# zbMATH — the first resource for mathematics

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
##### Keywords:
spectrum; homology; congruence complex hyperbolic manifolds
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.