On subadditivity of Kodaira dimension in positive characteristic

We show that for a surjective, separable morphism f of smooth projective varieties over an algebraically closed field of positive characteristic such that $f_* \mathcal{O}_X = \mathcal{O}_Y$ subadditivity of Kodaira dimension holds, provided the base is of general type and the Hasse-Witt matrix of the geometric general fiber is not nilpotent.

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