Albanese varieties with modulus over a perfect field

Russell, Henrik

doi:10.2140/ant.2013.7.853

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Albanese varieties with modulus over a perfect field

Henrik Russell

Vol. 7 (2013), No. 4, 853–892

DOI: 10.2140/ant.2013.7.853

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 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 ${CH}_{0} (X, D)$ of modulus $D$ and show that $Alb (X, D)$ can be viewed as a universal quotient of ${CH}_{0} {(X, D)}^{0}$ .

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

Keywords

Albanese with modulus, relative Chow group with modulus, geometric class field theory

Mathematical Subject Classification 2010

Primary: 14L10

Secondary: 11G45, 14C15

Milestones

Received: 18 February 2011

Revised: 7 April 2012

Accepted: 17 May 2012

Published: 29 August 2013

Authors

Henrik Russell
Freie Universität Berlin Mathematik und Informatik Arnimallee 3 14195 Berlin Germany

Vol. 7, No. 4, 2013