Colliot-Thélène–Kato–Saito conjecture on zero-cycles of degree one

Let $X$ be a smooth, proper, geometrically irreducible variety over a number field $k$. A zero-cycle of degree one on $X$ is a formal integer linear combination of closed points whose degree is $1$. The Brauer–Manin pairing on local zero-cycles is

$$\operatorname{Br}(X) \times \prod_{v\in\Omega}\operatorname{CH}_0(X\otimes_k k_v) \longrightarrow \mathbf{Q}/\mathbf{Z}.$$

There is no Brauer–Manin obstruction to a zero-cycle of degree $1$ if there are local zero-cycles of degree $1$ lying in the right kernel of this pairing. Colliot-Thélène–Kato–Saito conjecture. Every smooth, proper, geometrically irreducible variety $X$ over a number field $k$ has a zero-cycle of degree $1$ whenever there is no Brauer–Manin obstruction to the existence of such a zero-cycle. This is the zero-cycle analogue of the Brauer–Manin principle for rational points. The conjecture concerns a necessary local-global condition and remains unresolved in the stated generality.