Asymptotics of the N\'eron height pairing

TL;DR

This paper investigates the asymptotic behavior of the Néron height pairing in degenerating families of Riemann surfaces and proves the effectiveness of a related height jumping divisor, linking complex and non-archimedean geometry.

Contribution

It establishes the leading asymptotic term of the Néron height pairing under unipotent monodromy and proves Hain's conjecture on the effectiveness of the height jumping divisor.

Findings

Asymptotic behavior of Néron height pairing is controlled by non-archimedean pairing.

The height jumping divisor on moduli space is effective.

Degeneration of the canonical metric on the Poincaré bundle explains these phenomena.

Abstract

The aim of this paper is twofold. First, we study the asymptotics of the N\'eron height pairing between degree-zero divisors on a family of degenerating compact Riemann surfaces parametrized by an algebraic curve. We show that if the monodromy is unipotent the leading term of the asymptotics is controlled by the local non-archimedean N\'eron height pairing on the generic fiber of the family. Second, we prove a conjecture of R. Hain to the effect that the `height jumping divisor' related to the normal function $(2g-2)x-K$ on the moduli space $\mm_{g,1}$ of 1-pointed curves of genus $g \geq 2$ is effective. Both results follow from a study of the degeneration of the canonical metric on the Poincar\'e bundle on a family of jacobian varieties.