[Paper Review] On the Albanese map for smooth quasi-projective varieties

M. Spieß, Tamás Szamuely|ArXiv.org|Sep 1, 2000

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper generalizes Roitman's theorem on the Albanese map to smooth quasi-projective varieties with a smooth compactification, proving that the map from degree-zero algebraic singular homology to the generalized Albanese variety induces an isomorphism on prime-to-p torsion subgroups. It extends Kato-Saito's result over finite fields and uses motivic homotopy theory and duality in Voevodsky's category to establish the isomorphism via comparison of singular and étale cohomology groups.

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.

Motivation & Objective

• To extend Roitman's theorem on the Albanese map from proper varieties to smooth quasi-projective varieties with a smooth compactification.
• To establish the bijectivity of the Albanese map on torsion subgroups in the non-proper case using generalized Albanese varieties.
• To generalize Kato-Saito's result over finite fields to the quasi-projective setting, proving the Albanese map is an isomorphism of torsion groups of finite corank.
• To provide a conceptual proof using Voevodsky's triangulated category of effective motivic complexes and duality theorems.

Proposed method

• Uses Suslin's algebraic singular homology groups $h_0(X)^0$ as the domain of the Albanese map, which generalizes $CH_0(X)^0$ in the proper case.
• Interprets the generalized Albanese variety as an object in Voevodsky's category $DM^{eff}_{-}(k)$, leveraging its functoriality and homotopy invariance.
• Applies the fundamental isomorphism between $h^1(X, ζ/nζ)$ and $H^1_{ét}(X, ζ/nζ)$ from Suslin and Voevodsky to relate singular and étale cohomology.
• Uses duality between the generalized Albanese and Picard varieties to establish the structure of the target group.
• Employs tamely ramified class field theory and Galois coinvariants to analyze the structure of the Albanese group over finite fields.
• Reduces the problem to curves via hyperplane section arguments and uses known results on curve Albanese maps to deduce the general case.

Experimental results

Research questions

• RQ1 Does the Albanese map from $h_0(X)^0$ to $Alb_X(k)$ induce an isomorphism on prime-to-p torsion subgroups for smooth quasi-projective varieties with a smooth compactification?
• RQ2 Can Kato-Saito's bijectivity result for the Albanese map over algebraically closed fields of positive characteristic be extended to the non-proper case?
• RQ3 Is the generalized Albanese map compatible with the reciprocity law in tamely ramified class field theory for varieties over finite fields?
• RQ4 Can the Albanese map be interpreted as a morphism in Voevodsky's triangulated category of effective motivic complexes?

Key findings

• The Albanese map $alb_X: h_0(X)^0 o Alb_X(k)$ induces an isomorphism on prime-to-p torsion subgroups for any smooth quasi-projective variety $X$ over an algebraically closed field $k$ with a smooth compactification.
• Over algebraically closed fields of positive characteristic (the algebraic closure of a finite field), the map $alb_X$ is an isomorphism of torsion groups of finite corank.
• The proof relies on identifying the Albanese map as a morphism in Voevodsky's category $DM^{eff}_{-}(k)$, enabling the use of motivic duality and cohomological comparison theorems.
• The key technical step is the isomorphism between $h^1(X, ζ/nζ)$ and $H^1_{ét}(X, ζ/nζ)$, which allows the transfer of cohomological information to the Albanese group.
• The result generalizes Roitman's theorem to the quasi-projective setting and extends Kato-Saito's result beyond the proper case.
• A by-product is that the natural map $h_0(X_{ε}) o h_0(X)$ has a finite kernel isomorphic to the group $T$ from the reciprocity sequence, confirming a structural property of the homology group.