TITLE: Algebraic cycles of dimension 2 on the square of a generic abelian surface
QUESTION [9 upvotes]: I would like to know, what is known on algebraic cycles of dimension 2
modulo algebraic or rational equivalence
on the square of a generic abelian surface.
First, let $A$ be a generic abelian surface (generic abelian variety of dimension 2)
over $\mathbb{C}$.
Then the group of cycles of dimension 1 (divisors)
up to rational equivalence is known, it its the Picard group of $A$,
see e.g. the answers to this question
and Fulton, Intersection Theory, Chapter 19.
Still I have a stupid question: Can one "write down" all the (positive?) divisors on $A$?
The real question concerns the square $A^2=A\times_{\mathbb{C}} A$ of a generic abelian surface $A$.
This is an abelian variety of dimension 4.
I am interested in algebraic cycles of dimension 2 on $A^2$.
I think I know the group of algebraic cycles of dimension 2 on $A^2$
modulo homological equivalence and modulo torsion, it is  $\mathbb{Z}^6$
(because the space of invariants of $\mathrm{Sp}_{4,\mathbb{Q}}$
in $\wedge^4(\mathbb{Q}^4\oplus\mathbb{Q}^4)$ is of dimension 6).
What is known about the group of algebraic cycles of dimension 2 on $A^2$ modulo rational or algebraic equivalence?
In particular, what is known about the Griffiths group?
Again a stupid question: Can one "write down" all the cycles of dimension 2 on $A^2$ (in some sense)?

REPLY [2 votes]: Not an answer per se, but you might be interested in http://arxiv.org/abs/1003.3183, where similar questions are investigated.