Lawson Homology for Abelian Varieties

TL;DR

This paper develops a Fourier-Mukai transform for Lawson homology on abelian varieties, establishing decomposition theorems, proposing conjectures, and extending duality results to deepen understanding of Lawson homology and its relation to Chow groups.

Contribution

It introduces a Fourier-Mukai transform for Lawson homology, proves an inversion theorem, and proposes conjectures linking Lawson homology with Chow theory for abelian varieties.

Findings

Decomposition of Lawson homology groups with rational coefficients

An analogue of Beauville's conjecture for Lawson homology

A refined Friedlander-Lawson duality theorem for abelian varieties

Abstract

In this paper we introduce the Fourier-Mukai transform for Lawson homology of abelian varieties and prove an inversion theorem for the Lawson homology as well as the morphic cohomology of abelian varieties. As applications, we obtain the direct sum decomposition of the Lawson homology and the morphic cohomology groups with rational coefficients, inspired by Beauville's works on the Chow theory. An analogue of the Beauville conjecture for Chow groups is proposed and is shown to be equivalent to the (weak) Suslin conjecture for Lawson homology. A filtration on Lawson homology is proposed and conjecturally it coincides to the filtration given by the direct sum decomposition of Lawson homology for abelian varieties. Moreover, a refined Friedlander-Lawson duality theorem is obtained for abelian varieties. We summarize several related conjectures in Lawson homology theory in the appendix for convenience.