Generalized algebraic Morse inequalities and jet differentials

TL;DR

This paper provides an algebraic proof of the existence of jet differentials on complex projective manifolds of general type, introducing a new algebraic Morse inequality to replace the holomorphic version.

Contribution

It introduces a novel algebraic Morse inequality and applies it to prove a key theorem of Demailly regarding jet differentials.

Findings

Established an algebraic version of Morse inequalities.

Proved the existence of Green-Griffiths jet differentials algebraically.

Provided a new proof of a fundamental theorem in complex geometry.

Abstract

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 then use in our proof as an algebraic counterpart to Demailly's and Bonavero's holomorphic Morse inequalities.