{
  "id": "2002.06026",
  "version": "v1",
  "published": "2020-02-14T13:23:44.000Z",
  "updated": "2020-02-14T13:23:44.000Z",
  "title": "Successive minima and asymptotic slopes in Arakelov Geometry",
  "authors": [
    "François Ballaÿ"
  ],
  "comment": "34 pages",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let $X$ be a normal and geometrically integral projective variety over a global field $K$ and let $\\overline{D}$ be an adelic Cartier divisor on $X$. We prove a conjecture of Chen, showing that the essential minimum $\\zeta_{\\mathrm{ess}}(\\overline{D})$ of $\\overline{D}$ equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that $\\zeta_{\\mathrm{ess}}(\\overline{D})$ can be read on the Okounkov body of the underlying divisor $D$ via the Boucksom--Chen concave transform. This gives a new interpretation of Zhang's inequalities on successive minima and a criterion for equality generalizing to arbitrary projective varieties a result of Burgos Gil, Philippon and Sombra concerning toric metrized divisors on toric varieties. When applied to a projective space $X = \\mathbb{P}_K^d$, our main result has several applications to the study of successive minima of hermitian vector spaces. We obtain an absolute transference theorem with a linear upper bound, answering a question raised by Gaudron. We also give new comparisons between successive slopes and absolute minima, extending results of Gaudron and R\\'emond.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-14T13:23:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G40",
      "11G50",
      "11H06"
    ],
    "keywords": [
      "successive minima",
      "arakelov geometry",
      "asymptotic slopes",
      "mild positivity assumptions",
      "adelic cartier divisor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}