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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2506.22980 |
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| _version_ | 1866908683068243968 |
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| author | Zhang, Hui |
| author_facet | Zhang, Hui |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22980 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Abelianized Descent Obstruction for 0-Cycles Zhang, Hui Algebraic Geometry 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. |
| title | Abelianized Descent Obstruction for 0-Cycles |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2506.22980 |