  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.
