Hodge type theorems for arithmetic manifolds associated to orthogonal groups

TL;DR

This paper demonstrates that special cycles significantly contribute to the cohomology of arithmetic manifolds linked to orthogonal groups, establishing new cases where the Hodge conjecture holds, contingent on recent automorphic representation classifications.

Contribution

It proves that classes of totally geodesic submanifolds generate cohomology in certain degrees and confirms the Hodge conjecture for specific classes on Shimura varieties, under recent automorphic classification assumptions.

Findings

Special cycles generate large parts of cohomology.

Hodge conjecture verified for certain classes on Shimura varieties.

Results depend on recent automorphic representation classification.

Abstract

We show that special cycles generate a large part of the cohomology of locally symmetric spaces associated to orthogonal groups. We prove in particular that classes of totally geodesic submanifolds generate the cohomology groups of degree $n$ of compact congruence $p$-dimensional hyperbolic manifolds "of simple type" as long as $n$ is strictly smaller than $\frac{p}{3}$. We also prove that for connected Shimura varieties associated to $\OO (p,2)$ the Hodge conjecture is true for classes of degree $< \frac{p+1}{3}$. The proof of our general theorem makes use of the recent endoscopic classification of automorphic representations of orthogonal groups by \cite{ArthurBook}. As such our results are conditional on the hypothesis made in this book, whose proofs have only appear on preprint form so far; see the second paragraph of subsection \ref{org2} below.