  In this paper one proves a special case of a conjecture by Nicolas Bergeron.
This conjecture is a kind of automorphic Lefschetz property. It relates the
primitive cohomology of a locally symmetric manifolds modeled on $U(p,q+r)$ to
the primitive cohomology of some of its totally geodesic submanifolds that are
locally symmetric and modeled on $U(p,q)$.
