Hyperbolicity of generic hypersurfaces of polynomial degree via Green-Griffiths jet differentials

TL;DR

This paper proves the Kobayashi and Green-Griffiths-Lang conjectures for generic hypersurfaces in projective space with polynomial degree bounds, using Green-Griffiths jet differentials and holomorphic Morse inequalities.

Contribution

It introduces a new approach based on classical Green-Griffiths jet spaces, improving previous results on hyperbolicity of hypersurfaces.

Findings

Proves conjectures for generic hypersurfaces with polynomial degree bounds

Uses classical Green-Griffiths jet spaces instead of invariant jet differentials

Employs holomorphic Morse inequalities to establish existence of jet differential equations

Abstract

We give a new version of a recent result of B{\'e}rczi-Kirwan, proving the Kobayashi and Green-Griffiths-Lang conjectures for generic hypersurfaces in the projective space , with a polynomial lower bound on the degree. Our strategy again relies on Siu's technique of slanted vector fields and the use of holomorphic Morse inequalities to prove the existence of a jet differential equation with a negative twist -- however, instead of using a space of invariant jet differentials, we base our computations on the classical Green-Griffiths jet spaces.