LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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Local Hartogs extensions propagate along chains and glue uniquely

Statement

Let $Ω \subseteq C^{m}$ be open, let $G \subseteq Ω$ be a connected open set, and let $U_{1}, \dots, U_{N} \subseteq Ω$ be domains with $U_{j} \cap G \neq \emptyset$ for every $j$ . Assume that for each $j$ and every $g \in O (U_{j} \cap G)$ there is $E_{g} \in O (U_{j})$ satisfying $E_{g} = g$ on all of $U_{j} \cap G$ , and that after reordering every connected component of

$U_{j} \cap (G \cup U_{1} \cup \dots \cup U_{j - 1}) (2 \leq j \leq N) .$

meets $G$ .

Then every holomorphic function on $G$ extends uniquely to a holomorphic function on $G \cup U_{1} \cup \dots \cup U_{N}$ .

Facts & Assumptions

Given: A connected open set $G$ , open sets $U_{1}, \dots, U_{N}$ , and the local extension property stated above.

[L1]

The local-extension hypothesis here explicitly requires agreement on the whole set $U_{j} \cap G$ , which is stronger than agreement on one open overlap in the general extension convention (Holomorphic extension and domains of holomorphy in several variables).

[L2]

Coordinate shell neighborhoods are one class of open sets with the stated local extension property (Hartogs figures give local extension across polydisc shells).

[L3]

Holomorphic functions on a connected open set agree everywhere once they agree on one nonempty open subset (A holomorphic function vanishing on a nonempty open subset of a domain vanishes identically).

Proof

technique · direct

1.1L1given

Let $f \in O (G)$ . By the explicit hypothesis for $j = 1$ , there is $F_{1} \in O (U_{1})$ with $F_{1} = f$ on all of $U_{1} \cap G$ . Hence $f$ and $F_{1}$ glue to a holomorphic function on $G \cup U_{1}$ .

2.1L3step 1.1

Assume inductively that we have already obtained a holomorphic extension $F_{j - 1}$ on $G \cup U_{1} \cup \dots \cup U_{j - 1}$ . By hypothesis, the restriction of $F_{j - 1}$ to $U_{j} \cap G$ extends holomorphically to some $E_{j}$ on $U_{j}$ . Let $C$ be a connected component of $U_{j} \cap (G \cup U_{1} \cup \dots \cup U_{j - 1})$ . The hypothesis makes $C \cap G$ nonempty, and since both $C$ and $G$ are open, $C \cap G$ is a nonempty open subset of $C$ . On that open set, $E_{j}$ and $F_{j - 1}$ both agree with $f$ . Therefore [L3] makes them equal on the whole connected set $C$ . This holds for every overlap component, so $E_{j}$ and $F_{j - 1}$ glue to a holomorphic function $F_{j}$ on $G \cup U_{1} \cup \dots \cup U_{j}$ .

3.1step 2.1L2L3discharge-construct∎

Repeating step 2.1 for $j = 2, \dots, N$ yields a holomorphic extension on the whole union $G \cup U_{1} \cup \dots \cup U_{N}$ . Uniqueness at each stage follows from [L3], so the final extension is unique. The shell lemma [L2] identifies the geometric neighborhoods used later.

Depends on

Used by

Hartogs extension across a connected compact hole with a finite shell cover Theorem

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

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