CounterexampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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The bidisc minus the origin is not holomorphically convex

Statement refuted

The punctured bidisc

$Ω = {(z_{1}, z_{2}) \in C^{2} : ∣ z_{1} ∣ < 1, ∣ z_{2} ∣ < 1} ∖ {(0, 0)}$

is holomorphically convex.

Facts & Assumptions

Given: The punctured bidisc $Ω$ .

[L1]

A holomorphic function on a punctured several-variable domain extends across the missing point (An isolated puncture is removable in complex dimension at least two).

[L2]

A domain of holomorphy admits no common larger overlap extending every holomorphic function, and for domains in $C^{m}$ that condition is equivalent to holomorphic convexity (Holomorphic extension and domains of holomorphy in several variables, Cartan-Thullen theorem).

Counterexample

technique · direct

1.1

Every holomorphic function on $Ω$ extends to the full bidisc by [L1]. Thus the whole bidisc is a common larger domain across the missing origin for every holomorphic function on $Ω$ .

L1given

2.1

The extension statement in step 1.1 contradicts the domain-of-holomorphy condition in [L2], so $Ω$ is not a domain of holomorphy. Applying the equivalence in [L2], $Ω$ is not holomorphically convex either. This refutes the statement.

L2step 1.1∎

Depends on

Used by

FALSE: every domain in Cⁿ is a domain of holomorphy False statement

Dependency tree · two levels

13 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, §1.6 and §2.1 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, §3.2.3 (standard reference, not scraped)