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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.

32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
Full Text: DOI
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