  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 Poincar\'e 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.
