A note on the stable equivalence problem

We provide counterexamples to the stable equivalence problem in every dimension d ≥ 2. That means that we construct hypersurfaces H₁ , H₂ ⊂ C d+1 whose cylinders H₁ × C and H₂ × C are equivalent hypersurfaces in C d+2 , although H₁ and H₂ themselves are not equivalent by an automorphism of C d+1 . We also give, for every d ≥ 2, examples of two non-isomorphic algebraic varieties of dimension d which are biholomorphic.