{
  "id": "2008.08829",
  "version": "v1",
  "published": "2020-08-20T08:07:39.000Z",
  "updated": "2020-08-20T08:07:39.000Z",
  "title": "Basis divisors and balanced metrics",
  "authors": [
    "Yanir A. Rubinstein",
    "Gang Tian",
    "Kewei Zhang"
  ],
  "comment": "comments are welcome!",
  "categories": [
    "math.DG",
    "math.AG"
  ],
  "abstract": "Using log canonical thresholds and basis divisors Fujita--Odaka introduced purely algebro-geometric invariants $\\delta_m$ whose limit in $m$ is now known to characterize uniform K-stability on a Fano variety. As shown by Blum-Jonsson this carries over to a general polarization, and together with work of Berman, Boucksom, and Jonsson, it is now known that the limit of these $\\delta_m$-invariants characterizes uniform Ding stability. A basic question since Fujita-Odaka's work has been to find an analytic interpretation of these invariants. We show that each $\\delta_m$ is the coercivity threshold of a quantized Ding functional on the $m$-th Bergman space and thus characterizes the existence of balanced metrics. This approach has a number of applications. The most basic one is that it provides an alternative way to compute these invariants, which is new even for $\\mathbb{P}^n$. Second, it allows us to introduce algebraically defined invariants that characterize the existence of K\\\"ahler-Ricci solitons (and the more general $g$-solitons of Berman-Witt Nystr\\\"om), as well as coupled versions thereof. Third, it leads to approximation results involving balanced metrics in the presence of automorphisms that extend some results of Donaldson.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-20T08:07:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "balanced metrics",
      "invariants characterizes uniform ding stability",
      "basis divisors fujita-odaka",
      "th bergman space",
      "algebro-geometric invariants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}