Product of Brauer--Manin obstruction for 0-cycles over number fields and function fields

TL;DR

This paper investigates the Brauer--Manin obstruction's role in the existence of 0-cycles of degree 1 on varieties over number and function fields, establishing equivalences for product varieties.

Contribution

It proves that the Brauer--Manin obstruction for 0-cycles on product varieties is equivalent to the obstructions on each factor, extending to function fields of complex curves.

Findings

Equivalence of Brauer--Manin obstruction for product varieties and factors over number fields.

Analogous results established over function fields of complex curves.

Supports the conjecture that Brauer--Manin obstruction controls 0-cycle existence.

Abstract

It is conjectured that the Brauer--Manin obstruction is expected to control the existence of 0-cycles of degree 1 on smooth proper varieties over number fields. In this paper, we prove that the existence of Brauer--Manin obstruction to Hasse principle for 0-cycles of degree 1 on the product of smooth (non-necessarily proper) varieties is equivalent to the simultaneous existence of such an obstruction on each factor. We also prove an analogous statement for smooth varieties defined over function fields of $\mathbb{C}((t))$-curves.