{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "3CEKYF05P1"
    },
    {
      "field": "uses",
      "target": "589GJMH5EF"
    },
    {
      "field": "uses",
      "target": "AX46SNQCAA"
    },
    {
      "field": "uses",
      "target": "TDY6EY60DV"
    },
    {
      "field": "uses",
      "target": "VH4FD4ZHEQ"
    },
    {
      "field": "uses",
      "target": "VTN8JHGDS5"
    },
    {
      "field": "about",
      "target": "VTN8JHGDS5"
    }
  ],
  "body": "\n## Summary\n\nA deformation of the complex structure of a compact [[complex manifold|JSGWAN2RJV]] $X$ is a holomorphic family of compact complex manifolds.\nThe family is parametrized by a base, and its fibre over a chosen base point is $X$.\nBy Ehresmann's theorem all fibres are diffeomorphic, so the family varies only the complex structure on a fixed smooth manifold.\nKodaira and Spencer showed that first-order deformations are measured by $H^1(X, \\Theta_X)$, where $\\Theta_X$ is the sheaf of holomorphic vector fields, and obstructions lie in $H^2(X, \\Theta_X)$.\nThe theory is the analytic part of [[deformation theory|SWFR1GD6QT]] and leads to local [[moduli spaces|FJZRW9P0K3]].\n\n## Explanations\n\n### L1 human\n\nA complex manifold is a smooth manifold together with a complex structure, and one can try to change that structure a little.\nA deformation is a family of complex manifolds $X_t$ depending holomorphically on a parameter $t$, with $X_0 = X$.\nElliptic curves give the basic picture: the torus $\\C / (\\Z + \\tau \\Z)$ changes its complex structure as $\\tau$ moves in the upper half-plane, while staying the same smooth torus.\nSome manifolds, such as projective space, cannot be deformed at all; they are called rigid.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32G05",
      "14D15"
    ],
    "name": "Deformation of complex structure",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Kodaira%E2%80%93Spencer_map"
      }
    ],
    "specializes": [
      "SWFR1GD6QT"
    ],
    "tag": "VKQSM3BCE4"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/VKQSM3BCE4/",
  "links": [
    {
      "field": "specializes",
      "target": "SWFR1GD6QT"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/VKQSM3BCE4.md",
  "msc": [
    "32G05",
    "14D15"
  ],
  "name": "Deformation of complex structure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VKQ/VKQSM3BCE4.md",
  "status": null,
  "tag": "VKQSM3BCE4",
  "uri": "m2://VKQSM3BCE4",
  "url": "https://tamnd.github.io/math2/wiki/deformation-of-complex-structure/",
  "writing": {
    "tag": "VKQSM3BCE4",
    "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
      }
    ]
  }
}