A note on the stable equivalence problem

TL;DR

This paper constructs explicit counterexamples in all dimensions to the stable equivalence problem, showing that certain hypersurfaces become equivalent after taking cylinders, despite not being equivalent originally.

Contribution

It provides the first known counterexamples to the stable equivalence problem in every dimension, demonstrating that stable equivalence does not imply original hypersurface equivalence.

Findings

Counterexamples for all dimensions $d extgreater{}2$

Hypersurfaces with equivalent cylinders but non-equivalent original hypersurfaces

Examples of non-isomorphic algebraic varieties that are biholomorphic

Abstract

We provide counterexamples to the stable equivalence problem in every dimension $d\geq2$. That means that we construct hypersurfaces $H_1, H_2\subset\mathbb{C}^{d+1}$ whose cylinders $H_1\times\mathbb{C}$ and $H_2\times\mathbb{C}$ are equivalent hypersurfaces in $\mathbb{C}^{d+2}$, although $H_1$ and $H_2$ themselves are not equivalent by an automorphism of $\mathbb{C}^{d+1}$. We also give, for every $d\geq2$, examples of two non-isomorphic algebraic varieties of dimension $d$ which are biholomorphic.