∑math20

Algebraic and complex geometry · Result 058

Semialgebraic universal covers and bounded domains

2 papersLean

Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products \(D\times\mathbb C^m\times F\), with D bounded symmetric and F simply connected, normal, and projective. A universal cover is quasi-projective exactly when the bounded symmetric factor is absent. In particular, a smooth projective variety covered by ℂn has a finite étale cover by an abelian variety.

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The problem

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Papers 2

First page of Semialgebraic universal covers of normal projective varieties

Semialgebraic universal covers of normal projective varieties

OpenAI · September 24, 2026

We prove the Kollár–Pardon conjecture: the universal cover of a connected normal projective complex variety is biholomorphic to a semialgebraic open subset of a projective variety if and only if it is a product of a bounded symmetric domain, a complex affine space, and a simply connected normal projective variety.

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Cite
@misc{OAI:Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026,
  author = {{OpenAI}},
  title = {{Semialgebraic universal covers of normal projective varieties}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/paper.pdf}{OAI:Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026}},
  year = {2026}
}
First page of Symmetry of semialgebraic bounded domains with compact quotient

Symmetry of semialgebraic bounded domains with compact quotient

OpenAI · September 24, 2026

Every nonempty connected semialgebraic bounded open subset of a complex affine variety admitting a properly discontinuous cocompact group of biholomorphisms is smooth and biholomorphic to a bounded symmetric domain. This answers the bounded-domain question of Kollár and Pardon affirmatively.

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Cite
@misc{OAI:Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026,
  author = {{OpenAI}},
  title = {{Symmetry of semialgebraic bounded domains with compact quotient}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026/paper.pdf}{OAI:Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result answers the paper's symmetry question affirmatively. A nonempty connected bounded semialgebraic subset, relatively open in a complex affine algebraic set, is smooth and biholomorphic to a bounded symmetric domain whenever a discrete group acts properly and holomorphically with compact quotient.

The action need not be free, and the ambient algebraic set may be singular.

Comparator links

ResultComparator statement
Symmetry of semialgebraic bounded domainsSymmetricDomains.lean

Scope note on GitHub →