Several complex variables and analytic spaces Definition

Biholomorphism

Builds on Holomorphic map Inverse function

Definition

Let $U$ and $V$ be open subsets of $\C^n$ . A biholomorphism from $U$ to $V$ is a holomorphic map $F \colon U \to V$ such that:

$F$ is a bijection from $U$ onto $V$ .
The inverse map $F^{-1} \colon V \to U$ is holomorphic.

The sets $U$ and $V$ are biholomorphic, or biholomorphically equivalent, if there is a biholomorphism from $U$ to $V$ .

Remarks

Example: $F(z_1, z_2) = (2 z_1, \ z_2 + z_1^2)$ 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. Non-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. In 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. This differs from the real case, where $x \mapsto x^3$ is a smooth bijection of $\R$ whose inverse is not differentiable at $0$ . The biholomorphisms of $U$ onto itself form a group, the holomorphic automorphism group $\Aut(U)$ .

Reading path

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Holomorphic map concept
Inverse function concept
Biholomorphism definition

Tag	`FDARANFQ0B`
URI	`m2://FDARANFQ0B`
Kind	definition
Writing	W0 stub
MSC	32H02
Defines	Biholomorphism
Uses	Holomorphic map, Inverse function

Biholomorphism

Definition

Remarks

Reading path

Further reading

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