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  "body": "\n## Statement\n\nLet $X$ be a compact [[complex manifold|JSGWAN2RJV]], and let $\\Theta_X$ be the sheaf of holomorphic vector fields on $X$.\nAssume that $H^2(X, \\Theta_X) = 0$, and let $m = \\dim_\\C H^1(X, \\Theta_X)$.\nThen there is a [[deformation|VKQSM3BCE4]] $\\varpi \\colon \\mathcal{X} \\to B$ of $X$ with $X_0 \\cong X$, over an open neighbourhood $B$ of $0$ in $\\C^m$.\nIts [[Kodaira–Spencer map|EWZ1G10MAY]] at $0$,\n$$\\rho_0 \\colon T_0 B \\to H^1(X, \\Theta_X)$$\nis an isomorphism.\n\n## Remarks\n\nThe proof writes a deformed complex structure as a vector-valued $(0,1)$-form $\\varphi(t)$ with $\\varphi(0) = 0$.\nIntegrability of the new structure is the equation $\\bar\\partial \\varphi + \\tfrac12 [\\varphi, \\varphi] = 0$.\nOne solves it by a power series in $t$ whose linear term runs over harmonic representatives of $H^1(X, \\Theta_X)$.\nAt each order the obstruction to the next step lies in $H^2(X, \\Theta_X)$, which is zero, and estimates for elliptic operators give convergence.\nThe Newlander–Nirenberg theorem then turns the integrable almost complex structures into complex manifolds.\n\nBy the completeness theorem of Kodaira and Spencer, a family with surjective Kodaira–Spencer map at $0$ is complete there.\nSo the family of the theorem contains all small deformations of $X$.\n\nExample: for a compact Riemann surface of genus $g \\ge 2$, $H^2(X, \\Theta_X) = 0$ for dimension reasons, and $\\dim H^1(X, \\Theta_X) = 3g - 3$ by Riemann–Roch and Serre duality.\nSo $X$ has a complete family with a smooth $(3g-3)$-dimensional base.\n\nWithout the hypothesis $H^2(X, \\Theta_X) = 0$, the conclusion can fail: the base of a complete family with bijective Kodaira–Spencer map may have to be singular.\n\n## Explanations\n\n### L1 human\n\nThe space $H^1(X, \\Theta_X)$ lists all the directions in which the complex structure of $X$ could move to first order.\nThe theorem says that, if a certain obstruction space vanishes, every such direction is realized by an actual family.\nSo the small deformations of $X$ are parametrized by a ball in $H^1(X, \\Theta_X)$.\n\n### L2 human\n\nThis is the first general existence result in the deformation theory of complex manifolds, from 1958.\nKuranishi later removed the hypothesis by allowing singular bases: his family lives over the zero set of an obstruction map from $H^1$ to $H^2$.\nThe theorem applies to curves and to many surfaces, and it is the local model behind moduli spaces of these objects.\nIt does not apply to complex tori of dimension at least $2$, where $H^2(X, \\Theta_X) \\ne 0$, although their deformations are still unobstructed.\n",
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