Algebraic cycles on the generic abelian fourfold with polarization of type (1,2,2,2)

TL;DR

This paper constructs a non-trivial element in the higher Griffiths group for a generic abelian fourfold with a specific polarization, using its realization as a generalized Prym variety to identify certain algebraic cycles.

Contribution

It introduces a novel construction of algebraic cycles in the Griffiths group for a specific class of abelian fourfolds, leveraging their Prym variety structure.

Findings

Identifies a non-trivial element in Griff^{3,2} for the given abelian fourfold.

Demonstrates the use of Prym varieties to study algebraic cycles.

Provides new insights into the structure of algebraic cycles in higher-dimensional abelian varieties.

Abstract

In this paper we construct a non-trivial element in the higher Griff{}iths group $Griff ^{3,2}$ for the generic abelian fourfold $A^4$ with polarization of type $(1,2,2,2)$. The key idea is to use that $A^4$ can be realized as a generalized Prym variety and for this reason contains in a natural way some curves i.e. dimension 1 cycles.