Hyperbolicity of smooth logarithmic and orbifold pairs in $\mathbb{P}^n$

TL;DR

This paper establishes a precise criterion for the ampleness of logarithmic cotangent bundles in projective space, extends the results to orbifold pairs, and applies these findings to hyperbolicity questions, improving previous work.

Contribution

It provides a necessary and sufficient condition for ampleness in the logarithmic and orbifold setting, advancing understanding of hyperbolicity in algebraic geometry.

Findings

Derived a criterion for ampleness of logarithmic cotangent bundles.

Extended results to orbifold pairs.

Improved upon previous results by Darondeau-Rousseau.

Abstract

We derive a necessary and sufficient condition on a hyperplane arrangement in $\mathbb{P}^n$ for the associated logarithmic cotangent bundle to be ample modulo boundary. We extend this result to the orbifold setting and give some applications concerning hyperbolicity of pairs. We improve significantly the results of Darondeau-Rousseau.