Browse › Algebraic and complex geometry › 058 Semialgebraic universal covers and bounded domains
Semialgebraic universal covers of normal projective varieties
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
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.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Related prior literature mentioned by others
- János Kollár and John Pardon, Algebraic varieties with semialgebraic universal cover (arXiv:1104.2309v2, 2011-12-15; Journal of Topology, 2012). The AI Proved What? family exposition identifies this as the source of both the product-classification conjecture (Conjecture 2) and the bounded-domain question addressed by the two manuscripts. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Alan M. Nadel, Semisimplicity of the group of biholomorphisms of the universal covering of a compact complex manifold with ample canonical bundle (Ann. of Math. 132 (1990), 193–211). Original source for the Nadel half of the "Nadel–Frankel splitting theorem", which the paper uses to identify the transverse domain M as bounded symmetric, citing it only through Farb–Weinberger.
- Sidney Frankel, Locally symmetric and rigid factors for complex manifolds via harmonic maps (Ann. of Math. 141 (1995), 285–300). Describes the complete holomorphic vector fields on universal covers of compact manifolds with c₁ < 0; this is the Frankel side of the splitting theorem the paper combines with its "spanning vector fields" step.
- Andrew Zimmer, Smoothly bounded domains covering compact manifolds (arXiv:1910.05288, 2019). Its ball-uniformization theorem for bounded domains with a compact manifold quotient and a C^{1,1} boundary provides a boundary-regularity comparison for this paper's claimed bounded symmetric factor.
Digests and expositions
- AI Proved What?, Three building blocks? Semialgebraic universal covers and bounded domains (episode 058, October 2026). This video and transcript outline both arguments, identify the classification's dependence on other release manuscripts, and distinguish the bounded-domain Lean result from the broader classification. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 058
Independent sites; their content is not reviewed here.
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}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/058.md. The Lean statements have not been checked by people against the paper's claims.
Family 058: Semialgebraic universal covers and bounded domains
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.
Other papers in this family:
- Symmetry of semialgebraic bounded domains with compact quotient · September 24, 2026