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

Abstract

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.