Algebraic families of higher dimensional $\mathbb{A}^{1}$-contractible affine varieties non-isomorphic to affine spaces
Adrien Dubouloz, Parnashree Ghosh
TL;DR
The paper constructs higher-dimensional, smooth affine $\mathbb{A}^1$-contractible varieties that are not isomorphic to affine spaces, for every field of characteristic zero and every dimension $n\ge4$. These arise as generalizations of deformed Koras–Russell threefolds via explicit equations $\underline{x}^{\underline{n}}y+z^q+t^r+x_0p(\underline{x})=0$, and their $\mathbb{A}^1$-contractibility is proved by induction using cofiber sequences and homotopy purity. The authors show that while the varieties $X_m(\underline{n},p)$ are pairwise non-isomorphic, their $\mathbb{A}^1$-cylinders are all isomorphic, providing infinite families of non-isomorphic bases with isomorphic $\mathbb{A}^1$-cylinders and thus counterexamples to the generalized Zariski Cancellation Problem in dimension $m+3$. They also construct a parameterized algebraic family over an affine base whose fibers are $\mathbb{A}^1$-contractible and pairwise non-isomorphic, highlighting the richness of exotic $\mathbb{A}^1$-contractible varieties and their relation to cancellation phenomena. Overall, the work combines invariant theory (Makar-Limanov and Derksen), affine blow-up techniques, and motivic homotopy methods to advance the understanding of $\mathbb{A}^1$-contractibility and cancellation problems in algebraic geometry.
Abstract
We construct algebraic families of smooth affine $\mathbb{A}^1$-contractible varieties of every dimension $n\geq 4$ over fields of characteristic zero which are non-isomorphic to affine spaces and potential counterexamples to the Zariski Cancellation Problem. We further prove that these families of varieties are also counter examples to the generalized Cancellation problem.