Towards the Green-Griffiths-Lang conjecture via equivariant localisation

TL;DR

This paper proves the Green-Griffiths-Lang conjecture for generic projective hypersurfaces of sufficiently high degree using equivariant localisation techniques on jet differentials.

Contribution

It introduces a novel application of equivariant localisation to the Demailly-Semple jet bundle, confirming the conjecture for a broad class of hypersurfaces.

Findings

Confirmed the conjecture for hypersurfaces with degree ≥ n^{9n}

Demonstrated the effectiveness of equivariant localisation in complex geometry

Provided explicit degree bounds for the validity of the conjecture

Abstract

The Green-Griffiths-Lang conjecture says that for every complex projective algebraic variety $X$ of general type there exists a proper algebraic subvariety of $X$ containing all nonconstant entire holomorphic curves $f:\mathbb{C} \to X$. Using equivariant localisation on the Demailly-Semple jet differentials bundle we give an affirmative answer to this conjecture for generic projective hypersurfaces $X \subset \mathbb{P}^{n+1}$ of degree $\mathrm{deg}(X) \ge n^{9n}$.