A brief note concerning hard Lefschetz for Chow groups

TL;DR

This paper proposes a conjectural hard Lefschetz property for Chow groups, proves it in specific cases involving finite-dimensional motives and Griffiths groups, and discusses related results from Vial.

Contribution

It introduces a conjectural Lefschetz property for Chow groups and proves it for certain classes of varieties, advancing understanding of algebraic cycles.

Findings

Hard Lefschetz property formulated for Chow groups

Proved in cases with finite-dimensional motive

Established for varieties with vanishing middle Griffiths group

Abstract

We formulate a conjectural hard Lefschetz property for Chow groups, and prove this in some special cases: roughly speaking, for varieties with finite-dimensional motive, and for varieties whose self-product has vanishing middle-dimensional Griffiths group. An appendix includes related statements that follow from results of Vial.