Hodge type theorems for arithmetic manifolds associated to orthogonal groups

Abstract

We show that special cycles generate a large part of the cohomology of locally symmetric spaces associated to orthogonal groups. We prove in particular that classes of totally geodesic submanifolds generate the cohomology groups of degree $n$ of compact congruence $p$-dimensional hyperbolic manifolds "of simple type" as long as $n$ is strictly smaller than $\frac{p}{3}$. We also prove that for connected Shimura varieties associated to $\OO (p,2)$ the Hodge conjecture is true for classes of degree $< \frac{p+1}{3}$. The proof of our general theorem makes use of the recent endoscopic classification of automorphic representations of orthogonal groups by \cite{ArthurBook}. As such our results are conditional on the hypothesis made in this book, whose proofs have only appear on preprint form so far; see the second paragraph of subsection \ref{org2} below.