Infinitesimal part of Weak Lefschetz using Milnor $K-$ theory

TL;DR

This paper investigates the infinitesimal aspect of the Weak Lefschetz conjecture for Chow groups of smooth projective varieties over characteristic zero fields, utilizing Milnor K-theory and Grothendieck's techniques.

Contribution

It proves the infinitesimal part of the Weak Lefschetz conjecture for Chow groups by applying Milnor K-theory and extending Grothendieck's methods.

Findings

Established the infinitesimal case of the conjecture

Connected Chow groups with Milnor K-theory via Bloch's formula

Extended classical techniques to a new setting

Abstract

Let $X$ be a smooth projective variety over a field of characteristic $0$ and $L$ an ample divisor. In this paper we study the Weak Lefschetz conjecture for Chow groups using the technique employed by Grothendieck in his study of the problem for Picard groups, and using Bloch's formula to interpret Chow groups in terms of Milnor $K$- theory we prove the infinitesimal part of the conjecture.