Arithmetic normal functions and filtrations on Chow groups

Let $X/\mathbb{C}$ be a smooth projective variety, and let $\textrm {CH}^r(X,m)$ be the higher Chow group defined by Bloch. Saito and Asakura defined a descending candidate Bloch-Beilinson filtration $\textrm {CH}^r(X,m;\mathbb{Q}) = F^0\supset \cdots \supset F^r\supset F^{r+1} = F^{r+2}=\cdots$, using the language of mixed Hodge modules. Another more geometrically defined filtration is constructed by Kerr and Lewis in terms of germs of normal functions. We show that under the assumptions (i) $X/\mathbb{C} = X_0\times \mathbb{C}$ where $X_0$ is defined over $\overline {\mathbb{Q}}$, and (ii) the general Hodge conjecture, that $F^{\bullet }\textrm {CH}^r(X,m;\mathbb{Q})$ coincides with the aforementioned geometric filtration. More specifically, it is characterized in terms of germs of reduced arithmetic normal functions.