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  "body": "\n## Summary\n\nThe Kodaira–Spencer map of a [[deformation|VKQSM3BCE4]] measures how fast the complex structure of the fibre changes.\nIt sends a tangent vector of the base at $t$ to a class in $H^1(X_t, \\Theta_{X_t})$.\nHere $\\Theta_{X_t}$ is the sheaf of holomorphic vector fields on the fibre $X_t$.\nIt vanishes identically for a trivial family, and it is the basic tool to decide whether a family is complete or effectively parametrized.\nIt is the first-order part of the deformation theory of compact complex manifolds.\n\n## Explanations\n\n### L1 human\n\nA family of complex manifolds is glued, locally over the base, from pieces of a product.\nMoving in the base changes the gluing maps, and the derivative of this change is a holomorphic vector field on each overlap.\nThese vector fields form a Čech cocycle, and its class in $H^1(X_t, \\Theta_{X_t})$ does not depend on the choices.\nIf the class is zero, the complex structure does not change to first order in that direction.\nFor the family of elliptic curves $\\C/(\\Z + \\tau\\Z)$, the map is nonzero at every $\\tau$.\n",
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