Non cancellation for smooth contractible affine threefolds

TL;DR

This paper constructs examples of contractible affine threefolds that are biholomorphic but not isomorphic, demonstrating a negative answer to the generalized Cancellation Problem in this context.

Contribution

It provides the first known examples of biholomorphic but non-isomorphic exotic affine 3-spaces, challenging previous assumptions in algebraic geometry.

Findings

Constructed two non-isomorphic contractible affine threefolds with isomorphic cylinders.

Proved that these threefolds are biholomorphic as complex analytic varieties.

Showed that the generalized Cancellation Problem does not hold in general for contractible affine threefolds.

Abstract

We construct two non isomorphic contractible affine threefolds X and Y with isomorphic cylinders, showing that the generalized Cancellation Problem has a negative answer in general for contractible affine threefolds. We also establish that X and Y are actually biholomorphic as complex analytic varieties, providing the first example of a pair of biholomorphic but not isomorphic exotic affine 3-spaces.