{
  "aliases": null,
  "backlinks": [
    {
      "field": "specializes",
      "target": "831N2Q4GCB"
    },
    {
      "field": "specializes",
      "target": "Z4AMJS5KA6"
    },
    {
      "field": "uses",
      "target": "AX46SNQCAA"
    },
    {
      "field": "uses",
      "target": "CYAWQF7AJZ"
    },
    {
      "field": "uses",
      "target": "JRTB9SHAFD"
    },
    {
      "field": "uses",
      "target": "K91VB1DXPR"
    },
    {
      "field": "uses",
      "target": "MPNGD4V271"
    },
    {
      "field": "uses",
      "target": "MREZVW7JTW"
    },
    {
      "field": "uses",
      "target": "MSX3VKM9Z5"
    },
    {
      "field": "uses",
      "target": "VA2JEWWB61"
    },
    {
      "field": "uses",
      "target": "W4TT4W95GR"
    },
    {
      "field": "uses",
      "target": "WKD2S2XA9C"
    },
    {
      "field": "uses",
      "target": "ZNCF16AENG"
    },
    {
      "field": "defines",
      "target": "R8GZAJ85YF"
    }
  ],
  "body": "\n## Summary\n\nA complex analytic space, or complex space, is a [[locally ringed space|B7FPT8JTXR]] that is locally modelled on an analytic set.\nThe model is the common zero set of finitely many [[holomorphic functions|507C34W0ZT]] on an open subset of $\\C^n$.\nIts structure sheaf is the quotient of the sheaf of holomorphic functions by the ideal that these functions generate.\nComplex manifolds and [[complex analytic varieties|GB7M582M5Y]] are complex spaces.\nThe structure sheaf may have nilpotent elements, which record multiplicities, as for the double point $z^2 = 0$.\nComplex spaces are the analytic counterpart of schemes and the setting of the global theory of several complex variables.\n\n## Explanations\n\n### L1 human\n\nA complex manifold looks locally like an open subset of $\\C^n$.\nA complex space may look locally like the solution set of holomorphic equations, which may have singular points.\nAn example is the curve $z_2^2 = z_1^3$ with its cusp at the origin.\nThe space remembers not only the points but also the equations, through the functions that live on it.\nSo the equation $z^2 = 0$ in $\\C$ gives a single point that remembers it was counted twice.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32C15"
    ],
    "name": "Complex analytic space",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Complex_analytic_variety"
      }
    ],
    "specializes": [
      "B7FPT8JTXR"
    ],
    "tag": "HXSZA44J5J"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/HXSZA44J5J/",
  "links": [
    {
      "field": "specializes",
      "target": "B7FPT8JTXR"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/HXSZA44J5J.md",
  "msc": [
    "32C15"
  ],
  "name": "Complex analytic space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HXS/HXSZA44J5J.md",
  "status": null,
  "tag": "HXSZA44J5J",
  "uri": "m2://HXSZA44J5J",
  "url": "https://tamnd.github.io/math2/wiki/complex-analytic-space/",
  "writing": {
    "tag": "HXSZA44J5J",
    "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
      }
    ]
  }
}