{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a compact connected [[Kähler manifold|X60JG9AEQJ]] of complex dimension $n$, and let $K_X$ be its [[canonical bundle|W05VYY4HBR]], the bundle of holomorphic $n$-forms.\nThe manifold $X$ is a *Calabi–Yau manifold* if $K_X$ is holomorphically trivial, that is, if $X$ has a holomorphic $n$-form that vanishes nowhere.\n\n## Remarks\n\nSeveral stricter definitions are in use.\nMany authors also ask that $H^0(X, \\Omega^i_X) = 0$ for $0 \u003c i \u003c n$, or that $X$ be simply connected, or that the holonomy group of a Ricci-flat Kähler metric be exactly $SU(n)$.\nExample: a smooth quintic hypersurface $X \\subset \\mathbb{P}^4$ has $K_X = \\mathcal{O}_X(5 - 5) = \\mathcal{O}_X$ by the adjunction formula, so it is Calabi–Yau.\nExample: an elliptic curve $\\C/\\Lambda$ has the nowhere vanishing form $dz$.\nNon-example: $\\mathbb{P}^1$ has $K = \\mathcal{O}(-2)$, which has no nonzero sections, so it is not Calabi–Yau.\nA complex torus of dimension $2$ has trivial canonical bundle, but $H^0(X, \\Omega^1) \\cong \\C^2$, so it fails the stricter definition.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "STYAB4PFDW",
    "kind": "definition",
    "msc": [
      "14J32",
      "32Q25"
    ],
    "name": "Calabi–Yau manifold",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold"
      }
    ],
    "tag": "YZMPRMQ6GA",
    "uses": [
      "W05VYY4HBR"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/YZMPRMQ6GA/",
  "links": [
    {
      "field": "defines",
      "target": "STYAB4PFDW"
    },
    {
      "field": "uses",
      "target": "W05VYY4HBR"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/YZMPRMQ6GA.md",
  "msc": [
    "14J32",
    "32Q25"
  ],
  "name": "Calabi–Yau manifold",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YZM/YZMPRMQ6GA.md",
  "status": null,
  "tag": "YZMPRMQ6GA",
  "uri": "m2://YZMPRMQ6GA",
  "url": "https://tamnd.github.io/math2/wiki/calabi-yau-manifold-definition/",
  "writing": {
    "tag": "YZMPRMQ6GA",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}