Étale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$



Banerjee, Kalyan and Guletskii, Vladimir
Étale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$.

[img] Text
generic_hyperplane_6.pdf - Submitted Version

Download (522kB)

Abstract

Let $k$ be an uncountable algebraically closed field of characteristic $0$, and let $X$ be a smooth projective connected variety of dimension $2p$, appropriately embedded into $\mathbb P^m$ over $k$. Let $Y$ be a hyperplane section of $X$, and let $A^p(Y)$ and $A^{p+1}(X)$ be the groups of algebraically trivial algebraic cycles of codimension $p$ and $p+1$ modulo rational equivalence on $Y$ and $X$ respectively. Assume that, whenever $Y$ is smooth, the group $A^p(Y)$ is regularly parametrized by an abelian variety $A$ and coincides with the subgroup of degree $0$ classes in the Chow group $CH^p(Y)$. In the paper we prove that the kernel of the push-forward homomorphism from $A^p(Y)$ to $A^{p+1}(X)$ is the union of a countable collection of shifts of a certain abelian subvariety $A_0$ inside $A$. For a very general section $Y$ either $A_0=0$ or $A_0$ coincides with an abelian subvariety $A_1$ in $A$ whose tangent space is the group of vanishing cycles $H^{2p-1}(Y)_{\rm van}$. Then we apply these general results to sections of a smooth cubic fourfold in $\mathbb P^5$.

Item Type: Article
Additional Information: 38 pages
Uncontrolled Keywords: math.AG, math.AG, 14C25, 14D05, 14F30, 14J30, 14J35
Depositing User: Symplectic Admin
Date Deposited: 05 Dec 2018 09:43
Last Modified: 29 May 2019 07:25
Related URLs:
URI: http://livrepository.liverpool.ac.uk/id/eprint/3029507
Repository Staff Access