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

TL;DR

The paper constructs a new height pairing CH^p_{\,hom}(X/S) × CH^q_{\,hom}(X/S) → \widetilde{\mathrm{Pic}}(S) valued in geometric adelic line bundles, capturing the asymptotic Beilinson--Bloch height in a function-field setting and matching BP height under stated conditions. It develops a comprehensive analytic framework via Berkovich and hybrid spaces, and proves an Analytification Criterion for adelic divisors, enabling a robust parametrization of height pairings by adelic data. The toric case provides explicit connections between adelic divisors, tropical geometry, and toroidal b-divisors, while the Global Monge–Ampère measure extends intersection theory to this adelic context. The main results include functoriality and comparison with Beilinson--Bloch height, establishing the new pairing as a natural, analytically grounded generalization of classical height pairings in the function-field setting and relating archimedean and non-archimedean contributions through geometric adelic lines."

Abstract

Let $ S $ be a quasi-projective smooth variety over complex field $ \mathbb{C} $. For a smooth projective morphism $ π: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.

Theorems & Definitions (82)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Corollary 1.7
• Proposition 2.1
• Definition 2.2
• Remark 2.3