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 $m\gg 0$ 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