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On Goren–Oort stratification for quaternionic Shimura varieties
Published online by Cambridge University Press: 21 September 2016
Abstract
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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 
$p$ .
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