Level $m$ stratifications of versal deformations of $p$-divisible groups
Author:
Adrian Vasiu
Journal:
J. Algebraic Geom. 17 (2008), 599-641
DOI:
https://doi.org/10.1090/S1056-3911-08-00495-5
Published electronically:
February 6, 2008
MathSciNet review:
2424922
Full-text PDF
Abstract |
References |
Additional Information
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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References
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- A. Vasiu, Reconstructing p-divisible groups from their truncations of small level, arxiv.org/abs/math/0607268.
- A. Vasiu, Manin problems for Shimura varieties of Hodge type, arxiv.org/abs/ math/0209410.
- A. Vasiu, Integral canonical models of unitary Shimura varieties, to appear in Asian J. Math.
- T. Wedhorn, The dimension of Oort strata of Shimura varieties of PEL-type, Moduli of abelian varieties (Texel Island, 1999), pp. 441–471, Progr. of Math., Vol. 195, Birkhäuser, Basel, 2001. MR 1827029 (2002b:14029)
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Additional Information
Adrian Vasiu
Affiliation:
Department of Mathematical Sciences, Binghamton University, Binghamton, New York 13902-6000
Email:
adrian@math.binghamton.edu
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