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Descent obstruction is equivalent to étale Brauer-Manin obstruction. (English) Zbl 1180.14017
The author completes the proof of the result stated in the title.
More precisely, for a smooth projective geometrically integral $$k$$-variety $$X$$ defined over a number field $$k$$ and a torsor $$f: Y\to X$$ under a linear $$k$$-group $$G$$ denote $X(\mathbb A_k)^f = \bigcup_{[\sigma ]\in H^1(k,G)} f^{\sigma }(Y^{\sigma }(\mathbb A_k)),$ where $$f^{\sigma}: Y^{\sigma }\to X$$ is the twist of $$f$$ by the 1-cocycle $$\sigma$$ and $$\mathbb A_k$$ stands for the ring of adèles of $$k$$. Then the descent obstruction $$X(\mathbb A_k)^{\text{{desc}}}$$ is defined as the intersection of $$X(\mathbb A_k)^f$$ where $$f$$ ranges over all torsors under all linear $$k$$-groups.
The author’s goal is to compare $$X(\mathbb A_k)^{\text{{desc}}}$$ with the étale Brauer–Manin obstruction, defined as $X(\mathbb A_k)^{\text{{ét, Br}}} = \bigcap_f\bigcup_{[\sigma ]\in H^1(k,F)} f^{\sigma }\left(Y^{\sigma }(\mathbb A_k)^{\text{{Br}}}\right),$ where $$f$$ ranges over all torsors under all finite $$k$$-groups.
The question is motivated by the fact that the étale Brauer–Manin obstruction explains the counter-example to the Hasse principle constructed by the author [Invent. Math. 135, No. 2, 399–424 (1999; Zbl 0951.14013)] but does not explain the counter-example constructed by B. Poonen [Insufficiency of the Brauer–Manin obstruction applied to étale covers, to appear in Ann. Math., cf. arXiv:0806.1312]. In the above cited paper Poonen asked whether in his counter-example the descent obstruction is empty or not, and, more generally, whether one always has the inclusion $X(\mathbb A_k)^{\text{ét, Br}} \subset X(\mathbb A_k)^{\text{desc}}.$ The latter inclusion was proved by C. Demarche [Algebra and Number Theory 3, No. 2, 237–254 (2009; Zbl 1247.11090)]. In the paper under review the opposite inclusion is proved. The key result is the following formula for the descent obstruction (Theorem 1.1): if $$f: Y\to X$$ is a torsor under a finite $$k$$-group scheme $$F$$, then $X(\mathbb A_k)^{\text{{desc}}} = \bigcup_{[\sigma ]\in H^1(k,F)} f^{\sigma }\left(Y^{\sigma }(\mathbb A_k)^{\text{{desc}}}\right).$ (Note that a similar property does not hold for the Brauer–Manin set $$X(\mathbb A_k)^{\text{{desc}}}$$: this is what happens in the above mentioned counter-example to the Hasse principle due to the author.)
The proof of Theorem 1.1 is based on a method of M. Stoll [Algebra and Number Theory 1, No. 4, 349–391 (2007; Zbl 1167.11024)] where a similar formula has been established for another variant of the descent obstruction, in which the intersection is taken over $$f$$ ranging over all torsors under all finite $$k$$-groups.
As an application, the author presents some results for surfaces of Kodaira dimension zero. In particular, he proves the following conditional statement (Corollary 3.3): if the Brauer–Manin obstruction is the only obstruction to weak approximation on $$K3$$ surfaces, then the descent obstruction is the only obstruction to weak approximation on Enriques surfaces.

##### MSC:
 14G05 Rational points 14G25 Global ground fields in algebraic geometry 14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations 14J28 $$K3$$ surfaces and Enriques surfaces 14L30 Group actions on varieties or schemes (quotients)
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##### References:
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