Abstract:  We know that a Frobenius pull back of a semistable bundle need not remain semistable. However, if X is a nonsingular projective curve of genus g and defined over a field of characterstic p > 0, then Shepherd-Barron and X. Sun proved (independently), that for a semistable vector bundle V of rank r, the instability degree of F^*V is bounded by  2(g-1)(r-1).

 This bound on the instability is useful in keeping a check on some of the  behaviour of a vector bundle afterFrobenius pullbacks.

For example one can prove that, for any vector bundle V and for large  p (in terms of degree of X and rank of V), the Harder-Narasimhan filtration of F^*V is a refinement of the  Frobenius pull back of the Harder-Narasimhan filtration of V.We give counterexamples to prove that some such conditions on p is necessary.

  We extend such results to vector bundles over  higher dimesional verieties.To prove these, we answer a question/conjecture of X. Sun (though  for p bigger than rank of E + dimension of X), which is an anaolgue  of  the above mentioned result of Shepherd-Barron and X. Sun in higher dimension.