{"schema":"postcutoff/event@1","as_of":"2026-10-12T07:22:00+02:00","url":"https://postcutoff.com/e/2026-10-08-holomorphic-cancellation-problem-counterexample/","md":"https://postcutoff.com/e/2026-10-08-holomorphic-cancellation-problem-counterexample/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"Adam Bicz","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-08-holomorphic-cancellation-problem-counterexample","date":"2026-10-08","date_precision":"day","short_title":"Holomorphic Cancellation Problem (1998) answered in the negative, conditional on a newly announced exotic affine space; proofs developed with generative AI (Gaofeng Huang, Harvard)","deck":null,"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.12581: A Counterexample to the Holomorphic Cancellation Problem","url":"https://arxiv.org/abs/2610.12581","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"OpenAI math release: An explicit failure of complex affine-space cancellation (23 Sept 2026)","url":"https://github.com/openai/math/blob/main/preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"}],"official":2,"filed":"2026-10-12","updated":"2026-10-12","orgs":["Harvard University"],"title":"Holomorphic Cancellation Problem (1998) answered in the negative, conditional on a newly announced exotic affine space; proofs developed with generative AI (Gaofeng Huang, Harvard)","summary":"On 8 Oct 2026 Gaofeng Huang (Harvard) posted arXiv 2610.12581. It proves that a contractible Stein manifold X with the density property that embeds properly in C^k satisfies X × C^k ≅ C^(n+k). Combined with a result announced by Auroux, Bai and Pomerleano on 28 Sept 2026, this gives, for some n between 4 and 8, a complex affine variety X with X × C biholomorphic to C^(n+1) although X is not biholomorphic to C^n: a negative answer to the Holomorphic Cancellation Problem of Derksen and Kutzschebauch (1998). The paper's AI disclosure says it \"was prepared with generative AI assistance to develop and write the proofs\". It also cites an OpenAI math-release preprint claiming a counterexample to Zariski cancellation for n = 4.","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"],"key_numbers":[],"tags":["math","complex-analysis","affine-geometry","cancellation-problem","counterexample","generative-ai","openai-math-release"],"science":{"field":"mathematics","subfield":"complex analysis / affine algebraic geometry","problem":"Holomorphic Cancellation Problem (Derksen–Kutzschebauch 1998): if X × C is biholomorphic to C^(n+1), is X biholomorphic to C^n?","result":"No, for some n in {4,…,8}, assuming the exotic affine space announced by Auroux, Bai and Pomerleano.","open_since":"1998","ai_system":["unnamed generative AI"],"human_role":"AI-assisted: generative AI helped develop and write the proofs and search the literature; the author checked the content","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprint; conditional on an announced, unpublished result"},"status":"pending","shock":"","short":"Holomorphic Cancellation Problem"},"body_md":"## What happened\n\nThe cancellation problem asks whether a space X must be ordinary n-dimensional space when X times a line is ordinary\n(n+1)-dimensional space. The algebraic version (Zariski) is open in characteristic zero for n = 3 and n ≥ 5; the holomorphic\nversion was posed by Derksen and Kutzschebauch in 1998. Huang shows that any contractible Stein manifold with the density\nproperty becomes ordinary complex space after multiplying by enough copies of C. If one such manifold is not itself ordinary\nspace, which is what Auroux, Bai and Pomerleano announced in a September lecture, then going up one dimension at a time must\nproduce a counterexample somewhere between dimensions 4 and 8.\n\n## Why it matters\n\nA problem open for 28 years gets a negative answer in eight pages, but with two caveats: the key input is an announcement without\na paper, and the proofs were developed with an unnamed AI system and checked by one author. The paper also shows OpenAI's 6 Oct\nrelease being used and cited by mathematicians within two days: its n = 4 counterexample to Zariski cancellation is the algebraic\nsibling of this result.","disputed":[{"label":"Verification","text":"Unrefereed preprint; conditional on an announced, unpublished result","sources":[]}],"related":[{"id":"2026-10-09-arxiv-ai-disclosed-results-early-october-2026","url":"https://postcutoff.com/e/2026-10-09-arxiv-ai-disclosed-results-early-october-2026/","date":"2026-10-09","date_precision":"day","short_title":"arXiv, 5–9 October 2026","deck":"More conjectures settled with disclosed AI help (Quillen's local parameter conjecture in dimension 4, Laughlin gap, poset reconstruction), next to human-only results","takeaway":"The catalogue shows what the everyday use of frontier models in research mathematics looked like in October 2026: routine, disclosed in a sentence or two, and uneven in reliability.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":14,"filed":"2026-10-12","updated":"2026-10-12","orgs":["OpenAI","Anthropic","various mathematicians"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":74,"official":13,"filed":"2026-10-07","updated":"2026-10-12","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-12","type":"filed","text":"Created from the arXiv PDF"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-12","run":null,"sources_read":"The arXiv PDF","updated":"2026-10-12","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":161,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":100,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":191,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":130,"in_training_data":false}],"short_url":null}