Griffiths groups of supersingular abelian varieties

The Griffiths group Gr^r(X) of a smooth projective variety X over an algebraically dosed field is defined to be the group of homologically trivial algebraic cycles of codimension r on X modulo the subgroup of algebraically trivial algebraic cycles. The main result of this paper is that the Griffiths group Gr² (A_k̄) of a supersingular abelian variety A_k̄ over the algebraic closure of a finite field of characteristic p is at most a p-primary torsion group. As a corollary the same conclusion holds for supersingular Fermat threefolds. In contrast, using methods of C. Schoen it is also shown that if the Tare conjecture is valid for all smooth projective surfaces and all finite extensions of the finite ground field k of characteristic p > 2, then the Griffiths group of any ordinary abelian threefold A_k̄ over the algebraic closure of k is non-trivial; in fact, for all but a finite number of primes ℓ ≠ p it is the case that Gr² (A_k̄) ⊗ ℤ_ℓ ≠ 0.