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  "body": "\n## Summary\n\nA holomorphic automorphism of $\\C^n$ is a [[biholomorphism|FX92PP3HZR]] of $\\C^n$ onto itself.\nUnder composition these maps form the group $\\Aut(\\C^n)$, an [[automorphism group|0K1SCPBNKJ]].\nFor $n = 1$ this group consists only of the affine maps $z \\mapsto az + b$ with $a \\ne 0$.\nFor $n \\ge 2$ it is very large, and its study is a central topic of complex analysis in several variables.\n\n## Explanations\n\n### L1 human\n\nAn automorphism of $\\C^n$ is a holomorphic way to rearrange all of complex space, with a holomorphic way back.\nIn one variable only stretching, rotating and shifting are possible.\nIn two variables one can also shear: the map $(z, w) \\mapsto (z, w + g(z))$ is an automorphism for every entire function $g$, with inverse $(z, w) \\mapsto (z, w - g(z))$.\nComposing shears with linear maps already gives a huge, infinite-dimensional group.\n",
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