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

Abstract

Let $\mathrm{\Gamma }\setminus G∕K$ be a compact Hermitian locally symmetric space, where G is simple. We study the components of a de Rham cohomology class of $\mathrm{\Gamma }\setminus G∕K$ with respect to the Matsushima decomposition, where the class is obtained by taking the Poincaré 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 $G=\mathrm{SU}(p,q)$.