Math-2580 · mathematics · Algebraic geometry

Symmetry of bounded semialgebraic domains with compact quotient

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.

Let U be nonempty connected semialgebraic bounded open in a complex affine variety. If a group of biholomorphisms acts properly discontinuously and U has compact quotient, must U be smooth and biholomorphic to a bounded symmetric domain?

Short English statement

Sources

S1 Original question source or mathematical context. https://annals.math.princeton.edu/wp-content/uploads/annals-v176-n3-p04-p.pdf.
S2 Original question source or mathematical context. https://arxiv.org/abs/1808.03416.
S3 Original question source or mathematical context. https://doi.org/10.4171/owr/2018/37.
S4 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026/README.md.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2580-A1 · Literature report · LLM-generated · unknown · 2026-10-06

OpenAI claim summary for Math-2580

Manuscript 1 (family 058, 2026-09-24): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Symmetry of bounded semialgebraic domains with compact quotient

OpenAI's claim

With cocompact biholomorphisms, a connected semialgebraic bounded open set in a complex affine variety is claimed to be smooth and biholomorphically equivalent to a symmetric domain that is bounded. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.

Let $U$ be a nonempty connected bounded semialgebraic open subset of a complex affine variety in a fixed closed embedding into $\mathbb C^N$. Suppose a biholomorphism group $\Gamma$ acts properly discontinuously and $U/\Gamma$ is compact. OpenAI claims \[ U\text{ is smooth and biholomorphic to a bounded symmetric domain}. \] A bounded symmetric domain is one in which each point is isolated in the fixed locus of a holomorphic involution. Initial smoothness, freeness of the action, and algebraicity of the biholomorphisms are not assumed.

Outline of the argument

Manuscript 1 (family 058, 2026-09-24): Holomorphic supports and analytic discs select a noncollapsing boundary model despite possible ambient singularities. Joint normal-offset limits and weighted dilation give a quadratic domain; convergence of both maps and inverses identifies the original domain with it, and the homogeneous-domain theorem supplies symmetry.

Formal verification

For family 058, the scope page documents the following selections. The selected bounded-domain result allows a singular affine ambient set and nonfree proper cocompact group action, and concludes smoothness and symmetry. It does not select the entire projective universal-cover classification. The documented statement covers the main claim of manuscript 1. Comparator files: SymmetricDomains.lean. Declarations: OAI.Release061.main.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

OpenAI, Symmetry of semialgebraic bounded domains with compact quotient, 2026-09-24; citation key OAI:Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
Family 058: Lean scope.
Comparator SymmetricDomains.json.
Comparator SymmetricDomains.lean.
OpenAI catalogue entry.

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Record details

Subjects: Algebraic geometry
Collection: Top Open Problems
Mathematics rank: 2580