Arithmetic characteristic classes of automorphic vector bundles

Abstract

We develop a theory of arithmetic characteristic classes of (fully decomposed) automorphic vector bundles equipped with an invariant hermitian metric. These characteristic classes have values in an arithmetic Chow ring constructed by means of differential forms with certain log-log type singularities. We first study the cohomological properties of log-log differential forms, prove a Poincaré lemma for them and construct the corresponding arithmetic Chow groups. Then, we introduce the notion of log-singular hermitian vector bundles, which is a variant of the good hermitian vector bundles introduced by Mumford, and we develop the theory of arithmetic characteristic classes. Finally we prove that the hermitian metrics of automorphic vector bundles considered by Mumford are not only good but also log-singular. The theory presented here provides the theoretical background which is required in the formulation of the conjectures of Maillot-Roessler in the semi-abelian case and which is needed to extend Kudla's program about arithmetic intersections on Shimura varieties to the non-compact case.