Some results about zero-cycles on abelian and semi-abelian varieties

TL;DR

This paper extends results on zero-cycles on abelian and semi-abelian varieties, showing structural decompositions of Albanese kernels and constructing filtrations related to Suslin's homology and Somekawa K-groups.

Contribution

It proves the Albanese kernel decomposes into divisible and torsion parts for certain abelian varieties and constructs a filtration of Suslin's homology for semi-abelian varieties linked to Somekawa K-groups.

Findings

Albanese kernel decomposes into divisible and torsion parts for abelian varieties with good ordinary reduction.

Constructs a filtration of Suslin's homology with quotients isomorphic to Somekawa K-groups.

Extends previous results to broader classes of varieties and fields.

Abstract

In this short note we extend some results obtained in \cite{Gazaki2015}. First, we prove that for an abelian variety $A$ with good ordinary reduction over a finite extension of $\mathbb{Q}_p$ with $p$ an odd prime, the Albanese kernel of $A$ is the direct sum of its maximal divisible subgroup and a torsion group. Second, for a semi-abelian variety $G$ over a perfect field $k$, we construct a decreasing integral filtration $\{F^r\}_{r\geq 0}$ of Suslin's singular homology group, $H_0^{sing}(G)$, such that the successive quotients are isomorphic to a certain Somekawa K-group.