Generalized algebraic Morse inequalities and Hasse-Schmidt jet differentials

TL;DR

The paper provides an algebraic proof of the existence of Green-Griffiths jet differentials on complex projective manifolds of general type, introducing a new algebraic Morse inequalities applicable in positive characteristic.

Contribution

It introduces a novel algebraic version of Morse inequalities and applies them to prove the existence of jet differentials in both complex and positive characteristic settings.

Findings

Established algebraic proof of Green-Griffiths jet differentials existence.

Developed a new algebraic Morse inequalities applicable in positive characteristic.

Extended results to varieties over arbitrary algebraically closed fields.

Abstract

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.