{
  "id": "0802.1549",
  "version": "v2",
  "published": "2008-02-12T00:35:19.000Z",
  "updated": "2008-02-18T02:59:12.000Z",
  "title": "Metric Dependence and Asymptotic Minimization of the Expected Number of Critical Points of Random Holomorphic Sections",
  "authors": [
    "Benjamin Baugher"
  ],
  "comment": "19 pages, added references; also includes a Mathematica worksheet in both notebook and pdf form",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP"
  ],
  "abstract": "We prove the main conjecture from [M. R. Douglas, B. Shiffman and S. Zelditch, Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics. J. Differential Geom. 72 (2006), no. 3, 381-427] concerning the metric dependence and asymptotic minimization of the expected number \\mathcal{N}^{crit}_{N,h} of critical points of random holomorphic sections of the Nth tensor power of a positive line bundle. The first non-topological term in the asymptotic expansion of \\mathcal{N}^{crit}_{N,h} is the the Calabi functional multiplied by the constant \\be_2(m) which depends only on the dimension of the manifold. We prove that \\be_2(m) is strictly positive in all dimensions, showing that the expansion is non-topological for all m, and that the Calabi extremal metric, when it exists, asymptotically minimizes \\mathcal{N}^{crit}_{N,h}.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-02-18T02:59:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random holomorphic sections",
      "critical points",
      "metric dependence",
      "asymptotic minimization",
      "expected number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0802.1549B"
    }
  }
}