Several complex variables and analytic spaces Definition

Holomorphic automorphisms of C^n

Builds on Biholomorphism Holomorphic map Automorphism group

Definition

Let $n \ge 1$ . A holomorphic automorphism of $\C^n$ is a bijective holomorphic map $F \colon \C^n \to \C^n$ whose inverse $F^{-1}$ is also holomorphic. In other words, it is a biholomorphism of $\C^n$ onto itself. The set of these maps, with composition as the operation, is the group $\Aut(\C^n)$ .

Remarks

Example: for an entire function $g$ on $\C$ , the shear $F(z, w) = (z, w + g(z))$ is an automorphism of $\C^2$ . It is holomorphic, and $G(z, w) = (z, w - g(z))$ is holomorphic with $G \circ F = F \circ G = \id$ . Invertible affine maps $z \mapsto Az + b$ , with $A$ an invertible $n \times n$ complex matrix, are automorphisms as well.

Non-example: $F(z, w) = (z, zw)$ is holomorphic on $\C^2$ but not injective, since $F(0, w) = (0, 0)$ for all $w$ .

For a holomorphic map, bijectivity already implies that the inverse is holomorphic, so the last condition is automatic.

Reading path

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Automorphism group concept
Biholomorphism concept
Holomorphic map concept
Holomorphic automorphisms of C^n definition

Tag	`6QW3MQC3SK`
URI	`m2://6QW3MQC3SK`
Kind	definition
Writing	W0 stub
MSC	32M17, 32H02
Defines	Holomorphic automorphisms of C^n
Uses	Biholomorphism, Holomorphic map, Automorphism group

Holomorphic automorphisms of C^n

Definition

Remarks

Reading path

Further reading

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