Albanese varieties with modulus over a perfect field

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 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 CH0(X, D) of modulus D and show that Alb(X, D) can be viewed as a universal quotient of CH0(X, D)(0). As an application we can rephrase Lang's class field theory of function fields of varieties over finite fields in explicit terms.