Hodge type theorems for arithmetic manifolds associated to orthogonal groups

Nicolas Bergeron; John Millson; Colette Moeglin

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arXiv · 1110.3049

Hodge type theorems for arithmetic manifolds associated to orthogonal groups

Nicolas Bergeron·John Millson·Colette Moeglin

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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.

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Nicolas Bergeron, John Millson, Colette Moeglin. 2015-01-23. Hodge type theorems for arithmetic manifolds associated to orthogonal groups. https://arxiv.org/abs/1110.3049

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