Algebraic and complex geometry·OpenAI Math·Result 048
A characteristic-zero counterexample to Lipman–Zariski
One of the 372 results in OpenAI's public mathematics catalogue, filed under algebraic and complex geometry: 1 preprint, the latest dated 23 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a result titled ‘A characteristic-zero counterexample to Lipman–Zariski’ (algebraic and complex geometry) in a preprint dated 23 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More, is on YouTube.
Chapter 39 of 83 in Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · starts at 13:12 · watch from there on YouTube →
What OpenAI claims
Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.
Paper
A singular normal affine surface with free tangent sheaf
OpenAI's 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.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 23 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.
- Where is the proof?
- In the paper, A singular normal affine surface with free tangent sheaf (PDF).
- Is there a video?
- Yes, on YouTube (28:48, chapter 39).
Also in
- Chapter 9 of 10 in Nagata's Conjecture, Bloch's Conjecture and 8 More in Algebraic and Complex Geometry · starts at 2:45 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · Compilation · 28:48 · chapter 39 from 13:12 · description, sources and chapters →
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