{
  "id": "1110.3049",
  "version": "v3",
  "published": "2011-10-13T20:01:20.000Z",
  "updated": "2015-01-23T19:23:45.000Z",
  "title": "Hodge type theorems for arithmetic manifolds associated to orthogonal groups",
  "authors": [
    "Nicolas Bergeron",
    "John Millson",
    "Colette Moeglin"
  ],
  "comment": "83 pages",
  "categories": [
    "math.NT",
    "math.AG",
    "math.GT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-01-06T14:41:19.000Z",
      "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 $\\frac12 [\\frac{p}{2}]$. We also prove that for connected Shimura varieties associated to $\\OO (p,2)$ the Hodge conjecture is true for classes of degree $< 1/2 [\\frac{p+1}{2}]$. 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 stabilization of the trace formula for the (disconnected) groups $\\GL (N) \\rtimes <\\theta>$ and $\\SO(2n) \\rtimes <\\theta'>$ (where $\\theta$ and $\\theta'$ are the outer automorphisms), see \\cite[Hypothesis 3.2]{ArthurBook}. Unfortunately, at present the stabilization of the trace formula has been proved only for the case of {\\it connected} groups. The extension needed is part of work in progress by the Paris-Marseille team of automorphic form researchers. For more detail, see the second paragraph of subsection \\ref{org2} below.",
      "comment": "81 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-01-23T19:23:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "orthogonal groups",
      "hodge type theorems",
      "arithmetic manifolds",
      "trace formula",
      "totally geodesic submanifolds generate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 83,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.3049B"
    }
  }
}