{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "1XCBCB3MD7"
    },
    {
      "field": "uses",
      "target": "1XCBCB3MD7"
    },
    {
      "field": "defines",
      "target": "ZRGTKMHKX2"
    }
  ],
  "body": "\n## Summary\n\nAn open set $D \\subseteq \\C^n$ is holomorphically convex if the [[holomorphically convex hull|CD6SMMZSW8]] of every compact subset of $D$ is again compact.\nThis is the holomorphic analogue of convexity, where hulls are taken with respect to holomorphic functions on $D$ instead of linear functions.\nFor domains in $\\C^n$, holomorphic convexity is equivalent to being a domain of holomorphy.\nOn complex manifolds it is one of the conditions defining Stein manifolds.\n\n## Explanations\n\n### L1 human\n\nTake any compact piece $K$ of the domain and form its hull: all points where holomorphic functions cannot beat their maximum over $K$.\nIn a holomorphically convex domain these hulls never creep out to the boundary.\nEvery convex open set in $\\C^n$ has this property.\nThe punctured ball $\\set{0 \u003c \\abs{z} \u003c 1}$ in $\\C^2$ does not: the hull of a small sphere around the origin contains points arbitrarily close to the missing origin.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32E05",
      "32T05"
    ],
    "name": "Holomorphically convex domain",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Function_of_several_complex_variables"
      }
    ],
    "tag": "D5J6CP5V2Z"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/D5J6CP5V2Z/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/D5J6CP5V2Z.md",
  "msc": [
    "32E05",
    "32T05"
  ],
  "name": "Holomorphically convex domain",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/D5J/D5J6CP5V2Z.md",
  "status": null,
  "tag": "D5J6CP5V2Z",
  "uri": "m2://D5J6CP5V2Z",
  "url": "https://tamnd.github.io/math2/wiki/holomorphically-convex-domain/",
  "writing": {
    "tag": "D5J6CP5V2Z",
    "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
      }
    ]
  }
}