### Abstract

The Griffiths group Gr^{r}(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 Gr^{2} (A_{k̄}) of a supersingular abelian variety A_{k̄} 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_{k̄} 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 Gr^{2} (A_{k̄}) ⊗ ℤ_{ℓ} ≠ 0.

Original language | English (US) |
---|---|

Pages (from-to) | 213-219 |

Number of pages | 7 |

Journal | Canadian Mathematical Bulletin |

Volume | 45 |

Issue number | 2 |

State | Published - Jun 2002 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Canadian Mathematical Bulletin*,

*45*(2), 213-219.

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

Research output: Contribution to journal › Article

*Canadian Mathematical Bulletin*, vol. 45, no. 2, pp. 213-219.

}

TY - JOUR

T1 - Griffiths groups of supersingular abelian varieties

AU - Brent Gordon, B.

AU - Joshi, Kirti N

PY - 2002/6

Y1 - 2002/6

N2 - 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 (Ak̄) of a supersingular abelian variety Ak̄ 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 Ak̄ 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 (Ak̄) ⊗ ℤℓ ≠ 0.

AB - 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 (Ak̄) of a supersingular abelian variety Ak̄ 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 Ak̄ 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 (Ak̄) ⊗ ℤℓ ≠ 0.

KW - Beauville conjecture

KW - Chow group

KW - Griffiths group

KW - Supersingular Abelian variety

UR - http://www.scopus.com/inward/record.url?scp=0036623236&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036623236&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0036623236

VL - 45

SP - 213

EP - 219

JO - Canadian Mathematical Bulletin

JF - Canadian Mathematical Bulletin

SN - 0008-4395

IS - 2

ER -