The Lipman-Zariski conjecture in characteristic zero
On a smooth complex variety of dimension the tangent sheaf is locally free of rank . Lipman (1965), following a question of Zariski, proved that local freeness of the derivation module forces normality and asked for the converse of smoothness: if is a finitely generated integral -algebra and is locally free, is regular? It was known for hypersurfaces, local complete intersections, graded rings, isolated singularities in dimension at least three, klt and log canonical spaces, and several classes of normal surface singularities, but the surface case was open. Is every complex algebraic variety with locally free tangent sheaf nonsingular?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Algebraic geometry; singularities and derivation modules
- Posed by
- Joseph Lipman, after Oscar Zariski
- Year posed
- 1965
- Years open
- 61y
- 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
- 42 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: there are a finitely generated normal integral -algebra of dimension two and a maximal ideal with and not regular; the surface can be smooth away from that point. By Jorder's theorem no local tangent frame there commutes. The paper also answers negatively, in characteristic zero, the derivation-blowup question of Barajas, Chavez-Martinez and Romano-Velazquez (2026, Question 1). It does not address higher-dimensional non-isolated cases left open by earlier results or give a counterexample with a commuting frame (impossible by Jorder).
What the AI did
The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, 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). The manuscript is authored 'OpenAI' and names no human author. The manuscript (September 23, 2026) has no companion in the release.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the conjecture; it gives a finitely generated normal two-dimensional -algebra with globally free and a maximal ideal at which is not regular, which contradicts the conjecture as posed. The construction starts from an analytic Gorenstein surface singularity (exceptional curve of genus 21, self-intersection ) and algebraizes it; the paper checks that the example lies outside the Graf, Bergner-Graf, log canonical and Biswas-Gurjar-Kolte criteria. Not refereed. No Lean formalization in the release.