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)

Manuscript

An explicit failure of complex affine-space cancellation

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026}

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.

Result 047 · Algebraic and complex geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Agora members.