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Complex counterexamples to cancellation and affine fibrations
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GAME #047
Complex counterexamples to cancellation and affine fibrations
1 level of pure shapes made of equations!
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| 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. |
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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.
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