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In this paper, we prove that a smooth projective variety  X  of characteristic  p  &gt; 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   &gt; 2.

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