The variational Hodge conjecture for de Rham cohomology

Let $S$ be a smooth connected quasi-projective variety, let $\pi:\mathcal{X}\to S$ be a smooth projective morphism, and let $0\in S(\mathbb{C})$. Let $\beta\in H^{2p}(\mathcal{X}_0/\mathbb{C})$ be the cohomology class of a codimension $p$ algebraic cycle $Z_0$. Assume that $\beta$ extends to a section $\widetilde{\beta}$ of

$$\mathcal{H}^{2p}=\mathbb{R}^{2p}\pi_*\Omega^{\bullet}_{\mathcal{X}/S}$$

that is flat for the Gauss–Manin connection.

Variational Hodge conjecture for de Rham cohomology. For every complex point $s$ of $S$, the class $\widetilde{\beta}_s$ is the cohomology class of an algebraic cycle.

This is the de Rham formulation of the variational Hodge conjecture, using the Gauss–Manin connection to express flatness. Its general case is open.