Homology, Homotopy and Applications

Volume 16 (2014)

Number 2

Weak Lefschetz for Chow groups: Infinitesimal lifting

Pages: 65 – 84

DOI: http://dx.doi.org/10.4310/HHA.2014.v16.n2.a4

Authors

D. Patel (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

G. V. Ravindra (Department of Mathematics and Computer Science, University of Missouri, St. Louis, Missouri, U.S.A.)

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.

Keywords

$K$-theory, algebraic cycles

2010 Mathematics Subject Classification

14C25, 14C35

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