{
  "id": "0707.1352",
  "version": "v2",
  "published": "2007-07-10T02:09:05.000Z",
  "updated": "2008-02-19T03:36:12.000Z",
  "title": "Mixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations",
  "authors": [
    "Eduardo Cattani"
  ],
  "comment": "13 pages - Minor revisions. Final version to appear in International Mathematics Research Notices",
  "categories": [
    "math.AG"
  ],
  "abstract": "Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose restrictions on the cohomology algebra of a smooth compact K\\\"ahler manifold or on the intersection cohomology of a projective toric variety; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions on the possible $f$-vectors of convex polytopes. While the statements of these theorems depend on the choice of a K\\\"ahler class, or its analog, there is usually a cone of possible K\\\"ahler classes. It is then natural to ask whether the HLT and HRR remain true in a mixed context. In this note we present a unified approach to proving the mixed HLT and HRR, generalizing the previously known results, and proving it in new cases such as the intersection cohomology of non-rational polytopes.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-02-19T03:36:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32G20",
      "14F43",
      "32Q15",
      "52B20"
    ],
    "keywords": [
      "hodge-riemann bilinear relations",
      "mixed lefschetz theorems",
      "intersection cohomology",
      "hrr remain true",
      "hard lefschetz theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0707.1352C"
    }
  }
}