A note on the stable equivalence problem

We provide counterexamples to the stable equivalence problem in every dimension $dgeq2$. That means that we construct hypersurfaces $H_1, H_2subsetC^{d+1}$ whose cylinders $H_1timesC$ and $H_2timesC$ are equivalent hypersurfaces in $C^{d+2}$, although $H_1$ and $H_2$ themselves are not equivalent by an automorphism of $C^{d+1}$. We also give, for every $dgeq2$, examples of two non-isomorphic algebraic varieties of dimension $d$ which are biholomorphic.