Biholomorphism - Imported Knowlpedia Content

Definition

Let $X$ and $Y$ be complex manifolds. A biholomorphism from $X$ to $Y$ is a bijective holomorphic map

f:X\to Y

whose set-theoretic inverse $f^{-1}:Y\to X$ is also holomorphic. Two complex manifolds are biholomorphic if a biholomorphism exists between them. Biholomorphisms are precisely the isomorphisms in the category of complex manifolds and holomorphic maps; they preserve complex dimension, holomorphic functions, and the holomorphic-coordinate structure.

Local criterion

A holomorphic map is locally biholomorphic near a point exactly when its complex differential there is invertible, by the holomorphic inverse-function theorem. A globally bijective local biholomorphism is a biholomorphism. Bijectivity alone should not be substituted for the local differential condition without a theorem guaranteeing holomorphicity of the inverse.

Examples and invariants

Every invertible complex-affine map $z\mapsto Az+b$ on $\mathbb C^n$ is a biholomorphism. The exponential map $\exp:\mathbb C\to\mathbb C^\times$ is locally biholomorphic but not globally biholomorphic because it is not injective. Biholomorphic manifolds have isomorphic algebras of global holomorphic functions, though the converse need not hold without additional hypotheses.

Conventions and contrasts

A biholomorphism is stronger than a diffeomorphism of the underlying smooth manifolds because its differential must be complex linear in holomorphic coordinates. An antiholomorphic diffeomorphism is not a biholomorphism under this convention. The standard coordinate definition and its invariance are treated in Huybrechts, chapters “Local Theory” and “Complex Manifolds”.

References

Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: local holomorphic maps, complex manifolds, and the holomorphic inverse-function theorem.