Degeneration of the Archimedean Height Pairing

Abstract

We study one-parameter variation of the local archimedean height pairing and the global

Néron–Tate height pairing from a rather geometric perspective. The basic idea is to single

out the line bundle that encapsulates the pairing; the variation problem then translates

into an extension problem for this bundle. We investigate various aspects of this problem, including its Hodge-theoretic and intersection-theoretic ones. These line bundles and their extensions can be endowed with the structure of adelic line bundles, which provide finer information about the limiting behaviour of the height pairings.