Algebraic families of higher dimensional $\mathbb{A}^{1}$-contractible affine varieties non-isomorphic to affine spaces

TL;DR

The paper constructs higher-dimensional smooth affine varieties that are contractible in the -homotopy sense, are not isomorphic to affine spaces, and serve as potential counterexamples to the Zariski Cancellation Problem.

Contribution

It introduces algebraic families of -contractible affine varieties in all dimensions   that are non-isomorphic to affine spaces, challenging existing cancellation conjectures.

Findings

Constructed -contractible varieties in all dimensions  

Demonstrated these varieties are non-isomorphic to affine spaces

Proved these varieties counter the generalized Cancellation problem

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.