The singularities of the invariant metric on the line bundle of Jacobi forms

Abstract

A theorem by Mumford implies that every automorphic line bundle on a pure open Shimura variety, equipped with an invariant smooth metric, can be uniquely extended as a line bundle on a toroidal compactification of the variety, in such a way that the metric acquires only logarithmic singularities. This result is the key of being able to compute arithmetic intersection numbers from these line bundles. Hence it is natural to ask whether Mumford's result remains valid for line bundles on mixed Shimura varieties. In this paper we examine the simplest case, namely the sheaf of Jacobi forms on the universal elliptic curve. We show that Mumford's result cannot be extended directly to this case and that a new interesting kind of singularities appears. By using the theory of b-divisors, we show that an analogue of Mumford's extension theorem can be obtained. We also show that this extension is meaningful because it satisfies Chern-Weil theory and a Hilbert-Samuel type of formula.