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 p 3 .We also prove that for connected Shimura varieties associated to O(p, 2) the Hodge conjecture is true for classes of degree