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 $H^n_{\text{dR}}(Y/X)$ 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 $\mu \in X_*(G)$ and an $\mathbf{F}_p$-representation $r$ of $G$, we state a generalized Ogus Principle. If $\zeta:X \to \text{$G$-$\mathtt{Zip}$}^{\mu}$ 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.