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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 $p^e>2$.

A Characterization of Ordinary Abelian Varieties by the Frobenius Push-Forward of the Structure Sheaf II