Griffiths groups of supersingular abelian varieties

B. Brent Gordon, Kirti N Joshi

Research output: Contribution to journalArticle

Abstract

The Griffiths group Grr(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) of a supersingular abelian variety A 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 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) ⊗ ℤ ≠ 0.

Original languageEnglish (US)
Pages (from-to)213-219
Number of pages7
JournalCanadian Mathematical Bulletin
Volume45
Issue number2
StatePublished - Jun 2002
Externally publishedYes

Fingerprint

Abelian Variety
Algebraic Cycles
Threefolds
Trivial
Closure
Fermat
Projective Variety
Codimension
Torsion
Galois field
Modulo
Corollary
Subgroup
Valid

Keywords

  • Beauville conjecture
  • Chow group
  • Griffiths group
  • Supersingular Abelian variety

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Griffiths groups of supersingular abelian varieties. / Brent Gordon, B.; Joshi, Kirti N.

In: Canadian Mathematical Bulletin, Vol. 45, No. 2, 06.2002, p. 213-219.

Research output: Contribution to journalArticle

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