CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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.

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C^2 minus a complex line is a domain of holomorphy

Statement refuted

Refuted claim: removing a complex line from $C^{2}$ produces a domain that still cannot be a domain of holomorphy.

Facts & Assumptions

Given: The domain $Ω = {(z_{1}, z_{2}) \in C^{2} : z_{1} \neq 0}$ .

[L1]

A domain of holomorphy is tested by whether every holomorphic function extends through one fixed larger overlap (Holomorphic extension and domains of holomorphy in several variables).

Counterexample

technique · direct

1.1

The function $f (z_{1}, z_{2}) = 1 / z_{1}$ is holomorphic on $Ω$ .

givenalgebra

2.1

If $f$ extended holomorphically across any point of the missing hyperplane ${z_{1} = 0}$ , then $z_{1} f$ would extend holomorphically there as the constant function $1$ , forcing $f = 1 / z_{1}$ near that point and hence forcing a holomorphic function equal to $1 / z_{1}$ at $z_{1} = 0$ , which is impossible. So the missing complex line blocks extension, and $Ω$ is a domain of holomorphy rather than a Hartogs hole.

step 1.1L1algebra∎

Depends on

Holomorphic extension and domains of holomorphy in several variables

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

J. Lebl, Tasty Bits of Several Complex Variables, §2.1 (standard reference, not scraped)