Existence of global invariant jet differentials on projective hypersurfaces of high degree

Let X ⊂ Pn+1 be a smooth complex projective hypersurface. In this paper we show that, if the degree of X is large enough, then there exist global sections of the bundle of invariant jet differentials of order n on X , vanishing on an ample divisor. We also prove a logarithmic version, effective in low dimension, for the log-pair (Pn, D), where D is a smooth irreducible divisor of high degree. Moreover, these result are sharp, i.e. one cannot have such jet differentials of order less than n.