Arithmetic of 0-cycles on varieties defined over number fields

TL;DR

This paper explores the relationship between rational points and zero-cycles on rationally connected varieties over number fields, establishing implications among obstructions and exactness of certain sequences, with applications to homogeneous spaces.

Contribution

It demonstrates that the Brauer-Manin obstruction's role in rational points relates directly to zero-cycles, and proves the equivalence of certain local-global principles for these cycles.

Findings

Brauer-Manin obstruction is the only obstacle to weak approximation for rational points implies the same for zero-cycles.

Equivalence between the exactness of a local-global sequence for zero-cycles and the absence of obstructions.

Application of results to smooth compactifications of homogeneous spaces of linear algebraic groups.

Abstract

Let $X$ be a rationally connected algebraic variety, defined over a number field $k$. We find a relation between the arithmetic of rational points on $X$ and the arithmetic of zero-cycles. More precisely, we consider the following statements: (1) the Brauer-Manin obstruction is the only obstruction to weak approximation for $K$-rational points on $X_K$ for all finite extensions $K/k$; (2) the Brauer-Manin obstruction is the only obstruction to weak approximation in some sense that we define for zero-cycles of degree 1 on $X_K$ for all finite extensions $K/k$; (3) a certain sequence of local-global type for Chow groups of 0-cycles on $X_K$ is exact for all finite extensions $K/k$. We prove that (1) implies (2), and that (2) and (3) are equivalent. We also prove a similar implication for the Hasse principle. As an application, we prove the exactness of the sequence mentioned above for smooth compactifications of certain homogeneous spaces of linear algebraic groups.