Hodge cycles and the Leray filtration

TL;DR

This paper explores the relationship between Hodge cycles, the Leray filtration, and the Hodge and Tate conjectures, providing criteria and proving these conjectures for specific classes of algebraic varieties.

Contribution

It introduces a method to pull back the Leray filtration to Chow groups and applies it to verify the Hodge and Tate conjectures for certain fibered varieties and families of abelian varieties.

Findings

Hodge and Tate conjectures hold for desingularizations of self fiber products of elliptic surfaces.

Hodge and Tate conjectures are valid for families of abelian varieties over Shimura curves.

Criteria are established linking the Leray filtration to the validity of the conjectures.

Abstract

This is loosely a continuation of the author's previous paper arXiv:1802.09496. In the first part, 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. In the second part, we show that the Hodge conjecture holds for a good desingularization of a self fibre product of a non-isotrivial elliptic surface under appropriate conditions. We also show that the Hodge and Tate conjectures hold for natural families of abelian varieties parameterized by certain Shimura curves. This uses Zucker's description of the mixed Hodge structure on the cohomology of a variation of Hodge structures on a curve, along with appropriate "vanishing" theorems.