{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $Y$ be a [[complex manifold|JSGWAN2RJV]] with a distance function $d$ that induces its topology.\nThe manifold $Y$ has the *convex approximation property* if the following holds for every $n \\ge 1$.\nLet $K \\subseteq \\C^n$ be a [[compact|0R6EWFKRCF]] convex set, let $U$ be an open neighbourhood of $K$, and let $f \\colon U \\to Y$ be a [[holomorphic map|3MQTWKBBBT]].\nThen for every $\\varepsilon \u003e 0$ there is a holomorphic map $F \\colon \\C^n \\to Y$ with $d(F(z), f(z)) \u003c \\varepsilon$ for all $z \\in K$.\n\nThe manifold $Y$ is an *Oka manifold* if it has the convex approximation property.\n\n## Remarks\n\nThe property does not depend on the choice of $d$, since any two such distances are uniformly equivalent on a compact neighbourhood of the compact set $f(K)$.\nExample: $Y = \\C^m$ is Oka.\nA compact convex set is polynomially convex.\nSo by the [[Oka–Weil theorem|7AWZJNQGT2]] each component of $f$ is a uniform limit on $K$ of polynomials, and polynomial maps $F$ approximate $f$ on $K$.\nNon-example: the unit disc $D \\subseteq \\C$ is not Oka.\nTake $n = 1$, $K = \\set{\\abs{z} \\le 1/2}$ and $f(z) = z$; any holomorphic $F \\colon \\C \\to D$ is bounded, hence constant by Liouville's theorem, and a constant cannot be closer than $1/2$ to $z$ at both $z = 1/2$ and $z = -1/2$.\nForstnerič proved that the convex approximation property is equivalent to the full Oka principle for maps from Stein manifolds to $Y$, with approximation and interpolation; this justifies the definition.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "NXEAT9DMWD",
    "kind": "definition",
    "msc": [
      "32Q56"
    ],
    "name": "Oka manifold",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Stein_manifold"
      }
    ],
    "tag": "M0SSFESZQY",
    "uses": [
      "JSGWAN2RJV",
      "3MQTWKBBBT",
      "0R6EWFKRCF",
      "7AWZJNQGT2"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/M0SSFESZQY/",
  "links": [
    {
      "field": "defines",
      "target": "NXEAT9DMWD"
    },
    {
      "field": "uses",
      "target": "JSGWAN2RJV"
    },
    {
      "field": "uses",
      "target": "3MQTWKBBBT"
    },
    {
      "field": "uses",
      "target": "0R6EWFKRCF"
    },
    {
      "field": "uses",
      "target": "7AWZJNQGT2"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/M0SSFESZQY.md",
  "msc": [
    "32Q56"
  ],
  "name": "Oka manifold",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/M0S/M0SSFESZQY.md",
  "status": null,
  "tag": "M0SSFESZQY",
  "uri": "m2://M0SSFESZQY",
  "url": "https://tamnd.github.io/math2/wiki/oka-manifold-definition/",
  "writing": {
    "tag": "M0SSFESZQY",
    "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
      }
    ]
  }
}