{
  "id": "1511.02888",
  "version": "v1",
  "published": "2015-11-09T21:15:04.000Z",
  "updated": "2015-11-09T21:15:04.000Z",
  "title": "Hodge Theory for Combinatorial Geometries",
  "authors": [
    "Karim Adiprasito",
    "June Huh",
    "Eric Katz"
  ],
  "comment": "61 pages. Comments welcome!",
  "categories": [
    "math.CO",
    "math.AG"
  ],
  "abstract": "We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of Heron, Rota, and Welsh that postulates the log-concavity of the coefficients of the characteristic polynomial of M. We furthermore conclude that the f-vector of the independence complex of a matroid forms a log-concave sequence, proving a conjecture of Mason and Welsh for general matroids.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-09T21:15:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "combinatorial geometries",
      "hodge theory",
      "hodge-riemann relations",
      "hard lefschetz theorem",
      "matroid forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 61,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151102888A"
    }
  }
}