JournalscmhVol. 94, No. 3pp. 439–444

Counterexamples to the complement problem

  • Pierre-Marie Poloni

    Universität Bern, Switzerland
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We provide explicit counterexamples to the so-called Complement Problem in every dimension n3n\geq3, i.e. pairs of nonisomorphic irreducible algebraic hypersurfaces H1,H2CnH_1, H_2\subset\mathbb C^{n} whose complements CnH1\mathbb C^{n}\setminus H_1 and CnH2\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