, Volume 135, Issue 3, pp 221-244

On Cycles on Compact Locally Symmetric Varieties

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 We give a criterion to determine when the cycle class of a locally symmetric subvariety of a compact locally symmetric variety 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 which do generate the trivial module under the action of Hecke operators. We show that all Hodge classes (in degree ) on the locally symmetric variety associated to certain arithmetric subgroups Γ of are algebraic (provided that ).

Received 16 January 2001; in revised form 18 October 2001