Exactness of the Colliot–Thélène–Kato–Saito zero-cycle complex

Establish exactness, for every proper smooth geometrically integral variety over a number field, of the complex formed by the profinite completion of the Chow group of zero-cycles, the corresponding adelic zero-cycle quotient, and the Pontryagin dual of the Brauer group under the Brauer–Manin pairing.

Background

The paper recalls the Colliot–Thélène–Kato–Saito conjecture concerning the local–global principle and weak approximation for zero-cycles. For a proper smooth geometrically integral variety X over a number field, the conjecture predicts exactness of the standard Brauer–Manin complex involving the profinite completion of the Chow group of zero-cycles, the adelic zero-cycle group, and the Brauer–Manin pairing.

Unlike the corresponding conjecture for rational points, this conjecture is stated without a rational connectedness assumption. The paper studies symmetric powers as an intermediate object between rational points and zero-cycles and proves results about the Brauer–Manin obstruction for those symmetric powers, but the recalled general exactness conjecture itself remains a conjectural statement.