An Ogus Principle for Zip period maps: The Hasse invariant's vanishing order via `Frobenius and the Hodge filtration'

Abstract

This paper generalizes a result of Ogus that, under certain technical conditions, the vanishing order of the Hasse invariant of a family $Y/X$ of $n$-dimensional Calabi-Yau varieties in characteristic $p$ at a point $x$ of $X$ equals the "conjugate line position" of $H_{\text{dR}}^n(Y/X)$ at $x$, i.e. the largest $i$ such that the line of the conjugate filtration is contained in ${\text{Fil}}^i$ of the Hodge filtration. For every triple $(G,\mu ,r)$ consisting of a connected, reductive ${\mathbf{F}}_p$-group $G$, a cocharacter $\mu \in X_{\ast }(G)$ and an ${\mathbf{F}}_p$-representation $r$ of $G$, we state a generalized Ogus Principle. If $ζ:X \to \text{$G$-$\mathtt{Zip}$}^μ$ is a smooth morphism (=`Zip period map'), then the group theoretic Ogus Principle implies an Ogus Principle on $X$. We deduce an Ogus Principle for several Hodge and abelian-type Shimura varieties and the moduli space of K3 surfaces.