Asymptotics of local height pairing
Yuta Nakayama
TL;DR
The paper analyzes the Archimedean component of Beilinson–Bloch height pairings in a strongly semistable degeneration over a curve. It develops a motivic and Hodge-theoretic framework that equates the Bloch–Seibold G_m-biextension with Hain’s Archimedean biextension through derived regulator maps to Deligne cohomology, and then compares their sections via Godement resolutions. In the smooth setting, it establishes a precise correspondence between the biextensions and their sections using Deligne cohomology, while in the semistable case it controls degenerations with mixed Hodge modules to derive asymptotics for the height, including a leading q log|t| term tied to monodromy and Picard-type data. The approach provides a purely algebraic pathway that extends to the ℓ-adic setting and relates to Gao–Zhang’s Northcott-type conjectures, with explicit descriptions of line bundles, regulators, and height asymptotics across smooth and semistable fibers.
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 Poincaré 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 Poincaré 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.