On cycles on compact locally symmetric varieties

We give a criterion to determine when the cycle class of a locally symmetric subvariety S<SUB>H</SUB>(Γ) of a compact locally symmetric variety S(Γ) generates a non-trivial module under the action of Hecke operators, and give several examples where this criterion is satisfied. We also exhibit examples of subvarieties S<SUB>H</SUB>(Γ) which do generate the trivial module under the action of Hecke operators. We show that all Hodge classes (in degree 4n - 4) on the locally symmetric variety S(Γ) associated to certain arithmetric subgroups Γ of SU(2, n) are algebraic (provided that n ≥ 5)