{
  "id": "1608.05658",
  "version": "v1",
  "published": "2016-08-19T16:28:18.000Z",
  "updated": "2016-08-19T16:28:18.000Z",
  "title": "Variance of the volume of random real algebraic submanifolds",
  "authors": [
    "Thomas Letendre"
  ],
  "comment": "60 pages, all comments are welcome",
  "categories": [
    "math.MG",
    "math.AG",
    "math.PR"
  ],
  "abstract": "Let $\\mathcal{X}$ be a complex projective manifold of dimension $n$ defined over the reals and let $M$ denote its real locus. We study the vanishing locus $Z\\_{s\\_d}$ in $M$ of a random real holomorphic section $s\\_d$ of $\\mathcal{E} \\otimes \\mathcal{L}^d$, where $ \\mathcal{L} \\to \\mathcal{X}$ is an ample line bundle and $ \\mathcal{E}\\to \\mathcal{X}$ is a rank $r$ Hermitian bundle. When $r \\in \\{1,\\dots , n -- 1\\}$, we obtain an asymptotic of order $d^{r-- \\frac{n}{2}}$, as $d$ goes to infinity, for the variance of the linear statistics associated to $Z\\_{s\\_d}$, including its volume. Given an open set $U \\subset M$, we show that the probability that $Z\\_{s\\_d}$ does not intersect $U$ is a $O$ of $d^{-\\frac{n}{2}}$ when $d$ goes to infinity. When $n\\geq 3$, we also prove almost sure convergence for the linear statistics associated to a random sequence of sections of increasing degree. Our framework contains the case of random real algebraic submanifolds of $\\mathbb{RP}^n$ obtained as the common zero set of $r$ independent Kostlan--Shub--Smale polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-19T16:28:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random real algebraic submanifolds",
      "random real holomorphic section",
      "linear statistics",
      "ample line bundle",
      "common zero set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}