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Commentarii Mathematici Helvetici

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Volume 94, Issue 3, 2019, pp. 439–444
DOI: 10.4171/CMH/464

Published online: 2019-09-25

Counterexamples to the complement problem

Pierre-Marie Poloni[1]

(1) Universit├Ąt Bern, Switzerland

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.

Keywords: Affine algebraic geometry, complements of hypersurfaces, Danielewski surfaces

Poloni Pierre-Marie: Counterexamples to the complement problem. Comment. Math. Helv. 94 (2019), 439-444. doi: 10.4171/CMH/464