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We show that in positive characteristic, the Albanese morphism of normalproper varieties $X$ with $\kappa_S(X, \omega_X) = 0$ is separable, surjective,has connected fibers, and the generic fiber $F$ also satisfies $\kappa(F,\omega_F) = 0$. As a corollary, we deduce a new case of Iitaka's subadditivityconjecture for fibrations over abelian varieties, when the generic fiber hasnon-nilpotent Hasse-Witt matrix.

On the Albanese morphism of varieties with Frobenius stable Kodaira dimension zero