  Let $\Gamma\backslash G/K$ be a compact Hermitian locally symmetric space,
where $G$ is simple. We study the components of a de Rham cohomology class of
$\Gamma\backslash G/K$, with respect to the Matsushima decomposition, where the
class is obtained by taking Poincar\'e dual of a totally geodesic complex
analytic submanifold. Using an extension of the vanishing result of Kobayashi
and Oda, we specify the existence of certain components of such cohomology
classes when $G=\text{SU}(p,q), 5\le p\le q$.
