TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-31

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The germ projection is a local biholomorphism

Statement

Let $p : R (ξ_{0}, Ω) \to Ω$ be the projection of the Riemann surface of a complete analytic function. Then $p$ is a local biholomorphism.

Facts & Assumptions

Given: The projection $p ([f]_{z}) = z$ on the germ surface $R (ξ_{0}, Ω)$ .

[L1]

The germ neighborhoods form a holomorphic atlas, and on each basis element $N (f, U)$ the chart $ϕ_{f, U} ([f]_{z}) = z$ is a homeomorphism onto $U$ (The germ neighborhoods form a Hausdorff, second-countable Riemann-surface atlas).

[L2]

A map is locally biholomorphic when every point has neighbourhoods on which the map restricts to a biholomorphism (Biholomorphic maps between complex domains).

Proof

technique · direct

1.1

Let $ξ = [f]_{z}$ be a point of the germ surface. By [L1], the basis neighbourhood $N (f, U)$ of $ξ$ is mapped by the projection $p$ exactly as the chart $ϕ_{f, U}$ , namely $p ∣_{N (f, U)} = ϕ_{f, U} : N (f, U) \to U$ , and its inverse is $u \mapsto [f]_{u}$ .

L1given

2.1

Fact [L1] makes both $p ∣_{N (f, U)}$ and its inverse holomorphic in the chosen charts. Therefore $p ∣_{N (f, U)}$ is a biholomorphism onto the open set $U$ , and [L2] shows that $p$ is a local biholomorphism.

step 1.1L1L2∎

Depends on

Used by

Dependency tree · two levels

11 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

Curtis T. McMullen, Riemann Surfaces, Theorem 4.3 (standard reference, not scraped)
Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 8 §1.3 (standard reference, not scraped)