{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "1X3EV2TZNN"
    },
    {
      "field": "about",
      "target": "VEV5E2077C"
    },
    {
      "field": "uses",
      "target": "1X3EV2TZNN"
    },
    {
      "field": "uses",
      "target": "AD9GD9EM9R"
    },
    {
      "field": "uses",
      "target": "NXTV4Y81ZJ"
    },
    {
      "field": "uses",
      "target": "VEV5E2077C"
    },
    {
      "field": "defines",
      "target": "6V6WDRHRTF"
    }
  ],
  "body": "\n## Summary\n\nA complex manifold is Kobayashi hyperbolic when its [[Kobayashi pseudodistance|DY8QQ5GTTX]] is a true distance, that is, it separates points.\nBounded domains in $\\C^n$ and compact Riemann surfaces of genus at least $2$ are hyperbolic.\nThe spaces $\\C^n$, the projective spaces and the complex tori are not.\nA hyperbolic manifold admits no nonconstant holomorphic map from $\\C$, and for compact manifolds this property characterises hyperbolicity by [[Brody's lemma|VEV5E2077C]].\nHyperbolicity is linked with the geometry of the canonical bundle and with the arithmetic of varieties, through the [[Kobayashi conjecture|1X3EV2TZNN]] and the [[Green–Griffiths–Lang conjecture|AD9GD9EM9R]].\n\n## Explanations\n\n### L1 human\n\nA complex manifold is hyperbolic when holomorphic discs inside it cannot be too large.\nThen the Kobayashi construction, which measures distances with chains of discs, sees every pair of distinct points at a positive distance.\nIn $\\C$ there are discs of every radius, so all Kobayashi distances are $0$, and $\\C$ is not hyperbolic.\nThe unit disc is hyperbolic, and its Kobayashi distance is the usual hyperbolic distance.\nA domain such as $\\C$ minus two points is also hyperbolic, which is one way to see the little Picard theorem.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32Q45",
      "32F45"
    ],
    "name": "Kobayashi hyperbolic manifold",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Kobayashi_metric"
      }
    ],
    "tag": "HRPAX8V7HA"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/HRPAX8V7HA/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/HRPAX8V7HA.md",
  "msc": [
    "32Q45",
    "32F45"
  ],
  "name": "Kobayashi hyperbolic manifold",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HRP/HRPAX8V7HA.md",
  "status": null,
  "tag": "HRPAX8V7HA",
  "uri": "m2://HRPAX8V7HA",
  "url": "https://tamnd.github.io/math2/wiki/kobayashi-hyperbolic-manifold/",
  "writing": {
    "tag": "HRPAX8V7HA",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}