Kollár–Pardon classification of semialgebraic universal covers
Follow the latest progress by email.
For a connected normal complex projective variety, is its universal cover biholomorphic to a semialgebraic open subset of a projective variety exactly when it has the form D×Cᵐ×F, with D bounded symmetric and F simply connected normal projective?
Why it matters and what counts as progress
Why it matters: It seeks a geometric classification of normal projective universal covers admitting a semialgebraic realization.
What counts as progress: Prove both directions of the semialgebraic universal-cover classification for normal projective varieties, with the product factors as stated.
How the problem developed
- 12 Apr 2011 · Foundational result [1]
János Kollár and John Pardon Studied projective varieties with semialgebraic universal covers, providing the prior work from which the later product-classification conjecture arose.
- 24 Sept 2026 · Classification [2]
OpenAI Reported the iff classification for connected normal projective complex varieties: the universal cover has the stated semialgebraic form exactly for products of a bounded symmetric domain, affine space, and a simply connected normal projective variety.
AI contribution
Semialgebraic universal covers of normal projective varieties
A connected normal complex projective variety has a universal cover biholomorphic to a semialgebraic open subset of a projective variety exactly when that cover is D×C^m×F, with D a bounded symmetric domain and F simply connected, normal and projective.
Sources
- János Kollár and John Pardon, Algebraic varieties with semialgebraic universal cover
- OpenAI, Semialgebraic universal covers of normal projective varieties
- Semialgebraic universal covers of normal projective varieties
- Semialgebraic universal covers of normal projective varieties
- OpenAI mathematics release account
- Lean scope note for bounded-domain symmetry
- Lean bounded-domain symmetry proof
Website timeline
Publication dates and editorial changes.