C-pairs and their morphisms

TL;DR

This paper surveys Campana's theory of C-pairs in complex geometry, providing foundational definitions and a well-behaved notion of morphism to facilitate future research in hyperbolicity, rational points, and entire curves.

Contribution

It introduces a systematic framework for C-pairs and their morphisms, improving behavior in singular cases and connecting to minimal model theory.

Findings

Defines a new notion of morphism for C-pairs

Establishes functorial properties relating to minimal models

Serves as a comprehensive reference for future applications

Abstract

This paper surveys Campana's theory of C-pairs (or "geometric orbifolds") in the complex-analytic setting, to serve as a reference for future work. Written with a view towards applications in hyperbolicity, rational points, and entire curves, it introduces the fundamental definitions of C-pair-theory systematically. In particular, it establishes an appropriate notion of "morphism", which agrees with notions from the literature in the smooth case, but is better behaved in the singular setting and has functorial properties that relate it to minimal model theory.