Tentative d'\'epuisement de la cohomologie d'une vari\'et\'e de Shimura par restriction \`a ses sous-vari\'et\'es

Abstract

Let $G$ be a connected semisimple group over ${\Bbb Q}$. Given a maximal compact subgroup $K\subset G({\Bbb R})$ such that $X=G({\Bbb R})/K$ is a Hermitian symmetric domain, and a convenient arithmetic subgroup $\Gamma\subset G({\Bbb Q})$, one constructs a (connected) Shimura variety $S=S(\Gamma)=\Gamma\backslash X$. If $H\subset G$ is a connected semisimple subgroup such that $H({\Bbb R})\cap K$ is maximal compact, then $Y=H({\Bbb R})/K$ is a Hermitian symmetric subdomain of $X$. For each $g\in G({\Bbb Q})$ one can construct a connected Shimura variety $S(H,g)=(H({\Bbb Q})\cap g^{-1}\Gamma g)\backslash Y$ and a natural holomorphic map $j_g \colon S(H,g)\to S$ induced by the map $H({\Bbb A})\to G({\Bbb A}), h\mapsto gh$. Let us assume that $G$ is anisotropic, which implies that $S$ and $S(H,g)$ are compact. Then, for each positive integer $k$, the map $j_g$ induces a restriction map $$R_g \colon H^{k}(S, {\Bbb C})\to H^{k}(S(H,g), {\Bbb C}).$$ In this paper we focus on classical Hermitian domains and give explicit criterions for the injectivity of the product of the maps $R_g$ (for $g$ running through $G({\Bbb Q})$) when restricted to the strongly primitive (in the sense of Vogan and Zuckerman) part of the cohomology. In the holomorphic case we recover previous results of Clozel and Venkataramana. We also derive applications of our results to the proofs of new cases of the Hodge conjecture and of new results on the vanishing of the cohomology of some particular Shimura variety.