Counterexamples to the complement problem (Q2337227)

Summary: We provide explicit counterexamples to the so-called Complement Problem in every dimension \(n\geq 3\), 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.