CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

Griffiths Groups of Supersingular Abelian Varieties

  Published:2002-06-01
 Printed: Jun 2002
  • B. Brent Gordon
  • Kirti Joshi
Format:   HTML   LaTeX   MathJax   PDF   PostScript  

Abstract

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 $\Gr^2 (A_{\bar{k}})$ 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 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 $\Gr^2 (A_{\bar{k}}) \otimes \Z_\ell \neq 0$.
Keywords: Griffiths group, Beauville conjecture, supersingular Abelian variety, Chow group Griffiths group, Beauville conjecture, supersingular Abelian variety, Chow group
MSC Classifications: 14J20, 14C25 show english descriptions Arithmetic ground fields [See also 11Dxx, 11G25, 11G35, 14Gxx]
Algebraic cycles
14J20 - Arithmetic ground fields [See also 11Dxx, 11G25, 11G35, 14Gxx]
14C25 - Algebraic cycles
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/