Arithmetic of 0-cycles on varieties defined over number fields

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.

arXiv comment21 pages, part of the main result appeared in an old version of the author's preprint (arXiv:1011.5995), the proof here is simplified