Abstract
The generalization of Arakelov Theory to higher dimensions by Gillet and Soule is well suited to study vector bundles with smooth hermitian metrics. Nevertheless when working with non-compact Shimura varieties or with modular varieties, the metrics that appear naturally and that can be studied using group theoretical or modular methods almost never extend to a smooth metric on a compactification. Typically they extend to a log-singular hermitian vector bundle. In this series of lectures we will see an example of this phenomenon and we will describe an extension of Arakelov theory that can be used to study log-singular hermitian vector bundles.
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