Given a projective manifold, one can measure the positive directions in its tangent bundle by evaluating the slopes of its sub-sheaves with respect to movable classes. We show that the holomorphic foliations with positive slope and stable with respect to a movable class are algebraic. As a consequence, we infer that any quotient of an arbitrary tensor power of the cotangent bundle of the manifold has a pseudo-effective determinant, provided that the canonical class of the manifold is pseudo-effective. This represents a generalization of the celebrated generic semi-positivity theorem by Y. Miyaoka. A few other applications will be discussed. These results are part of a joint work with F. Campana. Speaker(s): Mihai Paun (University of Illinois, Chicago)