On Goren–Oort stratification for quaternionic Shimura varieties

Abstract

Let$F$be a totally real field in which a prime$p$is unramified. We define the Goren–Oort stratification of the characteristic-$p$fiber of a quaternionic Shimura variety of maximal level at$p$. We show that each stratum is a$(\mathbb{P}^{1})^{r}$-bundle over other quaternionic Shimura varieties (for an appropriate integer$r$). As an application, we give a necessary condition for the ampleness of a modular line bundle on a quaternionic Shimura variety in characteristic$p$.