The algebraic Green-Griffiths-Lang conjecture for complements of very general pairs of divisors

TL;DR

This paper proves that the complement of a very general pair of hypersurfaces in projective space is algebraically hyperbolic outside a proper subvariety, supporting key conjectures in complex geometry.

Contribution

It establishes algebraic hyperbolicity for complements of certain hypersurface pairs, extending prior results and providing evidence for major conjectures.

Findings

Complement of hypersurface pairs is algebraically hyperbolic outside a subvariety

Supports Lang-Vojta and Green-Griffiths conjectures

Extends previous work by Chen, Pacienza-Rousseau, and Riedl

Abstract

We prove that the complement of a very general pair of hypersurfaces of total degree $2n$ in $\mathbb{P}^n$ is algebraically hyperbolic modulo a proper closed subvariety. This provides evidence towards conjectures of Lang-Vojta and Green-Griffiths, and partially extends previous work of Chen, Pacienza-Rousseau, and Chen-Riedl and the third author.