Counterexamples to the Complement Problem

2016/05/17 by Pierre-Marie Poloni, Poloni, Pierre-Marie
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.1605.05169

to appear in Commentarii Mathematici Helvetici

arxiv created 2017/02/10 · arxiv updated 2017/02/13

Abstract

We provide explicit counterexamples to the so-called Complement Problem in every dimension n≥3, i.e. pairs of non-isomorphic irreducible hypersurfaces H₁, H₂⊂ℂⁿ whose complements ℂⁿ∖ H₁ and ℂⁿ∖ H₂ 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.

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