{
  "id": "0905.2212",
  "version": "v4",
  "published": "2009-05-13T22:24:17.000Z",
  "updated": "2011-12-12T09:20:39.000Z",
  "title": "Castelnuovo-Mumford Regularity and Computing the de Rham Cohomology of Smooth Projective Varieties",
  "authors": [
    "Peter Scheiblechner"
  ],
  "comment": "32 pages - filled a gap in Section 4.2, specific example added, minor improvements",
  "categories": [
    "math.AG",
    "cs.SC"
  ],
  "abstract": "We describe a parallel polynomial time algorithm for computing the topological Betti numbers of a smooth complex projective variety $X$. It is the first single exponential time algorithm for computing the Betti numbers of a significant class of complex varieties of arbitrary dimension. Our main theoretical result is that the Castelnuovo-Mumford regularity of the sheaf of differential $p$-forms on $X$ is bounded by $p(em+1)D$, where $e$, $m$, and $D$ are the maximal codimension, dimension, and degree, respectively, of all irreducible components of $X$. It follows that, for a union $V$ of generic hyperplane sections in $X$, the algebraic de Rham cohomology of $X\\setminus V$ is described by differential forms with poles along $V$ of single exponential order. This yields a similar description of the de Rham cohomology of $X$, which allows its efficient computation. Furthermore, we give a parallel polynomial time algorithm for testing whether a projective variety is smooth.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-12-12T09:20:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14Q15",
      "14Q20",
      "68W30"
    ],
    "keywords": [
      "projective variety",
      "rham cohomology",
      "smooth projective varieties",
      "castelnuovo-mumford regularity",
      "parallel polynomial time algorithm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.2212S"
    }
  }
}