The Galois-module structure of the space of holomorphic differentials of a curve.
We extend Ogus’notion of -crystal and -span to the context of Berthelot’s crystals of level and we study the generalization of Ogus’theorem on the equivalence between -crystals and -spans of width less than .
Let be a normal projective variety, and let be an ample Cartier divisor on . Suppose that is not the projective space. We prove that the twisted cotangent sheaf is generically nef with respect to the polarisation . As an application we prove a Kobayashi-Ochiai theorem for foliations: if is a foliation such that , then is at most the rank of .