Zariski cancellation and affine fibrations over the complex numbers

1 · The essentials

Can adding one extra coordinate make two different algebraic spaces look the same? The manuscript gives an explicit example over the complex numbers where this happens. It constructs a four-dimensional space that is not equivalent to ordinary four-dimensional coordinate space, although adding one free coordinate makes it equivalent to ordinary five-dimensional coordinate space. Here, equivalence means matching by algebraic rules in both directions, not merely looking alike. This gives a negative answer to the four-dimensional case of Zariski’s affine-space cancellation problem, and the main result has a Lean formalization. The same construction also gives families whose individual pieces are all ordinary three-dimensional coordinate spaces, but which cannot be assembled as products even locally in the required algebraic sense.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine two different objects that become indistinguishable after each receives the same extra attachment. Removing that attachment would not let you conclude that the original objects matched. Here the attachment is a free coordinate, and the matching must obey algebraic rules.

Why it matters

The result shows that an extra affine coordinate can conceal a genuine algebraic difference, even over the complex numbers. The related families also show that having the same simple space in every fiber does not guarantee a locally product-shaped family. These are distinct failures of attempts to recover global structure from apparently simple information.

2 · Going further

Write $A^{n}$ for affine $n$ -space over $C$ . The cancellation question considered here asks whether $X \times A^{1} ≅ A^{5}$ forces $X ≅ A^{4}$ . The main manuscript constructs an explicit integral complex affine fourfold $X$ for which the first isomorphism holds and the second does not. This supplies a negative answer to Zariski’s affine-space cancellation problem over $C$ in dimension four.

The argument’s central construction, as described in the abstract, is a single space whose product with an affine line becomes affine space although the space itself is not affine space. The supplied abstract does not give its equations or explain how either assertion is established, so a more detailed account of the mechanism is not available from this source. The manuscript is marked Lean-formalized, and the main cancellation result has a Lean formalization.

The same construction yields smooth surjections $X \to A^{1}$ and $A^{5} \to A^{2}$ whose fibers over every residue field are isomorphic to affine three-space, but which are not Zariski-locally trivial. These are presented as counterexamples to the Dolgachev–Weisfeiler affine-fibration conjecture over these bases. The manuscript also claims a counterexample to the Stable Coordinate Conjecture in ambient dimension five. Only the main manuscript is supplied; no companion manuscripts or details of their relationship are available.

Useful notions

Complex numbers
Spaces described by polynomial equations
Algebraic isomorphisms
Products with an extra coordinate
Fibers of a map

What was known before / what is new

Before

The source frames the construction as answering Zariski’s affine-space cancellation problem over the complex numbers in dimension four. It also identifies the Stable Coordinate Conjecture and the Dolgachev–Weisfeiler affine-fibration conjecture as targets of the construction. No further history or earlier partial results are given.

New

The main result is an explicit integral complex affine fourfold $X \neq ≅ A^{4}$ with $X \times A^{1} ≅ A^{5}$ . The same construction is shown in the manuscript to produce non-Zariski-locally-trivial smooth families of affine three-spaces over $A^{1}$ and $A^{2}$ . The abstract also claims failure of the Stable Coordinate Conjecture in ambient dimension five.

Limits and assumptions

The cancellation example is over the complex numbers and has dimension four. The stated fibration counterexamples have the specific bases and total spaces $X \to A^{1}$ and $A^{5} \to A^{2}$ ; their hypotheses include smoothness, surjectivity, and affine-three-space fibers over every residue field. The main cancellation result has a Lean formalization, but the supplied text does not specify whether that formalization also covers the fibration and stable-coordinate claims. It provides neither construction equations nor a detailed argument, and it does not state the status of other dimensions or other fields.

Mini glossary

Affine space: Affine $n$ -space, written $A^{n}$ , is the algebraic space of $n$ freely chosen coordinates over the specified field.
Algebraic isomorphism: An algebraic isomorphism is an invertible algebraic map whose inverse is also algebraic.
Integral complex affine fourfold: An integral complex affine fourfold is a four-dimensional affine algebraic space over $C$ whose coordinate ring has no zero divisors.
Fiber: A fiber is the algebraic space lying over a point of the base, considered over that point’s residue field.
Zariski-locally trivial: A family is Zariski-locally trivial if the base can be covered by Zariski-open sets on each of which the family is isomorphic, over that set, to a product with its fiber.
Cancellation: Cancellation here is the proposed inference that a space must be affine space if its product with an affine line is affine space.

Keywords : Zariski cancellation · Affine fibrations · Complex algebraic geometry

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/047.md

Manuscripts in this family

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An explicit failure of complex affine-space cancellation

23/09/2026

Verified by Lean

We construct an explicit integral complex affine fourfold X with $X \times A^{1} ≅ A^{5}$ but $X \neq ≅ A^{4}$ . 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 GitHub folder Lean formalization