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  "body": "\n## Definition\n\nLet $U$ and $V$ be open subsets of $\\C^n$.\nA *biholomorphism* from $U$ to $V$ is a [[holomorphic map|3MQTWKBBBT]] $F \\colon U \\to V$ such that:\n\n1. $F$ is a bijection from $U$ onto $V$.\n2. The [[inverse map|0J2WAR83WP]] $F^{-1} \\colon V \\to U$ is holomorphic.\n\nThe sets $U$ and $V$ are *biholomorphic*, or biholomorphically equivalent, if there is a biholomorphism from $U$ to $V$.\n\n## Remarks\n\nExample: $F(z_1, z_2) = (2 z_1, \\ z_2 + z_1^2)$ is a biholomorphism of $\\C^2$, since it is polynomial, bijective, and its inverse $(w_1, w_2) \\mapsto (w_1/2, \\ w_2 - w_1^2/4)$ is polynomial.\nNon-example: $F(z) = z^3$ from $\\C$ to $\\C$ is holomorphic and onto, but not injective, since $F(1) = F(e^{2\\pi i/3})$; so clause 1 fails.\nIn fact clause 2 follows from clause 1: an injective holomorphic map from an open set $U \\subseteq \\C^n$ to $\\C^n$ has open image and a holomorphic inverse.\nThis differs from the real case, where $x \\mapsto x^3$ is a smooth bijection of $\\R$ whose inverse is not differentiable at $0$.\nThe biholomorphisms of $U$ onto itself form a group, the holomorphic automorphism group $\\Aut(U)$.\n",
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