Vol. 3, No. 3, 2021

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Lifting Chern classes by means of Ekedahl–Oort strata

Gerard van der Geer and Eduard Looijenga

Vol. 3 (2021), No. 3, 469–480
Abstract

The moduli space 𝒜g of principally polarized abelian varieties of genus g is defined over and admits a minimal compactification 𝒜g, also defined over . The Hodge bundle over 𝒜g has its Chern classes in the Chow ring of 𝒜g with -coefficients. We show that over 𝔽p, these Chern classes naturally lift to 𝒜g and do so in the best possible way: despite the highly singular nature of 𝒜g they are represented by algebraic cycles on 𝒜g 𝔽p which define elements in the bivariant Chow ring. This is in contrast to the situation in the analytic topology, where these Chern classes have canonical lifts to the complex cohomology of the minimal compactification as Goresky–Pardon classes, which are known to define nontrivial Tate extensions inside the mixed Hodge structure on this cohomology.

Keywords
Chern classes, Baily–Borel compactification, Ekedahl–Oort strata
Mathematical Subject Classification 2010
Primary: 14G35, 11G18
Milestones
Received: 3 January 2020
Revised: 20 June 2020
Accepted: 5 July 2020
Published: 13 May 2021
Authors
Gerard van der Geer
Korteweg-de Vries Instituut
Universiteit van Amsterdam
Netherlands
Yau Mathematical Sciences Center
Tsinghua University
Beijing
China
Eduard Looijenga
Yau Mathematical Sciences Center
Tsinghua University
Haidian District
Beijing
China
Mathematisch Instituut
Universiteit Utrecht
Netherlands