Curves of genus three on a general abelian threefold and the non-finite generation of the Griffiths group.

*(English)*Zbl 0703.14004
Arithmetic of complex manifolds, Proc. Conf., Erlangen/FRG 1988, Lect. Notes Math. 1399, 10-26 (1989).

[For the entire collection see Zbl 0675.00006.]

Let A be a general principally polarized abelian threefold defined over \({\mathbb{C}}\). First, the author produces in A infinitely many birationally distinct irreducible curves C of genus \(3,\) which are obtained by considering isogenies associated to some infinitely many elements of the rational symplectic group. The fact that these curves are birationally distinct cannot be extended when genus \(g>3\) [see the author and G. P. Pirola, Invent. Math. 95, No.2, 263-276 (1989; Zbl 0638.14025)].

Second, the author shows that for these curves C, the 1-cycles \(C-[-1]C\) are not algebraically equivalent to 0 in A, where \([-1]\) is the multiplication by the \(-1\) map of A. Finally by considering the monodromy action around the translates of the hyperelliptic locus in the Siegel upper half space of genus 3, the author proves that there are no relations of algebraic equivalence among these 1-cycles in A.

Let A be a general principally polarized abelian threefold defined over \({\mathbb{C}}\). First, the author produces in A infinitely many birationally distinct irreducible curves C of genus \(3,\) which are obtained by considering isogenies associated to some infinitely many elements of the rational symplectic group. The fact that these curves are birationally distinct cannot be extended when genus \(g>3\) [see the author and G. P. Pirola, Invent. Math. 95, No.2, 263-276 (1989; Zbl 0638.14025)].

Second, the author shows that for these curves C, the 1-cycles \(C-[-1]C\) are not algebraically equivalent to 0 in A, where \([-1]\) is the multiplication by the \(-1\) map of A. Finally by considering the monodromy action around the translates of the hyperelliptic locus in the Siegel upper half space of genus 3, the author proves that there are no relations of algebraic equivalence among these 1-cycles in A.

Reviewer: H.Maeda

##### MSC:

14C15 | (Equivariant) Chow groups and rings; motives |

14H45 | Special algebraic curves and curves of low genus |

14J30 | \(3\)-folds |

14H40 | Jacobians, Prym varieties |

14K02 | Isogeny |

14C25 | Algebraic cycles |