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

TL;DR

This paper investigates the structure of certain cohomology classes in compact Hermitian locally symmetric spaces, identifying specific components related to totally geodesic submanifolds using advanced decomposition techniques.

Contribution

It extends vanishing results to specify the existence of particular cohomology components in spaces associated with SU(p,q), for 5 ≤ p ≤ q.

Findings

Identifies non-vanishing components of cohomology classes in specific symmetric spaces.

Extends previous vanishing theorems to new cases involving SU(p,q).

Provides criteria for the existence of certain geometric cohomology components.

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