Family 047Algebraic and complex geometryLeanNot yet digested

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.

Manuscripts in this family · 1

An explicit failure of complex affine-space cancellation

Sep 23, 2026·paper.pdfLean

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 𝔸

^{3}

-fibrations over 𝔸

^{1}

and 𝔸

^{2}

that are not Zariski-locally trivial. These fibrations disprove the Dolgachev–Weisfeiler affine-fibration conjecture over these bases.

Full abstract

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