Finiteness theorems in the class field theory of varieties over local fields

We show that the geometrical part of the abelian étale fundamental group of a proper smooth variety over a local field is finitely generated over with finite torsion, and describe its rank by the special fiber of the Néron model of the Albanese variety. As an application, we complete the class field theory of curves over local fields developed by Bloch and Saito, in which the theorem concerning the p-primary part in the positive characteristic case has remained unproven.