{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[reduced complex spaces|831N2Q4GCB]].\nA holomorphic map $f \\colon X \\to Y$ is a *modification* if:\n\n1. $f$ is [[proper|15XEBGNGHC]] and surjective;\n2. there is a [[nowhere dense|0VQYCCQVJ5]] [[analytic subset|GB7M582M5Y]] $B \\subseteq Y$ such that $f^{-1}(B)$ is nowhere dense in $X$;\n3. $f$ restricts to a biholomorphic map $X \\setminus f^{-1}(B) \\to Y \\setminus B$.\n\n## Remarks\n\nExample: the [[blow-up|C4KGYAFPN7]] $f \\colon X \\to \\C^2$ of $\\C^2$ at the origin is a modification with $B = \\set{0}$.\nIt is proper and surjective, $f^{-1}(0)$ is a projective line, which is nowhere dense in the $2$-dimensional manifold $X$, and $f$ is biholomorphic off it.\n\nNon-example: let $X$ be the disjoint union of $\\C$ and a point $p$, and let $f \\colon X \\to \\C$ be the identity on $\\C$ and send $p$ to $0$.\nThen $f$ is proper and surjective, and with $B = \\set{0}$ clause 3 holds.\nBut $f^{-1}(B)$ contains the isolated point $p$, so it is not nowhere dense and clause 2 fails.\nNo other choice of $B$ helps, since $f$ is not injective over $0$, so $p$ must lie over $B$.\n\nThe set $f^{-1}(B)$ is the exceptional set.\nSome authors call these maps proper modifications.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "SB3MY7XQM9",
    "kind": "definition",
    "msc": [
      "32S45",
      "32H35"
    ],
    "name": "Modification of a complex space",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Proper_morphism"
      }
    ],
    "tag": "56V4WT0BT2",
    "uses": [
      "831N2Q4GCB",
      "15XEBGNGHC",
      "GB7M582M5Y",
      "0VQYCCQVJ5",
      "C4KGYAFPN7"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/56V4WT0BT2/",
  "links": [
    {
      "field": "defines",
      "target": "SB3MY7XQM9"
    },
    {
      "field": "uses",
      "target": "831N2Q4GCB"
    },
    {
      "field": "uses",
      "target": "15XEBGNGHC"
    },
    {
      "field": "uses",
      "target": "GB7M582M5Y"
    },
    {
      "field": "uses",
      "target": "0VQYCCQVJ5"
    },
    {
      "field": "uses",
      "target": "C4KGYAFPN7"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/56V4WT0BT2.md",
  "msc": [
    "32S45",
    "32H35"
  ],
  "name": "Modification of a complex space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/56V/56V4WT0BT2.md",
  "status": null,
  "tag": "56V4WT0BT2",
  "uri": "m2://56V4WT0BT2",
  "url": "https://tamnd.github.io/math2/wiki/modification-of-a-complex-space-definition/",
  "writing": {
    "tag": "56V4WT0BT2",
    "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
      }
    ]
  }
}