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

Wedhorn, Torsten; Ziegler, Paul

doi:10.2140/ant.2023.17.923

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

Torsten Wedhorn and Paul Ziegler

Vol. 17 (2023), No. 4, 923–980

DOI: 10.2140/ant.2023.17.923

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 characteristic $0$ , 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.

Keywords

Shimura varieties, Ekedahl–Oort strata, tautological ring

Mathematical Subject Classification

Primary: 11G18, 14C15, 14G35

Secondary: 14M15, 20G15, 20G40

Milestones

Received: 4 November 2021

Revised: 30 March 2022

Accepted: 10 May 2022

Published: 2 May 2023

Authors

Torsten Wedhorn
Fachbereich Mathematik Technische Universität Darmstadt Darmstadt Germany

Paul Ziegler
Department Mathematik Technische Universität München Garching Germany

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Vol. 17, No. 4, 2023