Hard Lefschetz for Chow groups of generalized Kummer varieties

TL;DR

This paper establishes a hard Lefschetz theorem for the Chow groups of generalized Kummer varieties and Hilbert schemes of abelian surfaces, providing new insights into their algebraic cycles using advanced decomposition techniques.

Contribution

It introduces a new hard Lefschetz result for Chow groups of these varieties, extending previous work with novel applications and methods.

Findings

Hard Lefschetz theorem proven for Chow groups of generalized Kummer varieties.

Results apply to Chow groups of Hilbert schemes of abelian surfaces.

Provides new structural information about algebraic cycles in these varieties.

Abstract

The main result of this note is a hard Lefschetz theorem for the Chow groups of generalized Kummer varieties. The same argument also proves hard Lefschetz for Chow groups of Hilbert schemes of abelian surfaces. As a consequence, we obtain new information about certain pieces of the Chow groups of generalized Kummer varieties, and Hilbert schemes of abelian surfaces. The proofs are based on work of Shen-Vial and Fu-Tian-Vial on multiplicative Chow-K\"unneth decompositions.