The Albanese of a C-pair

TL;DR

This paper develops a theory of Albanese maps for C-pairs in the context of hyperbolicity and rational points, introducing C-semitoric pairs and establishing existence criteria within Campana's framework.

Contribution

It introduces C-semitoric pairs as analogs of tori in Albanese theory for C-pairs and proves the existence of Albanese maps under certain conditions.

Findings

Existence of Albanese maps in relevant cases

Sharp criteria for Albanese map existence

Conjecture on unconditional weak Albanese existence

Abstract

Written with a view toward applications in hyperbolicity, rational points, and entire curves, this paper addresses the problem of constructing Albanese maps within Campana's theory of C-pairs (or "geometric orbifolds"). It introduces C-semitoric pairs as analogs of the (semi)tori used in the classic Albanese theory and follows Serre by defining the Albanese of a C-pair as the universal map to a C-semitoric pairs. The paper shows that the Albanese exists in relevant cases, gives sharp existence criteria, and conjectures that a "weak Albanese" exists unconditionally.