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  "body": "\n## Summary\n\nA holomorphic map from an open subset of $\\C^n$ to $\\C^m$ is a map whose $m$ components are [[holomorphic functions of several variables|507C34W0ZT]].\nHolomorphic maps are the morphisms of [[several complex variables|BH57D2FBNR]] and of complex manifolds.\nTheir derivative at each point is a complex linear map, given by the complex Jacobian matrix $(\\partial F_k / \\partial z_j)$.\nFor $n = m = 1$ they are the holomorphic functions of one variable, and [[holomorphic maps between Riemann surfaces|T19P7J53A1]] are the same notion for complex curves.\n\n## Explanations\n\n### L1 human\n\nA holomorphic function returns one complex number; a holomorphic map returns several, each depending holomorphically on the inputs.\nNear each point it looks like a complex linear map, up to a small error.\nFor example, $F(z_1, z_2) = (z_1 + z_2^2, \\ e^{z_1})$ is a holomorphic map from $\\C^2$ to $\\C^2$.\nCompositions of holomorphic maps are again holomorphic.\n",
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