# Does the sheaf of locally exact differential forms splitting in positive characteristic

Let k be an algebraically closed field of characteristic $p>2$, $X$ a smooth projective curve of genus $g>1$ over $k$, and $F_X:X\rightarrow X$ be the absolute Frobenius morphism. Let $B^1_X$ be the locally free sheaf of locally exact differential forms on $X$ defined by the exact sequence of locally free sheaves $$0\rightarrow\mathscr{O}_X\rightarrow{F_X}_*(\mathscr{O}_X)\rightarrow B^1_X.$$ Then $B^1_X$ is a vector bundle of rank $p-1$, and the Harder-Narasimhan filtration of ${F_X}_*(B^1_X)$ is $$0=V_p\subset V_{p-1}\subset\cdots\subset V_{l+1}\subset V_l\subset\cdots\subset V_1={F_X}_*(B^1_X)$$ such that $V_i/V_{i+1}\cong\mathrm{\Omega}^{\otimes i}_X$. My question is $${F_X}_*(B^1_X)\ncong\mathrm{\Omega}^{\otimes p-1}_X\oplus\mathrm{\Omega}^{\otimes p-2}_X\oplus\cdots\oplus\mathrm{\Omega}^1_X?$$ In particular, I am especially interested in the case $p=3$, i.e ${F_X}_*(B^1_X)\ncong\mathrm{\Omega}^{\otimes 2}_X\oplus\mathrm{\Omega}^1_X?$