On cycles on compact locally symmetric varieties (Q1601797)

Let \(G\) be a connected simply connected semisimple algebraic group defined and anisotropic over \(\mathbb Q\), and let \(\Gamma\) be a torsion-free cocompact arithmetic subgroup of \(G(\mathbb Q)\). Assuming that the Riemannian symmetric space \(X = G(\mathbb R)/K_\infty\) with \(K_\infty\) being a maximal compact subgroup has a \(G(\mathbb R)\)-invariant complex structure, it is well-known that the locally symmetric space \(S(\Gamma) = \Gamma \backslash X\) has the structure of a smooth projective variety. In this paper the author obtains a criterion to determine when the cycle class of a locally symmetric subvariety \(S_H(\Gamma)\) of \(S(\Gamma)\) generates a nontrivial module under the action of Hecke operators and provide examples of such subvarieties. He also proves that all Hodge classes in degree \(4n-4\) on \(S(\Gamma)\) associated to certain arithmetic subgroups \(\Gamma\) of \(\text{SU}(2,n)\) are algebraic provided that \(n \geq 5\).