Local-global principle for 0-cycles on fibrations over rationally connected bases

We study the Brauer-Manin obstruction to the Hasse principle and to weak approximation for 0-cycles on algebraic varieties that possess a fibration structure. The exactness of the local-to-global sequence $(E)$ of Chow groups of 0-cycles was known only for a fibration whose base is either a curve or the projective space. In the present paper, we prove the exactness of $(E)$ for fibrations whose bases are Ch\^{a}telet surfaces or projective models of homogeneous spaces of connected linear algebraic groups with connected stabilizers. We require that either all fibres are split and most fibres satisfy weak approximation for 0-cycles, or the generic fibre has a 0-cycle of degree $1$ and $(E)$ is exact for most fibres.Comment: 20 pages. The introduction has been rewritten. More details of the applications of the main results are given. The proof of Th. 2.4 is removed, instead, a sketch is given at the end of the pape

Towards the Brauer-Manin obstruction on varieties fibred over the projective line

Recently Dasheng Wei proved that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for 0-cycles of degree 1 on some fibrations over the projective line defined by bi-cyclic normic equations. In the present paper, we prove the exactness of the global-to-local sequence for Chow groups of 0-cycles of such varieties, which signifies that the Brauer-Manin obstruction is also the only obstruction to weak approximation for 0-cycles of arbitrary degree. Our main theorem also generalizes several existing results

The local-global exact sequence for Chow groups of zero-cycles

A local-global sequence for Chow groups of zero-cycles involving Brauer groups has been conjectured to be exact for all proper smooth algebraic varieties. We apply existing methods to construct several new families of varieties verifying the exact sequence. The examples are explicit, they are normic bundles over the projective space.Comment: 10 pages, new organization of the last section and minor modification

Non-invariance of weak approximation properties under extension of the ground field

For rational points on algebraic varieties defined over a number field $K$, we study the behavior of the property of weak approximation with Brauer-Manin obstruction under extension of the ground field. We construct K-varieties accompanied with a quadratic extension $L/K$ such that the property holds over $K$ (conditional on a conjecture) while fails over $L$. The result is unconditional when $K = \mathbb{Q}$ or $K$ is one of several quadratic number fields. Over $\mathbb{Q}$, we give an explicit example.Comment: Comments are welcom

Local-global principle for certain biquadratic normic bundles

Let $X$ be a proper smooth variety having an affine open subset defined by the normic equation $N_{k(\sqrt{a},\sqrt{b})/k}({x})=Q(t_{1},...,t_{m})^{2}$ over a number field $k$. We prove that : (1) the failure of the local-global principle for zero-cycles is controlled by the Brauer group of $X$; (2) the analogue for rational points is also valid assuming Schinzel's hypothesis.Comment: comments are welcom

Approximation faible pour les 0-cycles sur un produit de vari\'et\'es rationnellement connexes

Consider weak approximation for 0-cycles on a smooth proper variety defined over a number field, it is conjectured to be controlled by its Brauer group. Let $X$ be a Ch\^atelet surface or a smooth compactification of a homogeneous space of a connected linear algebraic group with connected stabilizer. Let $Y$ be a rationally connected variety. We prove that weak approximation for 0-cycles on the product $X\times Y$ is controlled by its Brauer group if it is the case for $Y$ after every finite extension of the base field. We do not suppose the existence of 0-cycles of degree $1$ neither on $X$ nor on $Y$.Comment: In French. Comments and suggestions are welcom

Compatibility of weak approximation for zero-cycles on products of varieties

Zero-cycles are conjectured to satisfy weak approximation with Brauer-Manin obstruction for proper smooth varieties defined over number fields. Roughly speaking, we prove that the conjecture is compatible for products of rationally connected varieties, K3 surfaces, Kummer varieties, and one curve.Comment: This is the first version, comments are welcom

Principe local-global pour les z\'ero-cycles sur certaines fibrations au-dessus d'une courbe : I

Let $X$ be a smooth projective variety over a number field, fibered over a curve, with geometrically integral fibers. We prove that, supposing the finiteness of \sha(Jac(C)), if the fibers over a generalised Hilbertian subset satisfy the Hasse principle (resp. weak approximation), then the Brauer-Manin obstruction coming from the base curve is the only obstruction to the Hasse principle (resp. to weak approximation) for zero-cycles of degree 1 on $X$.Comment: Some mistakes are corrected, unnecessary arguments are omitted. 20 page

Astuce de Salberger et z\'ero-cycles sur certaines fibrations

We prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and to the weak approximation for zero-cycles on certain fibrations over a smooth curve or over the projective space. The principal novelty is that the arithmetic hypotheses are supposed only on the fibres over a generalized Hilbertian subset.Comment: 21 pages, in French. Minor mistakes are correcte

Principe local-global pour les z\'ero-cycles sur certaines fibrations au-dessus de l'espace projectif

We study the local-global principle for zero-cycles of degree 1 on certain varieties fibered over the projective space. Among other applications, we prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation for zero-cycles of degree 1 on Severi-Brauer-variety bundles or Ch\^atelet-surface bundles over the projective space.Comment: In French, 27 pages. Comparing to the version 3, Lemma 4.1 was not correct and hence has been removed in this version, the hypothesis (abelienne-scindee) has been slightly changed and the corresponding proofs have been fixe