Result 047, Algebraic and complex geometry

Zariski cancellation and affine fibrations over the complex numbers

Constructs an integral complex affine fourfold $X\not\cong\mathbb A^4$ with $X\times\mathbb A^1\cong\mathbb A^5$ , disproving affine-space cancellation over ℂ in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections $X\to\mathbb A^1$ and $\mathbb A^5\to\mathbb A^2$ have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Can adding one freely varying coordinate make an unusual algebraic shape indistinguishable from ordinary coordinate space? This manuscript claims an example over the complex numbers, showing that a product can conceal the original shape's algebraic structure.

What changes?

The manuscript reports an integral complex affine variety X of dimension four: an irreducible, reduced space defined by polynomial equations. X is not algebraically isomorphic to affine four-space, yet its product with the affine line is isomorphic to affine five-space. It also reports smooth surjections from X to the affine line and from affine five-space to the affine plane. Every residue-field fiber is affine three-space over that field, but neither map is Zariski-locally trivial.

What does that help mathematicians do?

The cancellation example would rule out recovering affine four-space merely from knowing that adding one coordinate produces affine five-space. The fibrations expose a separate limitation: even smooth families whose fibers are all affine three-spaces need not become products over Zariski-open neighborhoods, the algebraic notion of local patches. Researchers therefore cannot infer such local product structure from smoothness and the shapes of the individual fibers alone.

Are there practical applications?

The immediate value is foundational, concerning how algebraic spaces are recognized and how families of them fit together. These claimed counterexamples would provide concrete tests for proposed classification arguments: an argument must distinguish X from affine four-space despite their identical products with a line, and must account for more than the fibers when asserting local triviality.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

An explicit failure of complex affine-space cancellation

September 23, 2026 29 pages Main result formalized in Lean

We construct an explicit integral complex affine fourfold X with $X\times\mathbb A^1\cong\mathbb A^5$ but $X\not\cong\mathbb A^4$ . This gives a negative answer to Zariski's affine-space cancellation problem over ℂ in dimension four. The same construction disproves the Stable Coordinate Conjecture in ambient dimension five and yields smooth 𝔸³-fibrations over 𝔸¹ and 𝔸² that are not Zariski-locally trivial. These fibrations disprove the Dolgachev–Weisfeiler affine-fibration conjecture over these bases.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026,
  author = {{OpenAI}},
  title = {{An explicit failure of complex affine-space cancellation}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/paper.pdf}{OAI:An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/047.md.

Zariski cancellation and affine fibrations over the complex numbers

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

An explicit failure of complex affine-space cancellation

Scope

Affine-space cancellation asks whether $X\times\mathbb A^1\cong\mathbb A^{n+1}$ forces $X\cong\mathbb A^n$ . The formalized result gives an explicit finite-type complex domain $A$ of Krull dimension four with $A[w]\cong\mathbb C[x_1,\ldots,x_5]$ but $A\not\cong\mathbb C[x_1,\ldots,x_4]$ . Thus adjoining one variable erases a genuine algebraic distinction.

The separate stable-coordinate and general line-bundle lifting consequences are not included.

Comparator links

Result	Comparator statement
Complex affine-space cancellation counterexample	ComplexCancellation.lean

At a glance

Subject: Algebraic and complex geometry
Manuscripts: 1
Pages: 29
Dated: Sep 23, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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On the map

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