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  "body": "\n## Statement\n\nLet $X$ be a complex [[projective variety|A4H4CZ2V35]] of general type, that is, one whose [[Kodaira dimension|ZW00Q0SAGR]] equals its dimension.\nThen there is a proper algebraic subvariety $Y \\subsetneq X$ such that every nonconstant holomorphic map $f \\colon \\C \\to X$ has image contained in $Y$.\n\n## Remarks\n\nGreen and Griffiths posed the problem for entire curves in varieties of general type.\nSerge Lang stated it as part of a group of conjectures that link hyperbolicity with the arithmetic of varieties.\nThe conjecture is open in general.\nFor curves it holds: a smooth projective curve of general type has genus at least $2$, so its universal covering is the unit disc.\nHence every holomorphic map from $\\C$ to it is constant.\nMichael McQuillan proved it for surfaces of general type with $c_1^2 \u003e c_2$.\nIt is also known for very general hypersurfaces of high degree in projective space.\nA related conjecture of Lang says that a projective variety is [[Kobayashi hyperbolic|HRPAX8V7HA]] if and only if all its subvarieties are of general type.\n\n## Explanations\n\n### L1 human\n\nA variety of general type is, in a precise sense, a variety with a lot of negative curvature.\nThe conjecture predicts that holomorphic copies of the complex plane in such a variety cannot spread everywhere: they all lie inside one smaller algebraic subvariety.\nIn other words, such a variety is [[Brody hyperbolic|Y7W36GJDJD]] outside a small exceptional set.\n",
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      "first_stated": "1980",
      "posed_by": "Mark Green and Phillip Griffiths"
    },
    "kind": "conjecture",
    "msc": [
      "32Q45",
      "14E99"
    ],
    "name": "Green–Griffiths–Lang conjecture",
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        "url": "https://en.wikipedia.org/wiki/Kobayashi_metric"
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  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/AD9GD9EM9R/",
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  "markdown": "https://tamnd.github.io/math2/node/AD9GD9EM9R.md",
  "msc": [
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    "14E99"
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  "name": "Green–Griffiths–Lang conjecture",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/AD9/AD9GD9EM9R.md",
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