{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "9TTJRYP841"
    },
    {
      "field": "uses",
      "target": "ESBGJHPEDT"
    },
    {
      "field": "uses",
      "target": "HWEED6YTNN"
    },
    {
      "field": "defines",
      "target": "B9X31R5EGG"
    }
  ],
  "body": "\n## Summary\n\nThe Chern connection is a canonical [[connection|5ZYRRVE8BW]] on a holomorphic vector bundle with a Hermitian metric.\nIt is the unique connection that preserves the metric and whose $(0,1)$-part is the $\\bar\\partial$-operator of the bundle.\nIt turns the pair (holomorphic structure, metric) into curvature, a $(1,1)$-form with values in endomorphisms.\nIts curvature is the basis of positivity notions, of vanishing theorems such as the [[Kodaira vanishing theorem|NY77DQ2AX8]], and of Chern–Weil representatives of [[Chern classes|PAD37GKDPD]].\n\n## Explanations\n\n### L1 human\n\nOn a bundle with a metric there are many ways to differentiate sections that keep lengths constant under parallel transport.\nOn a holomorphic bundle there is also a natural way to take the \"antiholomorphic part\" of a derivative.\nThe Chern connection is the only one that does both jobs at once.\nFor a line bundle with a local holomorphic section $e$ of squared length $H = \\abs{e}^2$, it is given by $\\nabla e = \\partial \\log H \\otimes e$.\nSo the metric alone, written in a holomorphic frame, determines the connection and its curvature.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "53C07",
      "32L05",
      "53C55"
    ],
    "name": "Chern connection",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hermitian_connection"
      }
    ],
    "specializes": [
      "5ZYRRVE8BW"
    ],
    "tag": "7RC9CN2Q49"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/7RC9CN2Q49/",
  "links": [
    {
      "field": "specializes",
      "target": "5ZYRRVE8BW"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/7RC9CN2Q49.md",
  "msc": [
    "53C07",
    "32L05",
    "53C55"
  ],
  "name": "Chern connection",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7RC/7RC9CN2Q49.md",
  "status": null,
  "tag": "7RC9CN2Q49",
  "uri": "m2://7RC9CN2Q49",
  "url": "https://tamnd.github.io/math2/wiki/chern-connection/",
  "writing": {
    "tag": "7RC9CN2Q49",
    "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
      }
    ]
  }
}