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.

We changed the title of the paper, and add one more chapter about parametrization of Beilinson--Bloch heights