DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-08-28

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Biholomorphic maps between open sets in $C^{m}$

Definition

Let $m \geq 1$ , and let $U, V \subseteq C^{m}$ be open. A map $F : U \to V$ is biholomorphic if it is bijective (Injection, surjection, bijection), holomorphic on $U$ , and its inverse $F^{- 1} : V \to U$ is holomorphic in the sense of Holomorphic maps $C^{m} \to C^{n}$ and the complex Jacobian matrix. A biholomorphic map is a biholomorphism from $U$ onto $V$ .

For local use, $F$ is biholomorphic between neighbourhoods of $a$ and $F (a)$ when there are open sets $U_{0} \subseteq U$ and $V_{0} \subseteq V$ with $a \in U_{0}$ , $F (a) \in V_{0}$ , and $F ∣_{U_{0}} : U_{0} \to V_{0}$ a biholomorphism.

Depends on

Used by

The holomorphic inverse function theorem in several complex variables Theorem

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jiří Lebl, Tasty Bits of Several Complex Variables, Chapter 5 (standard reference, not scraped)
Jaap Korevaar and Jan Wiegerinck, Several Complex Variables, Section 5.2 (standard reference, not scraped)

Biholomorphic maps between open sets in CmCm