The classical Hasse invariant is a modular form of weight p-1
in characteristic p which has a simple zero at each supersingular
point. In this talk, we will discuss how to generalize the Hasse
invariant to unitary Shimura varieties with signature (1,n-1)
using the idea of Ekedahl-Oort stratification. We will also discuss
an application to the l-adic cohomology of unitary Shimura
varieties with bad reduction (Iwahori level structure) using
integral models of Harris-Taylor-Yoshida and the weight
spectral sequence of Rapoport-Zink.