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arXiv · 2506.22980

Abelianized Descent Obstruction for 0-Cycles

Hui Zhang
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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.

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Hui Zhang. 2025-06-28. Abelianized Descent Obstruction for 0-Cycles. https://arxiv.org/abs/2506.22980

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