The de Rham–Betti conjecture for smooth projective varieties

Let $k$ be a number field, let $Y$ be a smooth projective $k$-variety, and fix an embedding $k\hookrightarrow\mathbf{C}$. For a codimension-$n$ de Rham–Betti cycle, take a pair $(\alpha,\beta)\in H^{2n}_{\operatorname{dR}}(Y/k)\times H^{2n}_B(Y,\mathbf{Q})$ such that, under the period isomorphism

$$H^{2n}_{\operatorname{dR}}(Y/k)\otimes_k\mathbf{C}\xrightarrow{\sim}H^{2n}_B(Y,\mathbf{Q})\otimes_{\mathbf{Q}}\mathbf{C},$$

$\alpha\otimes 1$ maps to $(2\pi i)^n\beta\otimes 1$. De Rham–Betti conjecture. Any de Rham–Betti cycle on $Y$ is algebraic. The cycle classes of algebraic cycles provide such cycles, and the conjecture asserts the converse; for products of elliptic curves, the paper discusses a proof, while the general statement remains a conjecture in the supplied context.