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

TL;DR

This paper investigates the Brauer-Manin obstruction to the Hasse principle and weak approximation for 0-cycles on fibrations over rationally connected bases, extending known results to new types of bases.

Contribution

It proves the exactness of the local-to-global sequence for 0-cycles on fibrations with Ch ext{"a}telet surface or homogeneous space bases, broadening previous cases.

Findings

Exactness of sequence (E) for fibrations over Ch ext{"a}telet surfaces.

Exactness of sequence (E) for fibrations over homogeneous spaces.

Conditions under which the sequence (E) is exact for most fibers.

Abstract

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.