Albanese kernels and Griffiths groups

TL;DR

This paper links the Griffiths group of a product of a curve and a surface to the Albanese kernel of the surface, proving nonvanishing results for certain modifications of embeddings in a Lefschetz pencil over characteristic zero fields.

Contribution

It establishes a new description of the Griffiths group for products of curves and surfaces and proves nonvanishing of specific elements in the Griffiths group for infinitely many pencil members.

Findings

Griffiths group of C×S is a quotient of the Albanese kernel of S over a function field.

Nonvanishing of modified graph embeddings in Griffiths group for infinitely many pencil members.

Results hold over characteristic zero fields with certain geometric conditions.

Abstract

We describe the Griffiths group of the product of a curve $C$ and a surface $S$ as a quotient of the Albanese kernel of $S$ over the function field of $C$. When $C$ is a hyperplane section of $S$ varying in a Lefschetz pencil, we prove the nonvanishing in $\text{Griff}(C\times S)$ of a modification of the graph of the embedding $C\hookrightarrow S$ for infinitely many members of the pencil, provided the ground field $k$ is of characteristic $0$, the geometric genus of $S$ is $>0$, and $k$ is large or $S$ is "of motivated abelian type".