Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Level $ m$ stratifications of versal deformations of $ p$-divisible groups

Author: Adrian Vasiu
Journal: J. Algebraic Geom. 17 (2008), 599-641
Published electronically: February 6, 2008
MathSciNet review: 2424922
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Abstract: Let $ k$ be an algebraically closed field of characteristic $ p>0$. Let $ c,d,m$ be positive integers. Let $ D$ be a $ p$-divisible group of codimension $ c$ and dimension $ d$ over $ k$. Let $ \mathscr{D}$ be a versal deformation of $ D$ over a smooth $ k$-scheme $ \mathscr{A}$ which is equidimensional of dimension $ cd$. We show that there exists a reduced, locally closed subscheme $ \mathfrak{s}_D(m)$ of $ \mathscr{A}$ that has the following property: a point $ y\in\mathscr{A}(k)$ belongs to $ \mathfrak{s}_D(m)(k)$ if and only if $ y^*(\mathscr{D})[p^m]$ is isomorphic to $ D[p^m]$. We prove that $ \mathfrak{s}_D(m)$ is regular and equidimensional of dimension $ cd-\dim(\pmb{\mathrm{Aut}}(D[p^m]))$. We give a proof of Traverso's formula which for $ m\gg0$ computes the codimension of $ \mathfrak{s}_D(m)$ in $ \mathscr{A}$ (i.e., $ \dim(\pmb{\mathrm{Aut}}(D[p^m]))$) in terms of the Newton polygon of $ D$. We also provide a criterion of when $ \mathfrak{s}_D(m)$ satisfies the purity property (i.e., it is an affine $ \mathscr{A}$-scheme). Similar results are proved for quasi Shimura $ p$-varieties of Hodge type that generalize the special fibres of good integral models of Shimura varieties of Hodge type in unramified mixed characteristic $ (0,p)$.

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Adrian Vasiu
Affiliation: Department of Mathematical Sciences, Binghamton University, Binghamton, New York 13902-6000

Received by editor(s): June 14, 2006
Received by editor(s) in revised form: May 15, 2007
Published electronically: February 6, 2008
Dedicated: To Carlo Traverso, for his 62nd birthday