Hyperbolicity of Fermat-type curves and their complements

TL;DR

This paper proves the hyperbolicity of Fermat-type curves and their complements in the complex projective plane, using generalized Borel theorems, and improves existing degree bounds established by previous researchers.

Contribution

It introduces new methods based on generalized Borel theorems to establish hyperbolicity and refines degree bounds for Fermat-type curves and their complements.

Findings

Fermat-type curves are hyperbolic in ^2.

Degree bounds for hyperbolicity are improved.

Methodology advances previous techniques in complex geometry.

Abstract

In this paper, by using the generalized Borel theorems in $\mathbb{CP}^2$, we show the hyperbolicity of Fermat type curves and their complement in $\mathbb{CP}^2$. This improves Noguchi-Shirosaki's and Demailly-El Goul's degree bounds.