In characteristic zero, Ueno's conjecture states that if X is a smooth projective variety with Kodaira dimension zero, then its Albanese morphism in an algebraic fiber space and the Kodaira dimension of the general fiber is again zero. This was proven by Cao and Păun in 2016.

Building on the generic vanishing techniques of C. Hacon and Zs. Patakfalvi, we prove a positive characteristic version of this result. We use it to deduce new cases of Iitaka's subadditivity conjecture of Kodaira dimensions (in positive characteristics, this is only the second result towards this conjecture which does not depend on any dimension hypothesis). The goal of this talk is to explain how these techniques work, and how we can use them to prove such results.