Colliot-Thélène–Sansuc–Kato–Saito conjecture for zero-cycles

Let $F$ be a number field and let $Y$ be a smooth projective geometrically connected variety over $F$. Let $\Omega$ be the set of places of $F$ and $\Omega_f$ the set of finite places. Denote by $\widehat{\operatorname{CH}_0(Y)}$ and $\widehat{\operatorname{CH}_{0,\mathbf{A}}(Y)}$ the relevant profinite completions of the global and adelic Chow groups of zero-cycles. The Brauer–Manin pairing gives a map from the adelic completion to $\operatorname{Hom}(\operatorname{Br}(Y),\mathbb{Q}/\mathbb{Z})$.

Colliot-Thélène–Sansuc–Kato–Saito conjecture. The complex

$$\widehat{\operatorname{CH}_0(Y)}\stackrel{\Delta}{\longrightarrow}\widehat{\operatorname{CH}_{0,\mathbf{A}}(Y)}\longrightarrow\operatorname{Hom}(\operatorname{Br}(Y),\mathbb{Q}/\mathbb{Z})$$

is exact.

This asserts that the Brauer–Manin obstruction is the only obstruction to weak approximation for zero-cycles. The source attributes the conjecture to Colliot-Thélène and Sansuc for geometrically rational varieties and to Kato and Saito for general smooth projective varieties; its resolution is not specified.