Let be a smooth proper variety over a perfect field of arbitrary characteristic. Let be an effective divisor on with multiplicity. We introduce an Albanese variety of of modulus 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 of modulus and show that can be viewed as a universal quotient of .
As an application we can rephrase Lang’s class field theory of function fields of varieties over finite fields in explicit terms.
Henrik Russell. "Albanese varieties with modulus over a perfect field." Algebra Number Theory 7 (4) 853 - 892, 2013. https://doi.org/10.2140/ant.2013.7.853