{
  "id": "1612.07299",
  "version": "v1",
  "published": "2016-12-21T20:00:25.000Z",
  "updated": "2016-12-21T20:00:25.000Z",
  "title": "The Kähler-Ricci flow and optimal degenerations",
  "authors": [
    "Ruadhaí Dervan",
    "Gábor Székelyhidi"
  ],
  "comment": "12 pages",
  "categories": [
    "math.DG",
    "math.AG"
  ],
  "abstract": "We prove that on Fano manifolds, the K\\\"ahler-Ricci flow produces a \"most destabilising\" special degeneration, with respect to a new stability notion related to the H-functional. This answers questions of Chen-Sun-Wang and He. We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the supremum of Perelman's {\\mu}-functional on Fano manifolds, resolving a conjecture of Tian-Zhang-Zhang-Zhu as a special case. Secondly, we use this to prove that if a Fano manifold admits a K\\\"ahler-Ricci soliton, then the K\\\"ahler-Ricci flow converges to it modulo the action of automorphisms, with any initial metric. This extends work of Tian-Zhu and Tian-Zhang-Zhang-Zhu, where either the manifold was assumed to admit a K\\\"ahler-Einstein metric, or the initial metric of the flow was assumed to be invariant under a maximal compact group of automorphism.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-21T20:00:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "kähler-ricci flow",
      "optimal degenerations",
      "initial metric",
      "fano manifold admits",
      "maximal compact group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}