A characterization of ordinary abelian varieties by the Frobenius   push-forward of the structure sheaf
  For an ordinary abelian variety $X$, $F^e_*\mathcal{O}_X$ 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. 
Akiyoshi Sannai, Hiromu Tanaka
math.AG