Rationally connected foliations after Bogomolov and McQuillan

This paper is concerned with a sufficient criterion to guarantee that a given foliation on a normal variety has algebraic and rationally connected leaves. Following ideas from a preprint of Bogomolov-McQuillan and using recent works of Langer and Graber-Harris-Starr, we give a clean, short and simple proof of previous results. Apart from a new vanishing theorem for vector bundles in positive characteristic, our proof employs only standard techniques of Mori theory and does not make any reference to the more involved properties of foliations in characteristic p. We apply the result to show that Q-Fano varieties with unstable tangent bundles always admit a sequence of partial rational quotients naturally associated to the Harder-Narasimhan filtration.