Birational stability of the cotangent bundle and generic semipositivity for orbifolds pairs

Smooth orbifold pairs $(X,\Delta)$ play a major role in the birational classification of projective manifolds, both in the LMMP and through the multiple fibres of fibre spaces.
For orbifold pairs, the usual geometric invariants of manifolds can be defined (cotangent sheaf, morphisms and birational maps in particular).
A major problem of birational classification consists in deriving positivity properties of the cotangent bundle from those of the canonical bundle. A central result in this direction is Miyaoka's generic positivity of the cotangent bundle for non-uniruled manifolds.
The aim of the talk is to extend this result to the orbifold context. The original proof of Miyaoka cannot however be adapted. Instead, a combination of Bogomolov-Mc Quillan and orbifold additivity theorem parmit to show that if $K_X+\Delta$ is pseudo-effective, then $\Omega^1(X,\Delta)$ is generically semi-positive. This is joint work with M. Paun.