Parametrization of geometric Beilinson--Bloch heights via adelic line bundles

Let $S$ be a quasi-projective smooth variety over complex field $\mathbb{C}$. For a smooth projective morphism $\pi :X\to S$, we will introduce a new height pairing \begin{align*} CH^p_{\hom}(X/S) \times CH^q_{\hom}(X/S) \to \widetilde{$\mathrm{P}\mathrm{i}\mathrm{c}$}(S) \end{align*} with $\widetilde{\mathrm{P}\mathrm{i}\mathrm{c}}(S)$ the group of geometric adelic line bundles in the sense of Yuan–Zhang. It essentially parametrizes the asymptotic height pairing introduced by Brosnan and Pearlstein. We will show that this asymptotic height pairing coincides with Beilinson–Bloch pairing under certain conditions.