On Hard Lefschetz Conjecture on Lawson Homology

TL;DR

This paper explores the Hard Lefschetz conjecture on Lawson homology, relating it to Suslin's conjecture, and demonstrates its validity for symmetric products of low-genus curves through equivalence with Beauville-type vanishing conjectures.

Contribution

It establishes the equivalence of the Hard Lefschetz conjecture with Beauville-type vanishing conjectures for abelian varieties and symmetric products of curves, proving Suslin's conjecture in certain cases.

Findings

Hard Lefschetz conjecture is equivalent to Beauville-type vanishing conjecture for abelian varieties.

Suslin conjecture holds for symmetric products of curves with genus ≤ 2.

Connections between Lawson homology, Suslin conjecture, and Beauville-type conjectures are clarified.

Abstract

Friedlander and Mazur proposed a conjecture of hard Lefschetz type on Lawson homology. We shall relate this conjecture to Suslin conjecture on Lawson homology. For abelian varieties, this conjecture is shown to be equivalent to a vanishing conjecture of Beauville type on Lawson homology. For symmetric products of curves, we show that this conjecture amounts to the vanishing conjecture of Beauville type for the Jacobians of the corresponding curves. As a consequence, Suslin conjecture holds for all symmetric products of curves with genus at most 2.