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  "body": "\n## Definition\n\nLet $\\varpi \\colon \\mathcal{X} \\to B$ be a complex analytic family of compact complex manifolds, as in a [[deformation of complex structure|VKQSM3BCE4]], and let $t \\in B$ with fibre $X_t$.\nFor a complex manifold $Y$ let $\\Theta_Y$ denote the sheaf of holomorphic sections of its holomorphic [[tangent bundle|WY6K3P4YCG]].\nSince $\\varpi$ is a submersion, there is a short exact sequence of sheaves on $X_t$\n$$0 \\to \\Theta_{X_t} \\to \\Theta_{\\mathcal{X}}|_{X_t} \\xrightarrow{d\\varpi} T_t B \\otimes \\mathcal{O}_{X_t} \\to 0.$$\nThe *Kodaira–Spencer map* of the family at $t$ is the connecting homomorphism of its long exact [[cohomology|PQ6Q143ARJ]] sequence,\n$$\\rho_t \\colon T_t B = H^0(X_t, T_t B \\otimes \\mathcal{O}_{X_t}) \\to H^1(X_t, \\Theta_{X_t}).$$\n\n## Remarks\n\nThe identification $H^0(X_t, T_t B \\otimes \\mathcal{O}_{X_t}) = T_t B$ uses that $X_t$ is compact and connected, so its global holomorphic functions are constants.\nIn Čech terms: cover $\\mathcal{X}$ near $X_t$ by charts $(z_i, s)$ with $\\varpi(z_i, s) = s$, glued by $z_i = f_{ij}(z_j, s)$.\nFor $v = \\partial / \\partial s$, the class $\\rho_t(v)$ is represented by the vector fields $\\sum_k \\frac{\\partial f_{ij}^k}{\\partial s}(z_j, t)\\, \\frac{\\partial}{\\partial z_i^k}$ on the overlaps.\n\nExample: for a trivial family $X \\times B \\to B$, the sequence splits, and $\\rho_t = 0$ for all $t$.\n\nExample: for the family $\\C/(\\Z + \\tau\\Z)$ over the upper half-plane, $\\Theta_{X_\\tau} \\cong \\mathcal{O}_{X_\\tau}$, so $H^1(X_\\tau, \\Theta_{X_\\tau})$ is one-dimensional, and $\\rho_\\tau$ is an isomorphism at every $\\tau$.\n\nA family with $\\rho_t$ surjective at $t$ is complete at $t$: every small deformation of $X_t$ is pulled back from it, by a theorem of Kodaira and Spencer.\n",
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