Counterexamples to the Complement Problem

TL;DR

This paper constructs explicit counterexamples in all dimensions greater than or equal to three, showing that non-isomorphic hypersurfaces can have isomorphic complements, challenging assumptions in algebraic geometry.

Contribution

It provides the first explicit counterexamples to the Complement Problem in all dimensions n≥3, including cases with singular and smooth hypersurfaces.

Findings

Counterexamples exist in every dimension n≥3

Hypersurfaces with isomorphic complements can be non-isomorphic

Counterexamples include both singular and smooth hypersurfaces

Abstract

We provide explicit counterexamples to the so-called Complement Problem in every dimension $n\geq3$, i.e. pairs of non-isomorphic irreducible 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.