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  "body": "\n## Statement\n\nLet $U \\subseteq \\C^n$ be an open set, let $F = (F_1, \\dots, F_n) \\colon U \\to \\C^n$ be a [[holomorphic map|3MQTWKBBBT]], and let $a \\in U$.\nSuppose that the complex [[Jacobian determinant|0ZCSMYHTBP]] $\\det \\left( \\frac{\\partial F_k}{\\partial z_j}(a) \\right)_{k, j}$ is not zero.\nThen there are open neighbourhoods $U_0 \\subseteq U$ of $a$ and $V_0$ of $F(a)$ such that $F$ restricts to a [[biholomorphism|FX92PP3HZR]] from $U_0$ onto $V_0$.\n\n## Remarks\n\nViewed as a map of $\\R^{2n}$, $F$ has real Jacobian determinant $\\abs{\\det (\\partial F_k / \\partial z_j)}^2$, which is not zero at $a$.\nThe real [[inverse function theorem|0K9HSJFCMJ]] gives a $C^1$ local inverse $G$.\nIts derivative at each point is the inverse of the $\\C$-linear map $DF$, hence $\\C$-linear, so $G$ is holomorphic.\nExample: $F(z_1, z_2) = (e^{z_1}, \\ z_2 e^{z_1})$ has complex Jacobian determinant $e^{2 z_1} \\ne 0$ everywhere, so it is a local biholomorphism near every point, though it is not injective on $\\C^2$, since $F(z_1 + 2\\pi i, z_2) = F(z_1, z_2)$.\nThe hypothesis is needed: $F(z) = z^2$ on $\\C$ has derivative $0$ at $0$, and it is not injective on any neighbourhood of $0$.\n\n## Explanations\n\n### L1 human\n\nIf the complex derivative of a holomorphic map is invertible at a point, the map can be undone near that point.\nMoreover, the inverse is again holomorphic.\nSo holomorphic maps with invertible derivative are local changes of complex coordinates.\n\n### L2 human\n\nThis theorem is the basic local tool for complex manifolds and holomorphic maps.\nTogether with the holomorphic implicit function theorem, it shows that the zero set of holomorphic functions with independent differentials is locally a complex submanifold.\nA stronger fact holds in the complex case: an injective holomorphic map between open subsets of $\\C^n$ automatically has nonzero Jacobian determinant, so it is a biholomorphism onto its image.\nThe real analogue fails, as $x \\mapsto x^3$ shows.\n",
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