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

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.