{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]], let $E \\to X$ be a [[holomorphic vector bundle|TKA958Z4KZ]], and let $h$ be a [[Hermitian metric|Q6X6W2N29G]] on $E$.\nLet $\\bar\\partial_E$ be the [[$\\bar\\partial$-operator|M1VKCW3RFW]] of $E$, which sends a smooth section to an $E$-valued $(0,1)$-form and kills exactly the holomorphic sections.\nThe *Chern connection* of $(E, h)$ is the [[connection|5ZYRRVE8BW]] $\\nabla$ on $E$ such that:\n\n1. $\\nabla$ is compatible with $h$: $d\\, h(s, t) = h(\\nabla s, t) + h(s, \\nabla t)$ for all smooth sections $s, t$.\n2. The $(0,1)$-part of $\\nabla$ is $\\bar\\partial_E$: $\\nabla^{0,1} = \\bar\\partial_E$.\n\nSuch a connection exists and is unique.\n\n## Remarks\n\nWrite $\\nabla = \\nabla^{1,0} + \\nabla^{0,1}$ for the splitting by type.\nIn a local holomorphic frame $e_1, \\dots, e_r$, let $H$ be the matrix with entries $H_{ij} = h(e_j, e_i)$.\nThen the connection matrix is $\\theta = H^{-1} \\partial H$ in a suitable index convention, and the curvature is $\\Theta = \\bar\\partial \\theta$, a $(1,1)$-form with values in $\\operatorname{End} E$.\n\nExample: let $L$ be a holomorphic line bundle with local holomorphic frame $e$, and let $H = h(e, e) \u003e 0$.\nPut $\\nabla e = \\partial \\log H \\otimes e$.\nClause 2 holds because $\\bar\\partial_E e = 0$ and $\\partial \\log H$ has type $(1,0)$.\nClause 1 holds because $h(\\nabla e, e) + h(e, \\nabla e) = (\\partial \\log H + \\bar\\partial \\log H) H = dH$.\nThe curvature is $\\Theta = d \\partial \\log H = -\\partial \\bar\\partial \\log H$.\n\nNon-example: on the trivial line bundle over $\\C$ with the standard metric, let $\\nabla s = ds + i s\\, dx$, where $x = \\operatorname{Re} z$.\nSince $i\\, dx$ is purely imaginary, clause 1 holds.\nBut $i\\, dx = \\tfrac{i}{2}(dz + d\\bar z)$, so $\\nabla^{0,1} = \\bar\\partial + \\tfrac{i}{2} d\\bar z \\ne \\bar\\partial$, and clause 2 fails.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "7RC9CN2Q49",
    "kind": "definition",
    "msc": [
      "53C07",
      "32L05",
      "53C55"
    ],
    "name": "Chern connection",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hermitian_connection"
      }
    ],
    "tag": "B9X31R5EGG",
    "uses": [
      "TKA958Z4KZ",
      "Q6X6W2N29G",
      "5ZYRRVE8BW",
      "M1VKCW3RFW",
      "JSGWAN2RJV"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/B9X31R5EGG/",
  "links": [
    {
      "field": "defines",
      "target": "7RC9CN2Q49"
    },
    {
      "field": "uses",
      "target": "TKA958Z4KZ"
    },
    {
      "field": "uses",
      "target": "Q6X6W2N29G"
    },
    {
      "field": "uses",
      "target": "5ZYRRVE8BW"
    },
    {
      "field": "uses",
      "target": "M1VKCW3RFW"
    },
    {
      "field": "uses",
      "target": "JSGWAN2RJV"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/B9X31R5EGG.md",
  "msc": [
    "53C07",
    "32L05",
    "53C55"
  ],
  "name": "Chern connection",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/B9X/B9X31R5EGG.md",
  "status": null,
  "tag": "B9X31R5EGG",
  "uri": "m2://B9X31R5EGG",
  "url": "https://tamnd.github.io/math2/wiki/chern-connection-definition/",
  "writing": {
    "tag": "B9X31R5EGG",
    "kind": "definition",
    "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
      }
    ]
  }
}