Vol. 7, No. 4, 2013

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Albanese varieties with modulus over a perfect field

Henrik Russell

Vol. 7 (2013), No. 4, 853–892

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.

Albanese with modulus, relative Chow group with modulus, geometric class field theory
Mathematical Subject Classification 2010
Primary: 14L10
Secondary: 11G45, 14C15
Received: 18 February 2011
Revised: 7 April 2012
Accepted: 17 May 2012
Published: 29 August 2013
Henrik Russell
Freie Universität Berlin
Mathematik und Informatik
Arnimallee 3
14195 Berlin