Aq-components of geometric classes in compact Hermitian locally symmetric spaces

Let Gamma\G/K be a compact Hermitian locally symmetric space, where G is simple. We study the components of a de Rham cohomology class of Gamma\G/K with respect to the Matsushima decomposition, where the class is obtained by taking the Poincare dual of a totally geodesic complex analytic submanifold. Using an improved version of the vanishing result of Kobayashi and Oda, we specify the existence of certain components of such cohomology classes when G =