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A theorem by Mumford implies that every automorphic line bundle on a pureopen Shimura variety, equipped with an invariant smooth metric, can be uniquelyextended as a line bundle on a toroidal compactification of the variety, insuch a way that the metric acquires only logarithmic singularities. This resultis the key of being able to compute arithmetic intersection numbers from theseline bundles. Hence it is natural to ask whether Mumford's result remains validfor line bundles on mixed Shimura varieties.  In this paper we examine the simplest case, namely the sheaf of Jacobi formson the universal elliptic curve. We show that Mumford's result cannot beextended directly to this case and that a new interesting kind of singularitiesappears.  By using the theory of b-divisors, we show that an analogue of Mumford'sextension theorem can be obtained. We also show that this extension ismeaningful because it satisfies Chern-Weil theory and a Hilbert-Samuel type offormula.

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