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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 the Kodaira dimension of X is non-negative, then X is an ordinary abelian variety.