Foliations on unitary Shimura varieties in positive characteristic

Ehud De Shalit, Eyal Z. Goren

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1 Scopus citations

Abstract

When is inert in the quadratic imaginary field and <![CDATA[$m, unitary Shimura varieties of signature and a hyperspecial level subgroup at , carry a natural foliation of height 1 and rank in the tangent bundle of their special fiber . We study this foliation and show that it acquires singularities at deep Ekedahl-Oort strata, but that these singularities are resolved if we pass to a natural smooth moduli problem , a successive blow-up of . Over the ( -)ordinary locus we relate the foliation to Moonen's generalized Serre-Tate coordinates. We study the quotient of by the foliation, and identify it as the Zariski closure of the ordinary-étale locus in the special fiber of a certain Shimura variety with parahoric level structure at . As a result, we get that this 'horizontal component' of , as well as its multiplicative counterpart, are non-singular (formerly they were only known to be normal and Cohen-Macaulay). We study two kinds of integral manifolds of the foliation: unitary Shimura subvarieties of signature , and a certain Ekedahl-Oort stratum that we denote . We conjecture that these are the only integral submanifolds.

Original languageAmerican English
Pages (from-to)2267-2304
Number of pages38
JournalCompositio Mathematica
Volume154
Issue number11
DOIs
StatePublished - 1 Nov 2018

Bibliographical note

Publisher Copyright:
© 2018 The Author.

Keywords

  • Ekedahl-Oort strata
  • Shimura varieties
  • foliations

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