Progress concerning the local-global principle for zero-cycles on algebraic varieties

Abstract

We consider zero-cycles on algebraic varieties defined over number fields. The Hasse principle and weak approximation property are obstructed by the Brauer group of the variety and it is conjectured to be the only obstruction for all proper smooth varieties by Colliot-Th\'el\`ene, Sansuc, Kato and Saito. We summarize the progress related to this conjecture, particularly for higher dimensional varieties such as fibrations and homogeneous varieties.