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

TL;DR

This work proves a Big Picard type theorem for holomorphic maps into smooth quasi-projective varieties $X$ of log-general type with maximal quasi-Albanese dimension, under a small-ramification condition on ramified coverings $\pi:Y\to \mathbb{C}_{>\delta}$. It also establishes a generalized Green-Griffiths-Lang conjecture for such $X$, linking log-general type with pseudo-Picard/Brody hyperbolicity via several nonhyperbolicity loci $\mathrm{Sp}_{\bullet}(X)$. The authors develop a robust Nevanlinna-theoretic framework on ramified coverings, including a refined log Bloch-Ochiai theorem and second main theorem type estimates with weak truncation, to control holomorphic maps from $Y$ to semi-abelian varieties and their subvarieties. A key component is a ramified-covering strategy that replaces the classical Poincaré reducibility, enabling extension results and hyperbolicity statements in the non-compact setting. These results lay essential groundwork for the series’ Parts II and III, which address hyperbolicity phenomena in more general quasi-projective contexts and with local systems.

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.

Theorems & Definitions (82)

• Definition 1: Special subsets
• Conjecture 2: Generalized Green-Griffiths-Lang
• Theorem 1
• Theorem 2: =\ref{['thm:20250911']}
• Theorem 3
• Lemma 1.1: NW13
• Lemma 1.2
• proof
• Lemma 1.3
• proof