{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $X$ be a compact [[complex manifold|JSGWAN2RJV]] with a [[Hermitian metric|CK1DCPC3D3]].\nLet $\\norm{\\cdot}$ denote the norm of tangent vectors of $X$ for this metric.\n\n1. If $X$ is not [[Kobayashi hyperbolic|HRPAX8V7HA]], then there is a nonconstant [[holomorphic map|3MQTWKBBBT]] $g \\colon \\C \\to X$ such that $\\norm{g'(t)} \\le \\norm{g'(0)} = 1$ for all $t \\in \\C$.\n2. Consequently, $X$ is Kobayashi hyperbolic if and only if it is [[Brody hyperbolic|Y7W36GJDJD]].\n\n## Remarks\n\nRobert Brody proved the theorem in 1978.\nProof idea: if a compact $X$ is not hyperbolic, then there are holomorphic maps $f_k \\colon \\mathbb{D} \\to X$ with $\\norm{f_k'(0)} \\to \\infty$.\nThe reparametrisation lemma recentres and rescales each $f_k$ at a point where $(1 - \\abs{t}^2) \\norm{f_k'(t)}$ is nearly maximal, which gives maps $g_k$ from discs of radii $R_k \\to \\infty$ with $\\norm{g_k'(0)} = 1$ and $\\norm{g_k'(t)} \\le R_k^2 / (R_k^2 - \\abs{t}^2)$.\nThese maps are equicontinuous, and $X$ is compact, so by the Arzelà–Ascoli theorem a subsequence converges locally uniformly to the required $g$.\nPart 2 follows: a hyperbolic manifold has no entire curve because $k_\\C \\equiv 0$, and part 1 gives the converse.\nExample: a smooth plane curve of degree $d \\ge 4$ has genus at least $3$, so it has no entire curve, and by the theorem it is hyperbolic.\nCompactness is necessary: there are noncompact complex manifolds that contain no entire curve but are not Kobayashi hyperbolic.\n\n## Explanations\n\n### L1 human\n\nTo decide whether a compact complex manifold is hyperbolic, it is enough to look for copies of the complex plane inside it.\nIf there are none, the manifold is hyperbolic.\nIf the manifold is not hyperbolic, the lemma even produces a copy of the plane whose speed is bounded everywhere by its speed at the origin.\n\n### L2 human\n\nThe lemma turns hyperbolicity, a metric property, into a statement about entire curves, which is much easier to test and to deform.\nIt implies that hyperbolicity is an open condition in families of compact complex manifolds: a limit of entire curves in nearby fibres gives an entire curve in the limit fibre.\nIt is the starting point of the strategy for the [[Kobayashi conjecture|1X3EV2TZNN]].\nThere one shows that entire curves in a generic hypersurface must satisfy algebraic differential equations, and then that no such curves exist.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "HRPAX8V7HA",
      "Y7W36GJDJD"
    ],
    "kind": "statement",
    "msc": [
      "32Q45",
      "32F45"
    ],
    "name": "Brody's lemma",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Kobayashi_metric"
      }
    ],
    "tag": "VEV5E2077C",
    "uses": [
      "JSGWAN2RJV",
      "3MQTWKBBBT",
      "DY8QQ5GTTX",
      "HRPAX8V7HA",
      "Y7W36GJDJD",
      "CK1DCPC3D3"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/VEV5E2077C/",
  "links": [
    {
      "field": "about",
      "target": "HRPAX8V7HA"
    },
    {
      "field": "about",
      "target": "Y7W36GJDJD"
    },
    {
      "field": "uses",
      "target": "JSGWAN2RJV"
    },
    {
      "field": "uses",
      "target": "3MQTWKBBBT"
    },
    {
      "field": "uses",
      "target": "DY8QQ5GTTX"
    },
    {
      "field": "uses",
      "target": "HRPAX8V7HA"
    },
    {
      "field": "uses",
      "target": "Y7W36GJDJD"
    },
    {
      "field": "uses",
      "target": "CK1DCPC3D3"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/VEV5E2077C.md",
  "msc": [
    "32Q45",
    "32F45"
  ],
  "name": "Brody's lemma",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VEV/VEV5E2077C.md",
  "status": null,
  "tag": "VEV5E2077C",
  "uri": "m2://VEV5E2077C",
  "url": "https://tamnd.github.io/math2/wiki/brodys-lemma/",
  "writing": {
    "tag": "VEV5E2077C",
    "kind": "statement",
    "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
      }
    ]
  }
}