Aftermath bot: New page on the Lipman-Zariski conjecture with the Lattice24 review and the Yau-Zhu-Zuo mention

2026-10-23T15:54:44Z

New page on the Lipman-Zariski conjecture with the Lattice24 review and the Yau-Zhu-Zuo mention

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'''The Lipman–Zariski conjecture''', also called the Zariski–Lipman conjecture, asserts that a complex algebraic variety whose tangent sheaf is locally free must be nonsingular; in affine terms, if the module of derivations of a finitely generated algebra over a field of characteristic zero is locally free, the algebra is regular. A manuscript dated 23 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]) claims a counterexample in characteristic zero. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; this page does not certify them.

== Background ==
* As recounted in the introduction of the manuscript:
** Lipman showed that local freeness of the tangent sheaf forces normality. In positive characteristic the assertion fails even for hypersurfaces, for example the surface ''xy'' − ''z''''p'' = 0.
** Positive results in characteristic zero include Hochster's theorem for positively graded algebras, van Straten and Steenbrink's proof for isolated singularities in dimension at least three, Scheja and Storch's for hypersurfaces, Källström's for locally complete intersections, Greb, Kebekus, Kovács and Peternell's for Kawamata log terminal spaces, Druel's and Graf and Kovács's for log canonical spaces, and Jörder's for normal spaces whose tangent sheaf has a local basis of commuting vector fields.
** For normal surfaces, Graf proved the conjecture when the geometric genus is at most one, and Bergner and Graf when ''p''''g'' − ''g'' − ''b'' ≤ 2 in terms of the exceptional curves. Biswas, Gurjar and Kolte proved smoothness for normal affine surfaces whose smooth locus has trivial tangent bundle and logarithmic Kodaira dimension at most one.

== The original result ==
* '''Claim.''' Theorem ''thm:main'' of ''A singular normal affine surface with free tangent sheaf'' states that there are a finitely generated normal integral '''C'''-algebra ''A'' of Krull dimension two and a maximal ideal '''m''' such that Der'''C'''(''A'') is free of rank two and ''A'''''m''' is not regular. The surface can be chosen smooth away from the marked point.
* '''Consequences, as stated by the manuscript.''' The singularity has one exceptional curve of genus 21 and self-intersection −10, canonical discrepancy −5 and geometric genus at least 33, so it lies outside the log canonical and geometric-genus criteria above; the smooth locus of the affine surface has logarithmic Kodaira dimension two. The manuscript also states a negative answer in characteristic zero to the derivation-blowup question of Barajas, Chávez-Martínez and Romano-Velázquez.
* ''Method, as described:'' an isolated Gorenstein analytic surface singularity is built by changing the gluing maps of a neighborhood of the zero section of a line bundle over a genus-21 curve, while keeping its normal bundle, so that two vector fields with a non-commuting frame exist. The germ is then contracted and algebraized by finite determinacy, and freeness of the derivation module descends from the completed local ring.
* '''Sources.''' At commit <code>1581d4c</code>: [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026 A singular normal affine surface with free tangent sheaf].
* '''Formalization.''' The repository's Lean documentation has no scope note for this family at that commit.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 24 October 2026)
|-
| 2026-10-08 || S. S.-T. Yau, Q. Zhu and H. Zuo, ''Nakai conjectures for isolated weighted homogeneous hypersurface singularities'' ([https://arxiv.org/abs/2604.24508v4 arXiv:2604.24508], version 4) || Remark (cites the claim) || Proves Nakai's conjecture, that a ring of differential operators generated by derivations forces regularity, for coordinate rings of isolated weighted homogeneous hypersurface singularities. Its introduction recalls that Becker and Rego showed that Nakai's conjecture implies the Zariski–Lipman conjecture, and mentions OpenAI's claimed counterexample, stating that the claim "has not yet been verified by the mathematical community". || arXiv preprint, version 4; version 1 is dated 27 April 2026. The paper states that OpenAI's Codex assisted in parts of the argument and the manuscript preparation.
|-
| 2026-10-16 || J. Jardine, Lattice24, referee review [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_404.json rank_404] of the manuscript in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] || Review (AI-assisted audit) || Reports no gap and no minor issue after reading all six sections, with hand and symbolic checks of the curve data, the trace and residue identities, the jet computations, the formal-function step and the constants of the finite-determinacy algebraization. It states that it did not independently check how the cited positive results are quoted. || Part of an audit of the collection's manuscripts that the audit describes as complete.
|}

== Notes ==
* The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]

The Lipman–Zariski conjecture - Revision history

Aftermath bot: New page on the Lipman-Zariski conjecture with the Lattice24 review and the Yau-Zhu-Zuo mention