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  "body": "\n## Summary\n\nThe Kuranishi family of a compact [[complex manifold|JSGWAN2RJV]] $X$ is a semi-universal [[deformation|VKQSM3BCE4]] of $X$.\nEvery small deformation of $X$ is pulled back from it, and the pullback map is unique to first order.\nKuranishi constructed it in 1962 over a germ of [[complex analytic space|HXSZA44J5J]], the Kuranishi space.\nThat space is the zero set of a holomorphic obstruction map from a neighbourhood of $0$ in $H^1(X, \\Theta_X)$ to $H^2(X, \\Theta_X)$.\nIt is the local model for moduli of compact complex manifolds.\n\n## Explanations\n\n### L1 human\n\nEvery compact complex manifold has one \"best\" small family of deformations that contains all others.\nIt is best in the sense that any other small deformation of $X$ is obtained from it by a change of parameter.\nThe parameter space can be singular, because some first-order directions may fail to extend to actual deformations.\nFor an elliptic curve the Kuranishi family is a small disc of the family $\\C / (\\Z + \\tau \\Z)$; for projective space it is a single point, because projective space is rigid.\n",
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