Abelianized Descent Obstruction for 0-Cycles

Classical descent theory of Colliot-Thélène 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.