When is the Albanese morphism an algebraic fiber space in positive characteristic?

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 in the case of 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.