A characterization of ordinary abelian varieties by the Frobenius push-forward of the structure sheaf II

In this paper, we prove that a smooth projective variety $X$ of characteristic $p>0$ is an ordinary abelian variety if and only if $K_X$ is pseudo-effective and $F^e_*\mathcal O_X$ splits into a direct sum of line bundles for an integer $e$ with $p^e>2$.