Brauer–Manin sufficiency for rationally connected varieties

Prove that every proper smooth geometrically integral rationally connected variety over a number field satisfies weak approximation with Brauer–Manin obstruction; equivalently, establish that the Brauer–Manin obstruction is the only obstruction to weak approximation for rational points on such varieties.

Background

The paper recalls the programmatic conjecture of Colliot-Thélène and Sansuc concerning rational points on rationally connected varieties. It asserts that, for proper smooth geometrically integral rationally connected varieties over number fields, the closure of the rational points in the adelic points is exactly the Brauer–Manin set.

The authors note that this conjecture has been studied extensively for four decades but remains far from resolved. It is included because it is an explicitly stated conjecture in the paper, although it is not the main unresolved problem addressed by the paper.