unitary groups in three variables, (1990), Princeton University Press · Zbl 0724.11031
[11] Rohlfs, J., Projective limits of locally symmetric spaces and cohomology, J. reine angew. math., 479, 149-182, (1996) · Zbl 0876.22021
[12] Shahidi, F., Automorphic L-functions: a survey, Perspect. math., 10, 415-437, (1990) · Zbl 0715.11067
[13] Venkataramana, T.N., Cohomology of compact locally symmetric spaces, Compositio math., 125, 221-253, (2001) · Zbl 0983.11027
[14] Rudnick, Z.; Luo, W.; Sarnak, P., On the generalized Ramanujan conjecture for GL(n), Proc. sympos. pure math., 66, 301-310, (1999) · Zbl 0965.11023
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