The Kollar-Pardon conjecture on projective varieties with semialgebraic universal cover

In dimension one uniformization leaves only $\mathbb P^1$ , $\mathbb C$ and the disc as universal covers. Claudon, Hoering and Kollar classified quasi-projective universal covers (assuming abundance), and Kollar and Pardon proved a lattice case with a bounded symmetric quotient. Kollar and Pardon (2012, Conjecture 2) proposed that semialgebraic openness forces a product structure. Let $X$ be a connected normal projective complex variety with universal cover $\widetilde X$ . If $\widetilde X$ is biholomorphic to a semialgebraic open subset of a projective variety, is $\widetilde X\cong D\times\mathbb C^m\times F$ with $D$ a bounded symmetric domain and $F$ a simply connected normal projective variety?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Complex algebraic geometry; universal covers and uniformization
Posed by: Janos Kollar and John Pardon (Algebraic varieties with semialgebraic universal cover, J. Topology 5 (2012), Conjecture 2)
Year posed: 2012
Years open: 14y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 28 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for a connected normal projective complex variety $X$ , $\widetilde X$ is biholomorphic to a semialgebraic open subset of a projective variety if and only if $\widetilde X\cong D\times\mathbb C^m\times F$ with $D$ bounded symmetric and $F$ simply connected normal projective (either may be a point); no algebraicity of deck transformations is assumed and the compact factor may be singular. The cover is quasi-projective exactly when $D$ is a point; a smooth projective variety covered by $\mathbb C^n$ is finitely etale covered by an abelian variety. The companion proves that a connected bounded semialgebraic open subset of an affine variety with a proper cocompact holomorphic group action is smooth and a bounded symmetric domain. Not covered: non-normal or non-projective (e.g. compact Kahler) $X$ .

What the AI did

Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.

Verification

No independent mathematician has checked this yet. Checked here: abstract, introduction, Theorem 1.1 and the history section, read against Conjecture 2 as the manuscript quotes it; the proof was not refereed. Conditional inputs from the same release: the group-theoretic step uses OpenAI's abelianity theorem for special compact Kahler manifolds (catalog entry campana-abelianity-conjecture), and the quasi-projective consequences use OpenAI's log-abundance theorem; neither has been independently checked. Lean: the release's Comparator challenge SymmetricDomains (solution module OAI.Analysis.SymmetricDomains.Main, both present at the pinned commit) formalizes only the companion's bounded-domain theorem (Kollar-Pardon Question 25), a step toward but not the headline classification, so the tier is unreviewed; that statement was read here and not rebuilt.

The Kollar-Pardon conjecture on projective varieties with semialgebraic universal cover

What was actually shown

What the AI did

Verification

Sources

Discussion