Noncancellation for contractible affine threefolds

Abstract

We construct two nonisomorphic contractible affine threefolds X X and Y Y with the property that their cylinders X × A 1 X\times \mathbb {A}^{1} and Y × A 1 Y\times \mathbb {A}^{1} are isomorphic, showing that the generalized Cancellation Problem has a negative answer in general for contractible affine threefolds. We also establish that X X and Y Y are actually biholomorphic as complex analytic varieties, providing the first example of a pair of biholomorphic but not isomorphic exotic A 3 \mathbb {A}^{3} ’s.