Bloch–Beilinson conjecture on Abel–Jacobi maps

Let $V/\overline{\mathbb Q}$ be a smooth quasiprojective variety. Let

$$\Phi_{k,\mathbb Q}:\operatorname{CH}^k_{\mathrm{hom}}(V/\overline{\mathbb Q};\mathbb Q)\longrightarrow J^k(V(\mathbb C))\otimes\mathbb Q$$

be the Abel–Jacobi map. Bloch–Beilinson conjecture on Abel–Jacobi maps. The map $\Phi_{k,\mathbb Q}$ is injective. This is presented as a variant of the Bloch–Beilinson conjecture and predicts that homologically trivial rational Chow cycles over $\overline{\mathbb Q}$ are detected by their Abel–Jacobi invariants.