Failure of Lefschetz hyperplane theorem

TL;DR

This paper provides counterexamples demonstrating the failure of the Lefschetz hyperplane theorem and Grothendieck-Lefschetz theorem in certain non-singular quasi-projective varieties, challenging their classical applicability.

Contribution

It shows that the Lefschetz hyperplane theorem does not extend to higher degree hypersurfaces and presents simple counterexamples in quasi-projective cases.

Findings

Counterexample in projective space minus points

Failure of Grothendieck-Lefschetz theorem in quasi-projective case

Limitations of hyperplane conditions in Lefschetz theorem

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.