  In this paper, we study a variation of a conjecture of Debarre on positivity
of cotangent bundles of complete intersections. We establish the ampleness of
Schur powers of cotangent bundles of generic complete intersections in
projective manifolds, with high enough explicit codimension and multi-degrees.
Our approach is naturally formulated in terms of flag bundles and allows one to
reach the optimal codimension. On complex manifolds, this ampleness property
implies intermediate hyperbolic properties. We give a natural application of
our main result in this context.
