The Shafarevich conjecture on holomorphic convexity of universal covers
Shafarevich asked whether the universal cover of a complete algebraic variety is holomorphically convex. In the usual smooth projective formulation: for every smooth connected projective complex variety , is the universal cover , with the lifted complex structure, holomorphically convex, i.e. is the holomorphic hull of every compact compact? A direct form for varieties with large fundamental group (Kollar) asks whether is Stein when it contains no positive-dimensional compact analytic subvariety. The conjecture was proved for elliptic surfaces, for virtually nilpotent fundamental groups, and for fundamental groups with a faithful linear representation (Eyssidieux-Katzarkov-Pantev-Ramachandran). Does it hold in general?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Complex algebraic geometry; universal covers and fundamental groups
- Posed by
- Igor Shafarevich
- Year posed
- 1974
- Years open
- 52y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 52 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Surface manuscript, Theorem 1.1: a smooth connected projective complex surface containing a connected nodal curve with smooth rational components such that has infinite image; a component of its preimage in is a closed noncompact union of compact rational curves, so hulls are noncompact and is not holomorphically convex. By the linear Shafarevich theorem has no faithful finite-dimensional complex representation. Fourfold manuscript, Theorem 1.1: a smooth projective fourfold with large fundamental group (no positive-dimensional compact subvariety in ) whose universal cover is not Stein, hence not holomorphically convex; this disproves the large-fundamental-group form as well. Neither paper addresses quasi-projective or Kahler variants beyond these.
What the AI did
The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts: the surface counterexample (September 23) and the fourfold with large fundamental group (October 5).
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the surface manuscript and Theorem 1.1 of the fourfold manuscript were read against the conjecture as posed. The surface counterexample is an infinite Nori string: a connected nodal curve of rational components with infinite fundamental-group image, whose lift is a noncompact connected union of compact curves. The fourfold argument is topological (a lifted surface with nonzero third homology cannot be Stein). Both rely on cited inputs (Stover-Toledo ramified covers, Zarhin, Hamm-Le Lefschetz). The proofs were not refereed. No Lean formalization exists for this family.