Abelianized Descent Obstruction for 0-Cycles

TL;DR

This paper introduces abelianized descent obstructions for 0-cycles on algebraic varieties, establishing their equivalence with the Brauer--Manin obstruction and connecting them to existing descent obstructions in specific geometric contexts.

Contribution

It defines abelianized descent obstructions for 0-cycles using Borovoi's abelian cohomology and proves their equivalence to the Brauer--Manin obstruction, extending classical descent theory.

Findings

Proves the equality between Brauer--Manin and abelianized descent obstructions for 0-cycles.

Shows the abelianized descent obstruction is the closure of previous descent obstructions in certain varieties.

Extends classical descent theory to the setting of 0-cycles using abelian cohomology.

Abstract

Classical descent theory of Colliot-Th\'el\`ene and Sansuc for rational points tells that, over a smooth variety $X$, the algebraic Brauer--Manin subset equals the descent obstruction subset defined by a universal torsor. Moreover, Harari shows that the Brauer--Manin subset equals the descent obstruction subset defined by torsors under connected linear groups. By using the abelian cohomology theory by Borovoi, we define abelianized descent obstructions for 0-cycles by torsors under connected linear groups. As an analogy, we show the equality between the Brauer--Manin obstruction and the abelianized descent obstruction for 0-cycles. We also show that the abelianized descent obstruction is the closure of the descent obstruction defined by Balestrieri and Berg when $X$ is a projective rationally connected variety or a projective K3 surface.