Browse › Algebraic and complex geometry › 048 A characteristic-zero counterexample to Lipman–Zariski
A singular normal affine surface with free tangent sheaf
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Abstract
We construct a singular normal affine complex surface whose tangent sheaf is free of rank two. This disproves the Lipman–Zariski conjecture in characteristic zero.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- Stephen S.-T. Yau, Qiwei Zhu, Huaiqing Zuo, Nakai conjectures for isolated weighted homogeneous hypersurface singularities (arXiv:2604.24508v4, 8 Oct 2026). The introduction of v4 mentions OpenAI's claimed counterexample without a citation, and says "this claim has not yet been verified by the mathematical community."
Possibly relevant literature (found by the agent, not checked)
- Carl Tipler, The Zariski-Lipman conjecture for toric varieties (arXiv:2202.02109, J. Algebra, 2022). Proves smoothness for complex toric varieties with locally free tangent sheaf, a positive case not among those the introduction surveys alongside Hochster's graded case.
- Patrick Graf, The generalized Lipman-Zariski problem (arXiv:1405.1244, Math. Ann. 362 (2015)). It asks whether freeness of the reflexive p-forms forces smoothness; on a normal variety the p=1 case has the same hypothesis as the statement Theorem 1.1 addresses.
- Patrick Graf, A note on Flenner's extension theorem (arXiv:1905.01983, manuscripta math. 165 (2021)). It restates and reproves Flenner's extension theorem for non-isolated singularities, which extends the van Straten–Steenbrink input the introduction cites for the higher-dimensional case.
- Hannah Bergner, On the Lipman-Zariski conjecture for logarithmic vector fields on log canonical pairs (arXiv:1712.04052, 2017). Studies locally free logarithmic tangent sheaves on pairs, which relates to the manuscript's use of the logarithmic tangent sheaf along its exceptional curve in the finite-jet and holomorphic-lifting construction.
Digests and expositions
- Gradient Damage, Nagata's Conjecture, Bloch's Conjecture and 8 More in Algebraic and Complex Geometry (video compilation, 2026-10-14). The chapter beginning at 2:45 introduces the Lipman–Zariski conjecture and the claimed counterexample in result 048, with animations described by the source as illustrations rather than proofs.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 048
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026,
author = {{OpenAI}},
title = {{A singular normal affine surface with free tangent sheaf}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/paper.pdf}{OAI:A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026}},
year = {2026}
}Family 048: A characteristic-zero counterexample to Lipman–Zariski
Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.
This is the only paper in this family.