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arXiv · 2301.04940

Failure of Lefschetz hyperplane theorem

Ananyo Dan
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Abstract

In this article, we give a counterexample to the Lefschetz hyperplane theorem for non-singular quasi-projective varieties. A classical result of Hamm-L\^{e} shows that Lefschetz hyperplane theorem can hold for hyperplanes in general position. We observe that the condition of ``hyperplane'' is strict in the sense that it is not possible to replace it by higher degree hypersurfaces. The counterexample is very simple: projective space minus finitely many points. Moreover, as an intermediate step we prove that the Grothendieck-Lefschetz theorem also fails in the quasi-projective case.

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Ananyo Dan. 2023-01-12. Failure of Lefschetz hyperplane theorem. https://arxiv.org/abs/2301.04940

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