Hodge Type Theorems for Arithmetic Hyperbolic Manifolds

This note is a variation on the lecture given by the first named author at the conference celebrating the work (and sixty-fifth anniversary) of Jean-Michel Bismut. In this lecture a proof of some new cases of the Hodge conjecture for Shimura varieties uniformized by complex balls was sketched following [3]. In this note we exemplify the main ideas of the proof on real hyperbolic manifolds. The Hodge conjecture does not make sense anymore but, somewhat analogously, we prove that classes of totally geodesic submanifolds generate the cohomology groups of degree k of compact congruence n-dimensional hyperbolic manifolds “of simple type” as long as k is strictly smaller than n/3. This is a particular case of the main result of [2].