# Jumps in the Archimedean Height

## TL;DR

This paper introduces the asymptotic height pairing on local intersection cohomology groups, demonstrating its role in extending line bundles and analyzing their metrics in the context of Hodge structures and normal functions.

## Contribution

It defines a new pairing on intersection cohomology, applies it to extend biextension line bundles, and establishes positivity and continuity properties related to Hodge theory.

## Key findings

- The asymptotic height pairing governs the extension of the biextension line bundle.
- The pairing exhibits positivity properties that generalize recent results.
- It naturally arises from Mumford-Grothendieck biextensions associated with normal functions.

## Abstract

We introduce a pairing on local intersection cohomology groups of variations of pure Hodge structure, which we call the asymptotic height pairing. Our original application of this pairing was to answer a question on the Ceresa cycle posed by R. Hain and D. Reed. (This question has since been answered independently by Hain.) Here we apply the pairing to show that a certain analytic line bundle, called the biextension line bundle, defined in terms of normal functions, always extends to any smooth partial compactification of the base. We show that the the pairing on intersection cohomology governs the extension of the natural metric on this line bundle studied by Hain and Reed (as well as, more recently, by several other authors). We also prove a positivity property of the asypmtotic height pairing, which generalize results of a recent preprint of J. Burgos Gill, D. Holmes and R. de Jong, along with a continuity property of the pairing in the normal function case. Moreover, we show that the asymptotic height pairing arises in a natural way from certain Mumford-Grothendieck biextensions associated to normal functions.