{
  "id": "1707.09771",
  "version": "v1",
  "published": "2017-07-31T09:04:42.000Z",
  "updated": "2017-07-31T09:04:42.000Z",
  "title": "Variance of the volume of random real algebraic submanifolds II",
  "authors": [
    "Thomas Letendre",
    "Martin Puchol"
  ],
  "comment": "52 pages, 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 $M$ be 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, $r \\in \\{1,\\dots, n\\}$. We establish the asymptotic of the variance of the linear statistics associated with $Z\\_{s\\_d}$, as $d$ goes to infinity. This asymptotic is of order $d^{r-\\frac{n}{2}}$. As a special case, we get the asymptotic variance of the volume of $Z\\_{s\\_d}$. The present paper extends the results of [20], by the first-named author, in essentially two ways. First, our main theorem covers the case of maximal codimension ($r = n$), which was left out in [20]. And second, we show that the leading constant in our asymptotic is positive. This last result is proved by studying the Wiener--It{\\=o} expansion of the linear statistics associated with the common zero set in $\\mathbb{RP}^n$ of $r$ independent Kostlan--Shub--Smale polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-31T09:04:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random real algebraic submanifolds",
      "linear statistics",
      "asymptotic",
      "random real holomorphic section",
      "independent kostlan-shub-smale polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}