Refined obstructions to local-global principles for 0-cycles

TL;DR

This paper introduces new refined obstructions to local-global principles for 0-cycles on algebraic varieties over number fields, linking them to classical obstructions and conjectures.

Contribution

It develops novel refined obstructions to local-global principles for 0-cycles, connecting them with existing obstructions and answering open questions.

Findings

Refined obstructions control the Hasse principle and weak approximation for certain varieties.

Answering Zhang's question, the paper relates Brauer--Manin and connected descent obstructions.

Conditional on the Section Conjecture, a refined obstruction set matches the set of global 0-cycles.

Abstract

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.