{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "34PT0GE8RA"
    },
    {
      "field": "uses",
      "target": "6EFEG5VJEF"
    },
    {
      "field": "uses",
      "target": "7ZQ9FWZMP1"
    },
    {
      "field": "uses",
      "target": "A2WHCVP6JN"
    },
    {
      "field": "uses",
      "target": "C2R5Q2Q717"
    },
    {
      "field": "uses",
      "target": "JPP8N6MZGP"
    },
    {
      "field": "uses",
      "target": "K18PKZY5RS"
    }
  ],
  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]], for example an open subset of $\\C^n$.\nA subset $A \\subseteq X$ is a *complex analytic variety* in $X$, or an analytic subset of $X$, if:\n\n1. $A$ is [[closed|0ZCK8DQYN1]] in $X$, and\n2. every point $a \\in X$ has an open neighbourhood $U$ and finitely many [[holomorphic functions|507C34W0ZT]] $f_1, \\dots, f_k$ on $U$ with\n$$A \\cap U = \\set{z \\in U : f_1(z) = \\cdots = f_k(z) = 0}.$$\n\n## Remarks\n\nFor points $a \\notin A$ the second condition holds with $U = X \\setminus A$ and $f_1 = 1$, so it only matters at points of $A$.\nExample: in $\\C^2$, $A = \\set{z_1 z_2 = 0}$ is an analytic variety, with $U = \\C^2$ and $f_1 = z_1 z_2$; it is the union of the two coordinate axes.\nNon-example: the closed unit disc $\\set{\\abs{z} \\le 1}$ in $\\C$ is closed but not analytic.\nLet $a$ be a point with $\\abs{a} = 1$ and let $U$ be a connected neighbourhood of $a$.\nA holomorphic function on $U$ that vanishes on the part of the disc in $U$ vanishes on an open set, hence on all of $U$.\nSo the common zero set would be $U$, not the part of the disc in $U$.\nA set that satisfies condition 2 only at its own points is called locally analytic.\nIt need not be closed, for example the punctured line $\\set{z_1 = 0} \\setminus \\set{0}$ in $\\C^2$.\nSome authors reserve \"variety\" for irreducible analytic sets; here no irreducibility is required.\nThe empty set and $X$ itself are analytic varieties, and finite unions and arbitrary intersections of analytic varieties are analytic varieties.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "GB7M582M5Y",
    "kind": "definition",
    "msc": [
      "32B15",
      "32C25",
      "32C15"
    ],
    "name": "Complex analytic variety",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Complex_analytic_variety"
      }
    ],
    "tag": "VMDTGTMN9P",
    "uses": [
      "JSGWAN2RJV",
      "507C34W0ZT",
      "0ZCK8DQYN1"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/VMDTGTMN9P/",
  "links": [
    {
      "field": "defines",
      "target": "GB7M582M5Y"
    },
    {
      "field": "uses",
      "target": "JSGWAN2RJV"
    },
    {
      "field": "uses",
      "target": "507C34W0ZT"
    },
    {
      "field": "uses",
      "target": "0ZCK8DQYN1"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/VMDTGTMN9P.md",
  "msc": [
    "32B15",
    "32C25",
    "32C15"
  ],
  "name": "Complex analytic variety",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VMD/VMDTGTMN9P.md",
  "status": null,
  "tag": "VMDTGTMN9P",
  "uri": "m2://VMDTGTMN9P",
  "url": "https://tamnd.github.io/math2/wiki/complex-analytic-variety-definition/",
  "writing": {
    "tag": "VMDTGTMN9P",
    "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
      }
    ]
  }
}