Abstract
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 ⩾ 10n−11 25 . The following parts of this Note concern homological applications of this result. We prove, in particular, that if Sh 0 H⊂Sh 0 G are two Shimura varieties of type U( n−1,1) and U( n,1), the natural map H 2 n−3 (Sh 0 H)→ H 2 n−3 (Sh 0 G) is injective, first step of a “Lefschetz theorem” for Shimura varieties. To cite this article: N. Bergeron, L. Clozel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 995–998.
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