{
  "id": "1408.2107",
  "version": "v2",
  "published": "2014-08-09T14:34:49.000Z",
  "updated": "2015-03-11T14:38:47.000Z",
  "title": "Expected volume and Euler characteristic of random submanifolds",
  "authors": [
    "Thomas Letendre"
  ],
  "comment": "48 pages, reorganised, partially rewritten",
  "categories": [
    "math.MG",
    "math.AG",
    "math.PR"
  ],
  "abstract": "In a closed manifold of positive dimension $n$, we estimate the expected volume and Euler characteristic for random submanifolds of codimension $r\\in \\{1,...,n\\}$ in two different settings. On one hand, we consider a closed Riemannian manifold and some positive $\\lambda$. Then we take $r$ independent random functions in the direct sum of the eigenspaces of the Laplace-Beltrami operator associated to eigenvalues less than $\\lambda^2$ and consider the random submanifold defined as the common zero set of these $r$ functions. We compute asymptotics for the mean volume and Euler characteristic of this random submanifold as $\\lambda$ goes to infinity. On the other hand, we consider a complex projective manifold defined over the reals, equipped with an ample line bundle $\\mathcal{L}$ and a rank $r$ holomorphic vector bundle $\\mathcal{E}$ that are also defined over the reals. Then we get asymptotics for the expected volume and Euler characteristic of the real vanishing locus of a random real holomorphic section of $\\mathcal{E}\\otimes\\mathcal{L}^d$ as $d$ goes to infinity. The same techniques apply to both settings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-09T14:34:49.000Z",
      "abstract": "In a closed manifold of positive dimension $n$, we estimate the expected volume and Euler characteristic for random submanifolds of codimension $r\\in \\{1,\\dots,n\\}$ in two different settings. On one hand, we consider a closed Riemannian manifold and some positive $\\lambda$. Then we take $r$ independent random functions in the direct sum of the eigenspaces of the Laplace-Beltrami operator associated to eigenvalues less than $\\lambda^2$ and consider the random submanifold defined as the common zero set of these $r$ functions. We compute asymptotics for the mean volume and Euler characteristic of this random submanifold as $\\lambda$ goes to infinity. On the other hand, we consider a complex projective manifold defined over the reals, equipped with an ample line bundle $\\mathcal{L}$ and a rank $r$ holomorphic vector bundle $\\mathcal{E}$ that are also defined over the reals. Then we get asymptotics for the expected volume and Euler characteristic of the vanishing locus of a random real holomorphic section of $\\mathcal{E}\\otimes\\mathcal{L}^d$ as $d$ goes to infinity. The same techniques apply to both settings.",
      "comment": "48 pages, comments are welcome",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-03-11T14:38:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "euler characteristic",
      "random submanifold",
      "expected volume",
      "random real holomorphic section",
      "holomorphic vector bundle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.2107L"
    }
  }
}