Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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FALSE: an everywhere-invertible complex Jacobian forces global injectivity

Statement

False claim: if the complex Jacobian of a holomorphic self-map is invertible at every point, then the map is globally injective.

Facts & Assumptions

Given: The claim above.

[L1]

The map (z1,z2)↦(ez1,z2) has invertible complex Jacobian at every point and is not injective (The map (z1,z2)↦(ez1,z2) has invertible complex Jacobian everywhere and is not injective).

Refutation

technique · direct
1.1

Fact [L1] provides a holomorphic map whose complex Jacobian determinant is never zero.

L1
2.1

The same fact [L1] also shows that this map identifies (z1,z2) and (z1+2πi,z2). So the claimed global injectivity conclusion fails.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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