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Hyperbolicity of Fermat-type curves and their complements

Anh Tuan Nguyen

TL;DR

The paper addresses hyperbolicity of Fermat-type curves and their complements in $CP^2$ by transporting generalized Borel theorems into the CP$^2$ setting via Nevanlinna theory. Using a refined CP$^2$-specific four-case degeneracy framework (Cartan SMT and truncated counting), it proves hyperbolicity for the complement of Noguchi-El Goul's curves when $d>8$ and for Noguchi-Shirosaki's Fermat-type curves under explicit parameter ranges. These results improve previous degree bounds and, as a by-product, strengthen Lang-Vojta conjecture bounds for families of hyperbolic hypersurfaces. The work provides new low-degree hyperbolic examples and sharpens the interaction between Nevanlinna theory and complex hyperbolicity in dimension two.

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.

Hyperbolicity of Fermat-type curves and their complements

TL;DR

The paper addresses hyperbolicity of Fermat-type curves and their complements in by transporting generalized Borel theorems into the CP setting via Nevanlinna theory. Using a refined CP-specific four-case degeneracy framework (Cartan SMT and truncated counting), it proves hyperbolicity for the complement of Noguchi-El Goul's curves when and for Noguchi-Shirosaki's Fermat-type curves under explicit parameter ranges. These results improve previous degree bounds and, as a by-product, strengthen Lang-Vojta conjecture bounds for families of hyperbolic hypersurfaces. The work provides new low-degree hyperbolic examples and sharpens the interaction between Nevanlinna theory and complex hyperbolicity in dimension two.

Abstract

In this paper, by using the generalized Borel theorems in , we show the hyperbolicity of Fermat type curves and their complement in . This improves Noguchi-Shirosaki's and Demailly-El Goul's degree bounds.
Paper Structure (4 sections, 1 theorem, 74 equations)

This paper contains 4 sections, 1 theorem, 74 equations.

Table of Contents

  1. Introduction
  2. Generalized Borel theorems
  3. Hyperbolicity of complement of Noguchi-El Goul's curves in CP2
  4. Hyperbolicity of Noguchi-Shirosaki's Fermat-type curves

Key Result

Corollary 1

Under the same conditions as in Theorem B, the hyperbolic hypersurface $X_n$ defined as in define hypersurface X_n of degree $d^n$ with $d>5$ satisfies the Lang-Vojta's conjecture.

Theorems & Definitions (3)

  • Corollary
  • proof
  • proof