{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "PZZNV9SEF1"
    }
  ],
  "body": "\n## Summary\n\nA holomorphic fiber bundle is a [[fiber bundle|CX4M8K61AB]] in the category of [[complex manifolds|JSGWAN2RJV]].\nThe total space, base and fibre are complex manifolds, and the local trivializations are biholomorphic.\n[[Holomorphic vector bundles|TKA958Z4KZ]] are the case where the fibre is $\\C^r$ and the transition maps are linear.\nOther examples are the Hopf map $\\C^2 \\setminus \\set{0} \\to \\mathbb{P}^1$ and the elliptic fibration of a Hopf surface over $\\mathbb{P}^1$.\n\n## Explanations\n\n### L1 human\n\nA holomorphic fiber bundle is a complex manifold that maps onto a base.\nSeen from close up over the base, it looks like a product of a piece of the base with a fixed complex manifold, the fibre.\nThe fibres over different points are all the same complex manifold, but they may be twisted together globally.\nFor example, sending a nonzero vector of $\\C^2$ to the line it spans has every fibre equal to a punctured line $\\C^*$.\nGlobally this is not a product, although locally it is.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32L05",
      "55R10"
    ],
    "name": "Holomorphic fiber bundle",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fiber_bundle"
      }
    ],
    "specializes": [
      "CX4M8K61AB"
    ],
    "tag": "GB0D4XMZV3"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/GB0D4XMZV3/",
  "links": [
    {
      "field": "specializes",
      "target": "CX4M8K61AB"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/GB0D4XMZV3.md",
  "msc": [
    "32L05",
    "55R10"
  ],
  "name": "Holomorphic fiber bundle",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GB0/GB0D4XMZV3.md",
  "status": null,
  "tag": "GB0D4XMZV3",
  "uri": "m2://GB0D4XMZV3",
  "url": "https://tamnd.github.io/math2/wiki/holomorphic-fiber-bundle/",
  "writing": {
    "tag": "GB0D4XMZV3",
    "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
      }
    ]
  }
}