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  "body": "\n## Definition\n\nLet $X$ be a compact [[complex manifold|JSGWAN2RJV]].\nA deformation of $X$ over a pointed [[complex analytic space|HXSZA44J5J]] $(S, s_0)$ consists of two pieces of data.\nThe first is a proper flat holomorphic map $\\varpi \\colon \\mathcal{X} \\to S$ whose fibres are compact complex manifolds.\nThe second is a biholomorphism $X \\cong \\varpi^{-1}(s_0)$.\nThis extends the notion of [[deformation|VKQSM3BCE4]] over manifolds.\nA *Kuranishi family* (semi-universal deformation) of $X$ is a deformation $\\varpi \\colon \\mathcal{X} \\to (S, s_0)$ of $X$ such that:\n\n1. It is complete: let $\\mathcal{Y} \\to (T, t_0)$ be any deformation of $X$.\n   Then there are a neighbourhood $T'$ of $t_0$ and a holomorphic map $f \\colon (T', t_0) \\to (S, s_0)$ such that $\\mathcal{Y}|_{T'} \\cong f^* \\mathcal{X}$.\n   The isomorphism is required to be compatible with the identifications of the central fibres with $X$.\n2. The differential $df_{t_0} \\colon T_{t_0} T \\to T_{s_0} S$ of any such $f$ is uniquely determined by the deformation $\\mathcal{Y}$.\n\n## Remarks\n\nHere $T_{s_0} S$ is the [[Zariski tangent space|E7A5DHFT01]] of $S$ at $s_0$.\nKuranishi's theorem states that every compact complex manifold has a Kuranishi family, unique up to a non-unique isomorphism of germs.\nIts [[Kodaira–Spencer map|EWZ1G10MAY]] identifies $T_{s_0} S$ with $H^1(X, \\Theta_X)$, and the germ $(S, s_0)$ is the zero set of a holomorphic map from a neighbourhood of $0$ in $H^1(X, \\Theta_X)$ to $H^2(X, \\Theta_X)$.\nIf $H^0(X, \\Theta_X) = 0$, the map $f$ in clause 1 is itself unique, and the family is universal.\n\nExample: $X = \\mathbb{P}^n$ has $H^1(X, \\Theta_X) = 0$, so its Kuranishi family lives over a single point: $\\mathbb{P}^n$ is rigid.\n\nNon-example: for an elliptic curve $E$, the trivial family $E \\times \\Delta \\to \\Delta$ over a disc is a deformation of $E$ but not complete.\nEvery pullback of it is again trivial, while the family $\\C/(\\Z + \\tau\\Z)$ near $\\tau_0$ contains fibres not biholomorphic to $E$, so it is not such a pullback.\n",
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