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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $X \\subset \\mathbb{P}^{n+1}$ be a very general hypersurface of degree $d$ in complex [[projective space|ZT9A6JK8QF]].\nVery general means that the defining polynomial lies outside a countable union of proper algebraic subsets of the space of polynomials of degree $d$.\n\n1. If $d \\ge 2n + 1$, then $X$ is [[Kobayashi hyperbolic|HRPAX8V7HA]].\n2. If $d \\ge 2n + 3$, then the complement $\\mathbb{P}^{n+1} \\setminus X$ is Kobayashi hyperbolic.\n\n## Remarks\n\nKobayashi stated these problems in his 1970 book on hyperbolic manifolds.\nThe hypothesis $n \\ge 2$ is needed in part 1 in this form.\nFor $n = 1$ the bound would be $d \\ge 3$, but a smooth plane cubic is an elliptic curve, a complex torus, which is not hyperbolic.\nFor $n = 1$ the true statement is that smooth plane curves of degree at least $4$ are hyperbolic, since they have genus at least $3$.\nPart 1 is known for all $d$ larger than a bound that depends on $n$; Yum-Tong Siu and Damian Brotbek gave proofs.\nLater work lowered the bound, which is still far above $2n + 1$ for large $n$.\nFor $d \\le 2n - 1$, every hypersurface of degree $d$ in $\\mathbb{P}^{n+1}$ contains a line, so it is not hyperbolic.\nSince $X$ is compact, part 1 asks equivalently that $X$ contains no entire curve, that is, that $X$ is [[Brody hyperbolic|Y7W36GJDJD]].\n\n## Explanations\n\n### L1 human\n\nA complex manifold is hyperbolic when it contains no copy of the complex plane, in a strong sense.\nThe conjecture predicts that a typical hypersurface of large enough degree in projective space is hyperbolic, and so is the complement of such a hypersurface.\nThe degree needs to be large because hypersurfaces of small degree contain lines, which carry many copies of the plane.\n",
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    "history": {
      "first_stated": "1970",
      "posed_by": "Shoshichi Kobayashi"
    },
    "kind": "conjecture",
    "msc": [
      "32Q45",
      "14J70"
    ],
    "name": "Kobayashi conjecture",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Kobayashi_metric"
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    ],
    "tag": "1X3EV2TZNN",
    "uses": [
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  "markdown": "https://tamnd.github.io/math2/node/1X3EV2TZNN.md",
  "msc": [
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  "name": "Kobayashi conjecture",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/1X3/1X3EV2TZNN.md",
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  "tag": "1X3EV2TZNN",
  "uri": "m2://1X3EV2TZNN",
  "url": "https://tamnd.github.io/math2/wiki/kobayashi-conjecture/",
  "writing": {
    "tag": "1X3EV2TZNN",
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    "level": "W0",
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