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  "body": "\n## Definition\n\nLet $n \\ge 1$.\nA *holomorphic automorphism* of $\\C^n$ is a bijective [[holomorphic map|3MQTWKBBBT]] $F \\colon \\C^n \\to \\C^n$ whose inverse $F^{-1}$ is also holomorphic.\nIn other words, it is a [[biholomorphism|FX92PP3HZR]] of $\\C^n$ onto itself.\nThe set of these maps, with composition as the operation, is the group $\\Aut(\\C^n)$.\n\n## Remarks\n\nExample: for an entire function $g$ on $\\C$, the shear $F(z, w) = (z, w + g(z))$ is an automorphism of $\\C^2$.\nIt is holomorphic, and $G(z, w) = (z, w - g(z))$ is holomorphic with $G \\circ F = F \\circ G = \\id$.\nInvertible affine maps $z \\mapsto Az + b$, with $A$ an invertible $n \\times n$ complex matrix, are automorphisms as well.\n\nNon-example: $F(z, w) = (z, zw)$ is holomorphic on $\\C^2$ but not injective, since $F(0, w) = (0, 0)$ for all $w$.\n\nFor a holomorphic map, bijectivity already implies that the inverse is holomorphic, so the last condition is automatic.\n",
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