Tautological rings of Shimura varieties and cycle classes of
Ekedahl-Oort strata

We define the tautological ring as the subring of the Chow ring of a Shimura
variety generated by all Chern classes of all automorphic bundles. We explain
its structure for the special fiber of a good reduction of a Shimura variety of
Hodge type and show that it is generated by the cycle classes of the
Ekedahl-Oort strata as a vector space. We compute these cycle classes. As
applications we get the triviality of l-adic Chern classes of flat automorphic
bundles in characterstic 0, an isomorphism of the tautological ring of smooth
toroidal compactifications in positive characteristic with the rational
cohomology ring of the compact dual of the hermitian domain given by the
Shimura datum, and a new proof of Hirzebruch-Mumford proportionality for
Shimura varieties of Hodge type.