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

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 HⁿdR(Y/X) at x, i.e. the largest i such that the line of the conjugate filtration is contained in Filⁱ of the Hodge filtration. For every triple (G,μ,r) consisting of a connected, reductive Fₚ-group G, a cocharacter μ ∈ X_*(G) and an Fₚ-representation r of G, we state a generalized Ogus Principle. If ζ:X →G$-$Zipμ 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.