Refined obstructions to local-global principles for 0-cycles

We introduce new `refined’ obstructions to local-global principles for 0-cycles on algebraic varieties over number fields. Assuming finiteness of relevant Tate–Shafarevich groups, we show that the Hasse principle and weak approximation for 0-cycles on generalised Kummer varieties and bielliptic surfaces are controlled by obstructions of this new type. As an additional application of our refined obstructions, we answer a question of Zhang about the relationship between the Brauer–Manin and connected descent obstructions for 0-cycles. We also show that a Corwin–Schlank style refined obstruction set coincides with the set of global 0-cycles, conditionally on the Section Conjecture.