Homology, Homotopy and Applications

Weak Lefschetz for Chow groups: Infinitesimal lifting

D. Patel and G. V. Ravindra

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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


Export citation