Asymptotics of local height pairing

TL;DR

This paper investigates the asymptotic behavior of the Archimedean component of the Arakelov intersection number, connecting homological algebra, motivic viewpoints, and biextensions to deepen understanding of height pairings.

Contribution

It introduces a homological algebra approach to the Archimedean height, compares different biextensions, and simplifies existing theories using derived and motivic methods.

Findings

Comparison of Hain and Brosnan--Pearlstein biextensions over complex numbers.

Simplification of Bloch and Seibold's biextension symmetry proof.

Enhanced understanding of height pairings via derived regulator maps and Hodge theory.

Abstract

We discuss the asymptotics of the Archimedean part of the Arakelov intersection number. The theorem is motivated by recent conjectures and their proof strategy by Gao and Zhang on the Northcott property of the Beilinson--Bloch height pairing. Our method involves a homological algebra interpretation of the Archimedean height by Hain. This interpretation allows us to introduce motivic viewpoints using Deligne cohomology, cycle class maps and higher Chow groups. Especially, we compare the biextension by Hain and Brosnan--Pearlstein over $\mathbb{C}$ based on Poincar\'e line bundle and Hodge theory with the $\mathbb{G}_{\mathrm{m}}$-biextension of Bloch and Seibold defined by two families of homologically trivial cycles on a generically smooth family of projective varieties over a smooth curve. Our comparison, a relative version of the work of Gorchinskiy, enhances his derived viewpoint on these biextensions. Especially when the family of varieties are smooth, the two constructions are related via derived regulator maps to Deligne cohomology, reinterpreted similarly to Beilinson's absolute Hodge cohomology, as well as the derived description of Hardouin's biextension that generalizes Poincar\'e line bundle by Hain. The comparison when the family defined over a smooth curve has a strongly semistable reduction further involves a simple monodromy computation using mixed Hodge modules. Along our discussion, we simplify the discussion of Bloch and Seibold, partly in the style of Gorchinskiy. For example, the symmetry of their biextension is proved more easily than their work.