Albanese varieties with modulus over a perfect field

TL;DR

This paper introduces a higher-dimensional Albanese variety with modulus for smooth proper varieties over perfect fields, generalizing classical Jacobians and connecting to class field theory.

Contribution

It defines Albanese varieties with modulus in higher dimensions, provides explicit descriptions via 1-motives, and relates them to Chow groups and class field theory.

Findings

Defined Albanese varieties with modulus for higher dimensions

Established duality and explicit descriptions of these varieties

Connected the construction to Lang's class field theory

Abstract

Let X be a smooth proper variety over a perfect field k of arbitrary characteristic. Let D be an effective divisor on X with multiplicity. We introduce an Albanese variety Alb(X, D) of X of modulus D as a higher dimensional analogon of the generalized Jacobian of Rosenlicht-Serre with modulus for smooth proper curves. Basing on duality of 1-motives with unipotent part (which are introduced here), we obtain explicit and functorial descriptions of these generalized Albanese varieties and their dual functors. We define a relative Chow group of zero cycles w.r.t. the modulus D and show that Alb(X, D) is a universal quotient of this Chow group. As an application we can rephrase Lang's class field theory of function fields of varieties over finite fields in explicit terms.