Albanese varieties with modulus over a perfect field

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 $Alb\left(X,D\right)$ of $X$ of modulus $D$ as a higher-dimensional analogue of the generalized Jacobian of Rosenlicht and 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 ${CH}_0\left(X,D\right)$ of modulus $D$ and show that $Alb\left(X,D\right)$ can be viewed as a universal quotient of ${CH}_0{\left(X,D\right)}^0$.

As an application we can rephrase Lang’s class field theory of function fields of varieties over finite fields in explicit terms.