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

For an ordinary abelian variety $X$, $F_{\ast }^e{\mathcal{O}}_X$ is decomposed into line bundles for every positive integer $e$. Conversely, if a smooth projective variety $X$ satisfies this property and its Kodaira dimension is non-negative, then $X$ is an ordinary abelian variety.