Frobenius pull backs of vector bundles in higher dimensions

We prove that for a smooth projective variety X of arbitrary dimension and for a vector bundle E over X, the Harder–Narasimhan filtration of a Frobenius pull back of E is a refinement of the Frobenius pull back of the Harder–Narasimhan filtration of E, provided there is a lower bound on the characteristic p (in terms of rank of E and the slope of the destabilizing sheaf of the cotangent bundle of X). We also recall some examples, due to Raynaud and Monsky, to show that some lower bound on p is necessary. We also give a bound on the instability degree of the Frobenius pull back of E in terms of the instability degree of E and well defined invariants of X.