## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 16, Number 2 (2014), 65-84.

### Weak Lefschetz for Chow groups: Infinitesimal lifting

#### Abstract

Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic zero, and let $Y \subset X$ be a smooth ample hyperplane section. The Weak Lefschetz conjecture for Chow groups states that the natural restriction map $\mathrm{CH}^p (X)_{\mathbb{Q}} \to \mathrm{CH}^p (Y)_{\mathbb{Q}}$ is an isomorphism for all $p \lt \dim (Y) / 2$. In this note, we revisit a strategy introduced by Grothendieck to attack this problem by using the Bloch-Quillen formula to factor this morphism through a continuous $\mathrm{K}$-cohomology group on the formal completion of $X$ along $Y$. This splits the conjecture into two smaller conjectures: one consisting of an algebraization problem and the other dealing with infinitesimal liftings of algebraic cycles. We give a complete proof of the infinitesimal part of the conjecture.

#### Article information

**Source**

Homology Homotopy Appl., Volume 16, Number 2 (2014), 65-84.

**Dates**

First available in Project Euclid: 22 August 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1408712335

**Mathematical Reviews number (MathSciNet)**

MR3234501

**Zentralblatt MATH identifier**

1360.14023

**Subjects**

Primary: 14C25: Algebraic cycles 14C35: Applications of methods of algebraic $K$-theory [See also 19Exx]

**Keywords**

K-theory algebraic cycles

#### Citation

Patel, D.; Ravindra, G. V. Weak Lefschetz for Chow groups: Infinitesimal lifting. Homology Homotopy Appl. 16 (2014), no. 2, 65--84. https://projecteuclid.org/euclid.hha/1408712335