{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "589GJMH5EF"
    },
    {
      "field": "uses",
      "target": "D4WF2BJE2M"
    },
    {
      "field": "uses",
      "target": "H5FQQWDJRV"
    },
    {
      "field": "uses",
      "target": "PRC4WH5WYJ"
    },
    {
      "field": "uses",
      "target": "QWXTB8VVXN"
    },
    {
      "field": "uses",
      "target": "VH4FD4ZHEQ"
    },
    {
      "field": "uses",
      "target": "WV1Q8N8YW7"
    },
    {
      "field": "uses",
      "target": "Z8R2D33WSM"
    },
    {
      "field": "about",
      "target": "QWXTB8VVXN"
    },
    {
      "field": "defines",
      "target": "XGZMG3QDHZ"
    }
  ],
  "body": "\n## Summary\n\nA Kähler manifold is a [[complex manifold|JSGWAN2RJV]] with a Hermitian metric whose associated $2$-form is closed.\nIts Riemannian, complex and symplectic structures are then compatible in the strongest way.\nComplex projective space and all its complex submanifolds are Kähler, so Kähler geometry contains smooth projective algebraic geometry.\nCompact Kähler manifolds satisfy strong topological restrictions, such as the Hodge decomposition of their cohomology and the evenness of odd Betti numbers.\n\n## Explanations\n\n### L1 human\n\nOn a complex manifold one can measure lengths with a Hermitian metric, and every such metric has a $2$-form attached to it.\nA Kähler metric is one where this $2$-form has zero derivative.\nThis condition means that, to second order at each point, the metric looks like the flat metric of $\\C^n$ in suitable holomorphic coordinates.\nThe flat space $\\C^n$, every Riemann surface and complex projective space with its Fubini–Study metric are Kähler.\nMany compact complex manifolds, such as Hopf manifolds, are not.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32Q15",
      "53C55"
    ],
    "name": "Kähler manifold",
    "same_as": {
      "wikidata": "Q1353916",
      "wikipedia": "Kähler manifold"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/K%C3%A4hler_manifold"
      }
    ],
    "specializes": [
      "CK1DCPC3D3"
    ],
    "tag": "X60JG9AEQJ"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/X60JG9AEQJ/",
  "links": [
    {
      "field": "specializes",
      "target": "CK1DCPC3D3"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/X60JG9AEQJ.md",
  "msc": [
    "32Q15",
    "53C55"
  ],
  "name": "Kähler manifold",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/X60/X60JG9AEQJ.md",
  "status": null,
  "tag": "X60JG9AEQJ",
  "uri": "m2://X60JG9AEQJ",
  "url": "https://tamnd.github.io/math2/wiki/kahler-manifold/",
  "writing": {
    "tag": "X60JG9AEQJ",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}