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 $F_X:X\rightarrow X$ be the absolute Frobenius morphism, and $\E$ a torsion free sheaf on $X$. We give a upper bound of instability of truncated symmetric powers $\mathrm{T}^l(\E)(0\leq l\leq\rk(\E)(p-1))$ in terms of $L_{\max}(\Omg^1_X)$, $\mathrm{I}(\Omg^1_X)$ and $\mathrm{I}(\E)$ (Theorem \ref{InstabTl}). As an application, We obtain a upper bound of Frobenius direct image ${F_X}_*(\E)$ and some sufficient conditions of slope semi-stability of ${F_X}_*(\E)$. In addition, we study the slope (semi)-stability of sheaves of locally exact (closed) forms $B^i_X$ ($Z^i_X$).