Nevanlinna Pair and Algebraic Hyperbolicity

TL;DR

This paper introduces the Nevanlinna pair concept for projective varieties with divisors, unifying various notions of hyperbolicity and extension theorems through advanced Nevanlinna theory on Riemann surfaces.

Contribution

It defines the Nevanlinna pair for pairs (X, D) and connects multiple hyperbolicity concepts using recent Nevanlinna theory developments.

Findings

Unifies complex hyperbolicity and algebraic hyperbolicity concepts.

Links Nevanlinna theory with hyperbolic properties of varieties.

Provides a framework for extension theorems in complex geometry.

Abstract

We introduce the notion of the $\textit{Nevanlinna pair}$ for a pair $(X, D)$, where $X$ is a projective variety and $D$ is an effective Cartier divisor on $X$. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard type extension theorem (more generally the Borel hyperbolicity), as well as the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by P\v{a}un and Sibony.