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Transactions of the American Mathematical Society

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Comparing descent obstruction and Brauer-Manin obstruction for open varieties


Authors: Yang Cao, Cyril Demarche and Fei Xu
Journal: Trans. Amer. Math. Soc. 371 (2019), 8625-8650
MSC (2010): Primary 14G05
DOI: https://doi.org/10.1090/tran/7567
Published electronically: March 7, 2019
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Abstract: We provide a relation between Brauer-Manin obstruction and descent obstruction for torsors over not necessarily proper varieties under a connected linear algebraic group or a group of multiplicative type. Such a relation is also refined for torsors under a torus. The equivalence between descent obstruction and étale Brauer-Manin obstruction for smooth projective varieties is extended to smooth quasi-projective varieties, which provides the perspective to study integral points.


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Additional Information

Yang Cao
Affiliation: Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
Address at time of publication: Department of Mathematics and Physics, Leibniz University Hannover, Germany
Email: yangcao1988@gmail.com

Cyril Demarche
Affiliation: Sorbonne Universités, UPMC Université Paris 06, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Université Paris Diderot, Sorbonne Paris Cité, F-75005, Paris, France – and – Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France
Email: cyril.demarche@imj-prg.fr

Fei Xu
Affiliation: School of Mathematical Sciences, Capital Normal University, 105 Xisanhuanbeilu, 100048 Beijing, People’s Republic of China
Email: xufei@math.ac.cn

DOI: https://doi.org/10.1090/tran/7567
Keywords: Linear algebraic group, torsor, descent obstruction, Brauer-Manin obstruction
Received by editor(s): November 18, 2016
Received by editor(s) in revised form: July 29, 2017, and November 28, 2017
Published electronically: March 7, 2019
Additional Notes: The first named author acknowledges the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-12-BL01-0005.
The second named author acknowledges the support of the French Agence Nationale de la Recherche (ANR) under references ANR-12-BL01-0005 and ANR-15-CE40-0002-01.
The third named author acknowledges the support of NSFC grants no. 11471219 and 11631009.
Article copyright: © Copyright 2019 American Mathematical Society