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A_mathfrak{q}-components of geometric classes in compact Hermitian locally symmetric spaces
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A_mathfrak{q}-components of geometric classes in compact Hermitian locally symmetric spaces
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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 $G=\text{SU}(p,q), 5\le p\le q$.
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