Holomorphic Cancellation Problem (1998) answered in the negative, conditional on a newly announced exotic affine space; proofs developed with generative AI (Gaofeng Huang, Harvard)
Awaiting review
The takeaway
A space can stop being distinguishable from ordinary complex n-space after one extra dimension is added. The short proof was written with AI help and rests on a result by Auroux, Bai and Pomerleano that has been announced in a lecture but not yet published.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 12 October 2026
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Key facts
- Paper: ‘A Counterexample to the Holomorphic Cancellation Problem’, Gaofeng Huang (Department of Mathematics, Harvard; funded by a Swiss National Science Foundation Postdoc.Mobility grant), arXiv 2610.12581 [math.CV], 8 pages, 8 Oct 2026
- Theorem 1.1: ‘There exist an integer n ∈ {4, …, 8} and a contractible affine variety X of dimension n such that X × C is biholomorphic to C^(n+1) but X is not biholomorphic to C^n.’ The exact n is not determined
- Theorem 1.2 (the paper’s own tool): if X is a contractible Stein manifold of dimension n with the density property and X embeds properly and holomorphically in C^k, then X × C^k is biholomorphic to C^(n+k). The method is Andersén–Lempert theory
- Conditional input: Auroux, Bai and Pomerleano ‘recently announced’ that a specific exotic affine space is not biholomorphic to C⁴; the only reference is Pomerleano’s lecture ‘Exotic affine spaces and Floer theory’ at the CMSA Harvard workshop on Lagrangian Floer theory, 28 Sept 2026. No written proof is cited
- AI disclosure: ‘This work was prepared with generative AI assistance to develop and write the proofs, to search the literature, and to edit the language. The author checked the mathematical content and references and takes responsibility for the content of the paper.’ The model is not named
- Link to OpenAI’s release: ‘The batch of preprints released by OpenAI on October 6 2026 contains a counterexample to the Zariski cancellation problem for n = 4’, citing ‘An explicit failure of complex affine-space cancellation’ (openai/math, dated 23 Sept 2026). Huang notes consequences such as a non-linearizable algebraic C*-action on C⁵ from that preprint
- Not found as of 12 Oct 2026: expert reactions; the Auroux–Bai–Pomerleano paper
What happened
The cancellation problem asks whether a space X must be ordinary n-dimensional space when X times a line is ordinary (n+1)-dimensional space. The algebraic version (Zariski) is open in characteristic zero for n = 3 and n ≥ 5; the holomorphic version was posed by Derksen and Kutzschebauch in 1998. Huang shows that any contractible Stein manifold with the density property becomes ordinary complex space after multiplying by enough copies of C. If one such manifold is not itself ordinary space, which is what Auroux, Bai and Pomerleano announced in a September lecture, then going up one dimension at a time must produce a counterexample somewhere between dimensions 4 and 8.
Why it matters
A problem open for 28 years gets a negative answer in eight pages, but with two caveats: the key input is an announcement without a paper, and the proofs were developed with an unnamed AI system and checked by one author. The paper also shows OpenAI’s 6 Oct release being used and cited by mathematicians within two days: its n = 4 counterexample to Zariski cancellation is the algebraic sibling of this result.
What is disputed or not yet verified
| Verification | Unrefereed preprint; conditional on an announced, unpublished result |
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Sources
2 sources from 2 sites. Numbers match the chips in the text.
2 sources: 2 primary
Primary
Changes
- Filed from the arXiv PDF