{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $X$ be a [[Stein manifold|HMCD0JBFST]].\n\n1. Two [[holomorphic vector bundles|TKA958Z4KZ]] on $X$ are holomorphically isomorphic if and only if they are isomorphic as topological complex [[vector bundles|P4DZM975BV]].\n2. Every topological complex vector bundle on $X$ is topologically isomorphic to a holomorphic vector bundle.\n\n## Remarks\n\nGrauert proved the theorem in 1958, for holomorphic principal bundles over Stein spaces whose structure group is a complex Lie group.\nVector bundles of rank $r$ are the case of the group $GL_r(\\C)$.\nFor line bundles it is the [[Oka principle|S1CJV1PA3D]] of 1939, which follows from the exponential sequence.\nFor higher rank the group $GL_r(\\C)$ is not commutative, and the proof needs a new, nonlinear method.\nGrauert deformed continuous transition functions to holomorphic ones by successive approximation over an exhaustion of $X$, using a Runge-type approximation for maps into complex Lie groups.\nExample: on $\\C^n$, which is contractible, every topological complex vector bundle is trivial, so every holomorphic vector bundle on $\\C^n$ is holomorphically trivial.\nThe Stein hypothesis is needed.\nOn the projective line, the rank $2$ bundle $\\mathcal{O}(1) \\oplus \\mathcal{O}(-1)$ and the trivial bundle of rank $2$ are topologically isomorphic, since both have first Chern class $0$.\nThey are not holomorphically isomorphic, since the first has a nonzero holomorphic section that vanishes at a point.\n\n## Explanations\n\n### L1 human\n\nA vector bundle attaches a vector space to each point of a space.\nHolomorphic vector bundles carry much more structure than continuous ones, and in general there are many more of them.\nGrauert's theorem says that on a Stein manifold this extra structure brings nothing new: the holomorphic classification is the same as the topological one.\n\n### L2 human\n\nThe theorem reduces the classification of holomorphic vector bundles on a Stein manifold to homotopy theory.\nBundles of rank $r$ correspond to homotopy classes of maps from $X$ to the classifying space of $GL_r(\\C)$.\nIt is the starting point of modern Oka theory.\nThis theory asks for which target manifolds $Y$ the same principle holds for holomorphic maps from Stein manifolds to $Y$; these targets are the [[Oka manifolds|NXEAT9DMWD]].\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "HMCD0JBFST",
      "TKA958Z4KZ"
    ],
    "kind": "statement",
    "msc": [
      "32Q56",
      "32L05"
    ],
    "name": "Grauert's Oka principle for vector bundles",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Stein_manifold"
      }
    ],
    "tag": "2CVH8BCHTF",
    "uses": [
      "HMCD0JBFST",
      "TKA958Z4KZ",
      "P4DZM975BV",
      "S1CJV1PA3D",
      "NXEAT9DMWD"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/2CVH8BCHTF/",
  "links": [
    {
      "field": "about",
      "target": "HMCD0JBFST"
    },
    {
      "field": "about",
      "target": "TKA958Z4KZ"
    },
    {
      "field": "uses",
      "target": "HMCD0JBFST"
    },
    {
      "field": "uses",
      "target": "TKA958Z4KZ"
    },
    {
      "field": "uses",
      "target": "P4DZM975BV"
    },
    {
      "field": "uses",
      "target": "S1CJV1PA3D"
    },
    {
      "field": "uses",
      "target": "NXEAT9DMWD"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/2CVH8BCHTF.md",
  "msc": [
    "32Q56",
    "32L05"
  ],
  "name": "Grauert's Oka principle for vector bundles",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2CV/2CVH8BCHTF.md",
  "status": null,
  "tag": "2CVH8BCHTF",
  "uri": "m2://2CVH8BCHTF",
  "url": "https://tamnd.github.io/math2/wiki/grauerts-oka-principle-for-vector-bundles/",
  "writing": {
    "tag": "2CVH8BCHTF",
    "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
      }
    ]
  }
}