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 issue is to establish the exactness of a local-to-global sequence (E) of Chow groups of 0-cycles. Recently, Harpaz and Wittenberg proved the exactness of (E) for rationally connected fibrations whose smooth fibres verify the exactness of (E) and whose base is either the projective space or a curve verifying the exactness of (E). In the present paper, we prove the exactness of (E) for fibrations whose bases are Châtelet surfaces or smooth proper models of homogeneous spaces of connected linear algebraic groups with connected stabilizers. We require that either the fibration is valuatively split in codimension 1 and most closed 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.