---
tag: 2J7JB3H3FD
kind: concept
name: Holomorphic automorphisms of C^n
msc: [32M17, 32H02]
sources:
  - url: https://en.wikipedia.org/wiki/Function_of_several_complex_variables
---

## Summary

A holomorphic automorphism of $\C^n$ is a [[biholomorphism|FX92PP3HZR]] of $\C^n$ onto itself.
Under composition these maps form the group $\Aut(\C^n)$, an [[automorphism group|0K1SCPBNKJ]].
For $n = 1$ this group consists only of the affine maps $z \mapsto az + b$ with $a \ne 0$.
For $n \ge 2$ it is very large, and its study is a central topic of complex analysis in several variables.

## Explanations

### L1 human

An automorphism of $\C^n$ is a holomorphic way to rearrange all of complex space, with a holomorphic way back.
In one variable only stretching, rotating and shifting are possible.
In 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 $(z, w) \mapsto (z, w - g(z))$.
Composing shears with linear maps already gives a huge, infinite-dimensional group.