We prove that torsion codimension $2$ algebraic cycles modulo rational equivalence on supersingular abelian varieties are algebraically equivalent to zero. As a consequence, we prove that homological equivalence coincides with algebraic equivalence for algebraic cycles of codimension $2$ on supersingular abelian varieties over the algebraic closure of finite fields.

Based on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence.

We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological K0-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory.

We compute our cobordism theory for some low dimensional varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordism theory.

The Griffiths group $\text{G}{{\text{r}}^{r}}\left( X \right)$ of a smooth projective variety $X$ over an algebraically closed 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 $\text{G}{{\text{r}}^{2}}\left( {{A}_{{\bar{k}}}} \right)$ of a supersingular abelian variety ${{A}_{{\bar{k}}}}$ 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 $\text{C}$. Schoen it is also shown that if the Tate 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 ${{A}_{{\bar{k}}}}$ over the algebraic closure of $k$ is non-trivial; in fact, for all but a finite number of primes $\ell \,\ne \,p$ it is the case that $\text{G}{{\text{r}}^{2}}\left( {{A}_{{\bar{k}}}} \right)\,\otimes \,{{\mathbb{Z}}_{\ell }}\,\ne \,0$.