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

TL;DR

This paper investigates the structure of algebraically trivial cycles on hyperplane sections of smooth projective varieties, especially cubic fourfolds, revealing the kernel of push-forward maps as unions of shifts of abelian subvarieties.

Contribution

It establishes a general description of the kernel of push-forward homomorphisms for algebraic cycles on hyperplane sections, linking it to abelian subvarieties and vanishing cycles, with applications to cubic fourfolds.

Findings

Kernel of push-forward is a union of shifts of an abelian subvariety.

For general sections, the abelian subvariety is either trivial or matches a subvariety with tangent space of vanishing cycles.

Results apply specifically to hyperplane sections of cubic fourfolds in projective space.

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 $\mathbb P^5$.