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

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.