Hodge cycles and the Leray filtration

Donu Arapura

Vol. 319 (2022), No. 2, 233–258

DOI: 10.2140/pjm.2022.319.233

Abstract

Given a fibered variety, we pull back the Leray filtration to the Chow group, and use this to give some criteria for the Hodge and Tate conjectures to hold for such varieties. We show that the Hodge conjecture holds for a good desingularization of a self fibre product of a nonisotrivial elliptic surface under appropriate conditions. We also show that the Hodge and Tate conjectures hold for natural families of abelian varieties parametrized by certain Shimura curves.

Keywords

Hodge cycles, elliptic surfaces, Shimura curves

Mathematical Subject Classification

Primary: 14C30

Milestones

Received: 4 July 2021

Revised: 13 June 2022

Accepted: 25 June 2022

Published: 11 September 2022