Grothendieck--Lefschetz for ample subvarieties

TL;DR

This paper proves a Grothendieck--Lefschetz theorem for smooth ample subvarieties and related cases, extending classical results to broader contexts with applications in algebraic geometry.

Contribution

It establishes a new Grothendieck--Lefschetz theorem for ample subvarieties and generalizes it to cases with small cohomological dimension, including in all characteristics.

Findings

Proved Grothendieck--Lefschetz theorem for smooth ample subvarieties.

Extended results to subvarieties with small cohomological complement.

Provided several applications in algebraic geometry.

Abstract

We establish a Grothendieck--Lefschetz theorem for smooth ample subvarieties of smooth projective varieties over an algebraically closed field of characteristic zero and, more generally, for smooth subvarieties whose complement has small cohomological dimension. A weaker statement is also proved in a more general context and in all characteristics. Several applications are included.