Purity of level m stratifications

We will discuss stratifications coming from truncations of Barsotti-Tate groups (also called p-divisible groups), generalizing the Ekedahl-Oort stratification. We will recall some background material about stratifications, explain the hierarchy of purity properties, and describe in detail the situation for level m stratifications. We will illustrate our main result in the case of the Siegel modular varieties, and mention how our techniques apply to good reductions of Shimura varieties of Hodge type. Joint work with A. Vasiu and T. Wedhorn.