Griffiths Groups of Supersingular Abelian Varieties

Gordon, B. Brent ; Joshi, Kirti

Canadian mathematical bulletin, Tome 45 (2002) no. 2, pp. 213-219

Résumé

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 $\text{G}{{\text{r}}^{2}}\left( {{A}_{{\bar{k}}}} \right)\,\otimes \,{{\mathbb{Z}}_{\ell }}\,\ne \,0$ .

Détail

DOI : 10.4153/CMB-2002-024-2

Mots-clés : 14J20, 14C25, Griffiths group, Beauville conjecture, supersingular Abelian variety, Chow group