Our aim is threefold. The basic objects are non-proper Shimura varieties (embeddable in the Siegel space),
automorphic bundles, their various Chern classes over compactifications of the Shimura varieties.
The goals:1) Construct Chern classes of automorphic vector bundles in l-adic cohomology of the minimal
compactification of a Shimura variety over barQ.2) Try to understand in which more arithmetic cohomology it is
possible to lift those l-adic Chern classes.3) If 1) and 2) were available, one could try to construct the
l-adic realization of the motivic mixed Tate extensions which are believed by many people to exist.