Griffiths groups of supersingular abelian varieties

Details Griffiths groups of supersingular abelian varieties Griffiths groups of supersingular abelian varieties

2001-10-05

arxiv.org/abs/math/0110067v2

Mathematics

Algebraic Geometry

AMSLaTeX; to appear in Canadian Math. Bull. only abstract replaced (author names in wrong order in the previous abstract)

Scientific paper

The Griffiths group $\Gr^r(X)$ 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 $\Gr^2(A_\kbar)$ of a supersingular abelian variety $A_\kbar$ 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 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 $A_\kbar$ over the algebraic closure of $k$ is non-trivial.

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