# Counterexamples to the complement problem

### Pierre-Marie Poloni

Universität Bern, Switzerland

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## Abstract

We provide explicit counterexamples to the so-called Complement Problem in every dimension $n\geq3$, 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.

## Cite this article

Pierre-Marie Poloni, Counterexamples to the complement problem. Comment. Math. Helv. 94 (2019), no. 3, pp. 439–444

DOI 10.4171/CMH/464