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

In this article, we present a conjectural formula describing the cokernel of the Albanese map of zero-cycles of smooth projective varieties over -adic fields in terms of the Néron-Severi group and provide a proof under additional assumptions on an integral model of . 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 for the Albanese cokernel; the abelian group on the 'local side' turns out to be a finite group.