On the existence of global orbifold jet differentials

We use holomorphic Morse inequalities to derive an existence theorem for global jet differentials on projective complex manifolds, in the logarithmic and orbifold contexts. The proof, which exhibits new phenomena in the orbifold case, also produces explicit bounds for cohomology groups of (orbifold) jet bundles. These results can in turn be used to investigate some fundamental conjectures such as the Green-Griffiths-Lang conjecture on entire holomorphic curves. This is joint work with F. Campana, L. Darondeau and E. Rousseau.