Instability of truncated symmetric powers of sheaves

Abstract

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 Lmax(ΩX1), I(ΩX1) 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 BXi (resp. ZXi).