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Progress concerning the local-global principle for zero-cycles on algebraic varieties. (English) Zbl 1358.11073

This article is a brief survey with no proofs, it deals mainly with the author’s recent contributions, see for instance [Ann. Sci. Éc. Norm. Supér. (4) 46, No. 1, 35–56 (2013; Zbl 1264.14033)]. The theme is the investigation of the nature of the Brauer-Manin obstruction to the Hasse principle, see [J.-L. Colliot-Thélène, J. Théor. Nombres Bordx. 7, No. 1, 51–73 (1995; Zbl 0870.14002)] and [E. Peyre, in: Séminaire Bourbaki. Volume 2003/2004, 165–193, Exp. No. 931 (2005; Zbl 1080.14026)]. The progress reported concerns certain higher dimensional varieties, in particular homogeneous varieties and fibrations. The author considers zero-cycles on algebraic varieties defined over number fields. The Hasse principle and weak approximation property are obstructed by the Brauer group of the variety and it is conjectured to be the only obstruction for all proper smooth varieties by Colliot-Thélène, Sansuc, Kato and Saito.
Given a proper smooth and geometrically integral algebraic variety \(X\) over a number field \(k\), the Brauer-Manin pairing for 0-cycles, discussed by { J. -L. Colliot-Thélène} in [loc. cit.], determines the complex below; the theme of this survey is the conjecture stating its exactness.
\[ \varprojlim_n \mathrm{CH}_0(X)/n\rightarrow\prod_{v\in\Omega}\varprojlim_n \mathrm{CH}_0'(X_{k_v})/n\rightarrow\text{Hom}(\text{Br}(X),\mathbb{Q}/\mathbb{Z}). \] Here \(\Omega\) denotes the set of places of \(k\), and \(k_v\) the completion of \(k\) with respect to \(v\in\Omega\). The modified Chow group \(\mathrm{CH}'(X_{k_v})\) is the usual Chow group \(\mathrm{CH}_0(X_{k_v})\) of 0-cycles for each non-Archimedean place \(v\), while \(\mathrm{CH}'(X_{k_v}):=\mathrm{CH}_0(X_{k_v})/N_{\bar{k}_v|k_v}\mathrm{CH}_0(X_{\bar{k}_v})\) for Archimedean places \(v\). A fibration being a dominant proper morphism \(f: X \to B\), the author explains that using the known exactness for the case when \(B\) is a projective spaces or a curve (under the hypothesis that the Tate- Shafarevich group is finite) then the fibration method he employs yields results for \(X\), granted that suitable conditions are assumed for the behaviour of the fibres. Several theorems are recounted which the author has found and which deal both with the conjecture and of its relations with the Hasse principle. The reader can thus get a useful glimpse of the contributions due to the author for higher dimensional varieties, in particular for homogeneous varieties and fibrations.

MSC:

11G35 Varieties over global fields
14C25 Algebraic cycles
14G25 Global ground fields in algebraic geometry
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References:

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