Stability of direct images under Frobenius morphism
  Let $X$ be a smooth projective variety over an algebraically field $k$ with ${\rm char}(k)=p>0$ and $F:X\to X_1$ be the relative Frobenius morphism. When ${\rm dim}(X)=1$, we prove that $F_*W$ is a stable bundle for any stable bundle $W$ (Theorem \ref{thm1.3}). As a step to study the question for higher dimensional $X$, we generalize the canonical filtration (defined by Joshi-Ramanan-Xia-Yu for curves) to higher dimensional $X$ (Theorem \ref{thm2.6}). 
Xiaotao Sun