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

Abstract

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{Pic}}(S) \end{align*} with $ \widetilde{\mathrm{Pic}}(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.