Counterexamples to the Complement Problem

Abstract

We provide explicit counterexamples to the so-called Complement Problem in every dimension $n\geq3$, i.e. pairs of non-isomorphic irreducible 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.