On the finiteness of maps into simple abelian varieties satisfying certain tangency conditions

TL;DR

This work proves the finiteness of non-constant morphisms from a normal variety $V$ to a simple abelian variety $A$ under tangency constraints encoded by a nonzero Campana orbifold divisor $\Delta$ on $A$, over fields finitely generated over $\mathbb{Q}$. The method reduces to the curve case and studies the morphism-scheme $\underline{\mathrm{Hom}}^{\mathrm{nc}}(C,(A,\Delta))$, establishing hyperbolicity and a quasi-finite, non-dominant map to $A$. By showing $(A,\Delta)$ is of general type and hyperbolic, the authors obtain finiteness of $K$-rational near-maps via Faltings's theorem on subvarieties of abelian varieties, and derive corollaries for root stacks. These results support conjectures on the Mordellicity of Campana orbifolds and clarify the relationship between Campana morphisms and root-stack morphisms, with implications for arithmetic and geometric hyperbolicity in this setting.

Abstract

We show that given a simple abelian variety $A$ and a normal variety $V$ defined over a finitely generated field $K$ of characteristic zero, the set of non-constant morphisms $V \to A$ satisfying certain tangency conditions imposed by a Campana orbifold divisor $Δ$ on $A$ is finite. To do so, we study the geometry of the scheme $\underline{\mathrm{Hom}}^{\mathrm{nc}}(C, (A, Δ))$ parametrizing such morphisms from a smooth curve $C$ and show that it admits a quasi-finite non-dominant morphism to $A$.

Theorems & Definitions (20)

• Theorem A
• Corollary B
• Theorem 1.1
• proof
• Theorem C
• Remark 1.2
• Remark 1.3
• Definition 2.1
• Definition 2.2
• Lemma 2.3