A higher dimensional generalization of Lichtenbaum duality in terms of the Albanese map

TL;DR

This paper proposes a conjectural formula for the cokernel of the Albanese map of zero-cycles on smooth projective varieties over p-adic fields, linking it to the Néron-Severi group, and proves it under certain assumptions.

Contribution

It introduces a conjectural description of the Albanese cokernel in terms of algebraic groups and provides a proof under specific conditions on the variety's integral model.

Findings

Conjectural formula relating Albanese cokernel to Néron-Severi group.

Proof of the formula under additional assumptions.

Finite nature of the local Albanese-cokernel group.

Abstract

We present a conjectural formula describing the cokernel of the Albanese map of zero-cycles of smooth projective varieties $X$ over $p$-adic fields in terms of the N\'eron-Severi group and provide a proof under additional assumptions on an integral model of $X$. The proof depends on a non-degeneracy result of Brauer-Manin pairing due to Saito-Sato and on Gabber-de Jong's comparison result of cohomological- and Azumaya-Brauer groups. We will also mention the local-global problem of the Albanese-cokernel; the abelian group on the "local side" turns out to be a finite group.