A Counterexample to the Holomorphic Cancellation Problem

In this note we show that if a contractible Stein manifold $X$ with the density property admits a proper holomorphic embedding into ${\mathbb{C}}^k$, then $X\times {\mathbb{C}}^k$ is biholomorphic to ${\mathbb{C}}^{n+k}$ with $n=\dim X$. Combined with an announced result of Auroux, Bai and Pomerleano, this shows that for some $4\le n\le 8$ there is a complex affine variety $X$ such that $X\times \mathbb{C}$ is biholomorphic to ${\mathbb{C}}^{n+1}$ and $X$ is not biholomorphic to ${\mathbb{C}}^n$. This settles the Holomorphic Cancellation Problem in the negative.