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In this paper, we prove that for a fibration f : X → Z from a smooth projective 3-fold to a smooth projective curve, over an algebraically closed field k with chark = p > 5, if the geometric generic fiber X η is smooth, then subadditivity of Kodaira dimensions holds, i.e. κ(X) ≥ κ(X η ) + κ(Z).

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In this paper, we study the Albanese morphisms in positive characteristic. We prove that the Albanese morphism of a variety with nef anti-canonical divisor is an algebraic fiber space, under the assumption that the general fiber is F -pure. Furthermore, we consider a notion of F -splitting for morphisms, and investigate it of the Albanese morphisms. We show that an F -split variety has F -split Albanese morphism, and that the F -split Albanese morphism is an algebraic fiber space. As an application, we provide a new characterization of abelian varieties.

We study the Iitaka-Kodaira dimension of nef relative anti-canonical divisors. As a consequence, we prove that given a complex projective variety with klt singularities, if the anti-canonical divisor is nef, then the dimension of a general fibre of the maximal rationally connected fibration is at least the Iitaka-Kodaira dimension of the anti-canonical divisor.