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

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 $(0,p)$.