Albanese and Picard 1-Motives in Positive Characteristic

TL;DR

This paper constructs 1-motives in positive characteristic that realize etale cohomology groups of varieties, extending previous work by incorporating compactly supported cohomology and addressing the more complex Albanese case.

Contribution

It introduces the definition of Albanese and Picard 1-motives in positive characteristic, utilizing advanced techniques like alterations and cycle class theory, and establishes independence of l results over finite fields.

Findings

Defined Picard 1-motives using standard techniques.

Extended the concept to Albanese 1-motives with new tools.

Proved independence of l results for varieties over finite fields.

Abstract

We define 1-motives of a variety X over a perfect field of positive characteristic which realize the etale cohomology groups of X in dimension and codimension one. This is the analogue in positive characteristic of previous results of Barbieri-Viale and Srinivas, except that we only consider the etale realization but also consider compactly supported cohomology. The dimension-1 case (called the Picard 1-motives) can be done by standard techniques, and indeed this case is probably well known. But the codimension-one case (Albanese 1-motive) requires stronger tools, namely a strong version of de Jong's alterations theorem and some cycle class theory on smooth Deligne-Mumford stacks which may be of independent interest. Unfortunately, we only succeed in defining the Albanese 1-motive for a variety X over an algebraically closed base field, and only up to isogeny. As a corollary to our definition of these 1-motives we deduce some independence of l results when X is a variety over a finite field.