The Beilinson–Hodge conjecture for geometric higher normal functions

Let $\mathcal{X}_{\mathcal{U}}$ be the total space of the family over $\mathcal{U}$, let $\mathcal{H}^{2p-r}_v(p)$ be the varying part of the relevant polarized variation of Hodge structure, and let

$$\mathrm{AJ}^v_{\phi}\colon \mathrm{CH}^p(\mathcal{X}_{\mathcal{U}},r-1)\longrightarrow \mathrm{ANF}(\mathcal{H}^{2p-r}_v(p))$$

be the projected Abel–Jacobi map. Beilinson–Hodge conjecture. If $\mathcal{X}_{\mathcal{U}}$ is defined over $\overline{\mathbb{Q}}$, then $\mathrm{AJ}^v_{\phi}$ is surjective. Equivalently, admissible and geometric higher normal functions with values in $\mathcal{H}^{2p-r}_v(p)$ coincide. This asserts that all admissible normal functions in the varying part arise from higher Chow cycles, a special case of the Beilinson–Hodge conjecture.