The Archimedean height pairing for differential forms on degeneration of Riemann surfaces

TL;DR

This paper introduces a new Archimedean height pairing for differential forms on degenerating Riemann surfaces, analyzes its asymptotics, and connects it to existing pairings, broadening geometric applications.

Contribution

It defines a novel height pairing for fiberwise cohomologically trivial forms and extends related pairings to more general geometric contexts.

Findings

Established the asymptotic behavior of the pairing during degeneration.

Connected the pairing to Filip--Tosatti's current-valued pairing.

Extended the applicability of the pairing to broader settings.

Abstract

We define the Archimedean height pairing for fiberwise cohomologically trivial differential forms on a one-parameter degeneration of Riemann surfaces, and we study its asymptotic behavior. The proof relies on recent work by Dai--Yoshikawa on the asymptotics of small eigenvalues. As an application, we relate this pairing to the current-valued pairing of Filip--Tosatti, extending their construction to broader geometric settings.