$A_{\mathfrak{q}}$-components of geometric classes in compact Hermitian locally symmetric spaces

Abstract

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 Poincaré 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 $G=\text{SU}(p,q),5\le p\le q$.

14 pages