This is a remastered and expanded version of a an earlier preprint of the author, in which we give a fully algebraic proof of an important theorem of Demailly, stating the existence of many Green-Griffiths jet differentials on a complex projective manifold of general type. To this end, we introduce a new algebraic version of the Morse inequalities, which we use in our proof as an algebraic counterpart to Demailly's and Bonavero's holomorphic Morse inequalities. This new version also applies to positive characteristic, giving the existence of Hasse-Schmidt jet differentials for a smooth projective variety of general type over an arbitrary algebraically closed field.
Benoît Cadorel (Tue,) studied this question.