On the Albanese map for smooth quasi-projective varieties

TL;DR

This paper generalizes Roitman's theorem to smooth quasi-projective varieties by replacing the Chow group with Suslin's algebraic singular homology and using the generalized Albanese, revealing the motivic nature of the Albanese map.

Contribution

It extends Roitman's theorem to quasi-projective varieties using Suslin's homology and introduces a new proof method that clarifies the motivic aspects of the Albanese map.

Findings

The generalized Albanese map is an isomorphism over algebraic closures of finite fields.

The proof method is new and makes the motivic nature of the Albanese transparent.

The result applies to varieties admitting a smooth compactification.

Abstract

Let k be an algebraically closed field and X a smooth projective k-variety. A famous theorem of A. A. Roitman states that the canonical map from the degree zero part of the Chow group of zero cycles on X to the group of k-points of its Albanese variety induces an isomorphism on torsion prime to the characteristic of k. In the present paper we prove a generalisation to quasi-projective varieties admitting a smooth compactification. As was first observed by Ramachandran, for such a generalisation one should replace the Chow group of zero cycles by Suslin's 0-th algebraic singular homology group and the Albanese variety by the generalised Albanese of Serre. The method of proof is new even in the projective case and makes the motivic nature of the Albanese transparent. We also prove that the generalised Albanese map is an isomorphism if k is the algebraic closure of a finite field.