# Mumford's theorem on mixed Shimura varieties: the case of a line bundle on the universal elliptic curve

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Jürg Kramer

created by risa on 23 Oct 2015

28 oct 2015
-- 14:30

Aula di Consiglio, Dip. Matematica, Università "La Sapienza", Roma

**Abstract.**

In the case of a Shimura variety X of non compact type, by a theorem of Mumford, automorphic vector bundles equipped with the natural invariant metric have a unique extension to vector bundles over a toroidal compactification of X equipped with a logarithmically singular hermitian metric. This result is crucial to define arithmetic Chern classes for these vector bundles. It is natural to ask whether Mumford's result remains valid for 'automorphic' vector bundles on mixed Shimura varieties. In our talk we will examine the simplest case, namely the Jacobi line bundle on the universal elliptic curve, whose sections are Jacobi forms. We will show that Mumford's result cannot be extended directly to this case since a new type of metric singularity appears. By using the theory of b-divisors, we show however that an analogue of Mumford's extension theorem can be obtained in this setting.