Kato–Saito and Colliot-Thélène's exactness conjecture for zero-cycles

Let $k$ be a number field and let $V$ be a proper smooth geometrically integral variety over $k$. Consider the complex

$$\varprojlim_n\operatorname{CH}_0(V)/n\longrightarrow\prod_{v\in\Omega_k}\varprojlim_n\operatorname{CH}'_0(V_{k_v})/n\longrightarrow\operatorname{Hom}(\operatorname{Br}(V),\mathbb{Q}/\mathbb{Z}),$$

where the modified Chow groups are the usual Chow groups at non-archimedean places, zero at complex places, and $\operatorname{CH}_0(V_{\mathbb{R}})/\operatorname{N}_{\mathbb{C}|\mathbb{R}}\operatorname{CH}_0(V_{\mathbb{C}})$ at real places. Kato–Saito and Colliot-Thélène's conjecture. The sequence above is exact for all proper smooth varieties. This conjecture asserts that the Brauer–Manin obstruction completely controls the local-global principle and weak approximation for zero-cycles on proper smooth varieties over number fields.