Hyperbolicity and fundamental groups of complex quasi-projective varieties (I): Maximal quasi-Albanese dimension by Nevanlinna theory

TL;DR

This paper proves a Big Picard theorem and the Green-Griffiths-Lang conjecture for certain complex quasi-projective varieties using Nevanlinna theory, focusing on those with maximal quasi-Albanese dimension and log-general type.

Contribution

It establishes a Big Picard theorem and confirms the Green-Griffiths-Lang conjecture for varieties of maximal quasi-Albanese dimension via Nevanlinna theory, advancing understanding of their hyperbolic properties.

Findings

Proved a Big Picard theorem for holomorphic maps to these varieties.

Confirmed the Green-Griffiths-Lang conjecture for the specified class of varieties.

Applied Nevanlinna theory to complex quasi-projective varieties of log-general type.

Abstract

This is the first part of a series of three papers. In this paper, we establish a Big Picard type theorem for holomorphic maps $f:Y \to X$, where $Y$ is a ramified covering of the punctured disc $\mathbb{D}^*$ with small ramification and $X$ is a complex quasi-projective variety of log-general type and of maximal quasi-Albanese dimension. As a byproduct, we prove the generalized Green-Griffiths-Lang conjecture for such $X$. This paper summarizes the parts of the three-paper series that are based primarily on Nevanlinna theory.