Entire curves in C-pairs with large irregularity

TL;DR

This paper generalizes the Bloch-Ochiai theorem to C-pairs with high irregularity, showing that entire curves cannot be dense in such pairs, extending classical results to more singular and complex settings.

Contribution

It introduces a Nevanlinna theory for C-pairs and broadens the scope of the Bloch-Ochiai theorem to include singular Kähler pairs with large irregularity.

Findings

No dense entire C-curves in C-pairs with large irregularity

Extension of Bloch-Ochiai theorem to singular Kähler pairs

Development of a Nevanlinna theory for C-pairs

Abstract

This paper extends the fundamental theorem of Bloch-Ochiai to the context of C-pairs: If (X, D) is a C-pair with large irregularity, then no entire C-curve in X is ever dense. In its most general form, the paper's main theorem applies to normal K\"ahler pairs with arbitrary singularities. However, it also strengthens known results for compact K\"ahler manifolds without boundary, as it applies to some settings that the classic Bloch-Ochiai theorem does not address. The proof builds on the work of Kawamata, Ueno, and Noguchi, recasting parabolic Nevanlinna theory as a "Nevanlinna theory for C-pairs". We hope the approach might be of independent interest.