Instability of truncated symmetric powers of sheaves

Let X be a smooth projective variety of dimension n over an algebraically closed field k of characteristic p>0. Let FX:X→X be the absolute Frobenius morphism, and E a torsion free sheaf on X. We give an upper bound of instability of truncated symmetric powers Tl(E)(0⩽l⩽rk(E)(p−1)) in terms of , and I(E) (Theorem 3.5). As an application, we obtain an upper bound of the instability of Frobenius direct image FX⁎(E) and some sufficient conditions of FX⁎(E) being slope semi-stable. In addition, we study the slope (semi)-stability of sheaves of locally exact (resp. closed) forms (resp. ).