A note on the stable equivalence problem

Abstract

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.