Counterexamples to the complement problem

Abstract

We provide explicit counterexamples to the so-called Complement Problem in every dimension $n\ge 3$, i.e. pairs of nonisomorphic irreducible algebraic hypersurfaces $H_1,H_2\subset {\mathbb{C}}^n$ whose complements ${\mathbb{C}}^n\setminus H_1$ and ${\mathbb{C}}^n\setminus H_2$ are isomorphic. Since we can arrange that one of the hypersurfaces is singular whereas the other is smooth, we also have counterexamples in the analytic setting.