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Hodge Type Theorems for Arithmetic Manifolds Associated to Orthogonal Groups

2016; Oxford University Press; Linguagem: Inglês

10.1093/imrn/rnw067

1687-0247

Nicolas Bergeron, John J. Millson, Colette Mœglin,

Geometry and complex manifolds

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 |$\mathcal{O} (p,2)$| 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 [6]. As such our results are conditional on the hypothesis made in this article, whose proofs have only appeared in preprint form so far; see the second paragraph of Section 1.3.1 below.