Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture

TL;DR

This paper investigates the existence of entire curves on complex varieties of general type, using holomorphic Morse inequalities and cohomology analysis to advance understanding of the Green-Griffiths-Lang conjecture.

Contribution

It introduces a novel approach combining Morse inequalities and probabilistic cohomology analysis to progress towards the Green-Griffiths-Lang conjecture.

Findings

Established new conditions for entire curves on varieties of general type.

Connected cohomological properties with algebraic differential equations.

Made progress towards a generalized Green-Griffiths-Lang conjecture.

Abstract

The goal of this work is to study the existence and properties of non constant entire curves f drawn in a complex irreducible n-dimensional variety X, and more specifically to show that they must satisfy certain global algebraic or differential equations as soon as X is projective of general type. By means of holomorphic Morse inequalities and a probabilistic analysis of the cohomology of jet spaces, we are able to reach a significant step towards a generalized version of the Green-Griffiths-Lang conjecture.