# Automorphic Lefschetz properties via $L^2$ cohomology

## TL;DR

This paper proves a special case of a conjecture relating the primitive cohomology of certain locally symmetric manifolds to that of their totally geodesic submanifolds, advancing understanding in automorphic Lefschetz properties.

## Contribution

It establishes a specific case of Bergeron's conjecture using $L^2$ cohomology techniques, linking primitive cohomology across different symmetric spaces.

## Key findings

- Proves a special case of Bergeron's automorphic Lefschetz conjecture.
- Connects primitive cohomology of $U(p,q+r)$-modeled manifolds to $U(p,q)$-modeled submanifolds.
- Utilizes $L^2$ cohomology methods to achieve the result.

## Abstract

In this paper one proves a special case of a conjecture by Nicolas Bergeron. This conjecture is a kind of automorphic Lefschetz property. It relates the primitive cohomology of a locally symmetric manifolds modeled on $U(p,q+r)$ to the primitive cohomology of some of its totally geodesic submanifolds that are locally symmetric and modeled on $U(p,q)$.