Lefschetz for local Picard groups

TL;DR

This paper proves a strengthened version of the Grothendieck-Lefschetz theorem for local Picard groups, using acyclicity results, and extends to a restriction theorem for higher rank bundles in positive characteristic.

Contribution

It establishes a new, stronger Lefschetz-type theorem for local Picard groups and introduces a restriction theorem for higher rank bundles in positive characteristic.

Findings

Proved a strengthened Lefschetz theorem for local Picard groups.

Developed a restriction theorem for higher rank bundles in positive characteristic.

Utilized acyclicity results for absolute integral closures.

Abstract

We prove a strengthening of the Grothendieck-Lefschetz hyperplane theorem for local Picard groups conjectured by Koll\'ar. Our approach, which relies on acyclicity results for absolute integral closures, also leads to a restriction theorem for higher rank bundles on projective varieties in positive characteristic.