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. (Lean)

Manuscripts (2)

Semialgebraic universal covers of normal projective varieties

September 24, 2026 · source · cite as \cite{OAI:Semialgebraic-universal-covers-of-normal-projective-varieties-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.

Symmetry of semialgebraic bounded domains with compact quotient

September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-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.

Result 058 · Algebraic and complex geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.