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

TL;DR

This paper proves the finiteness of non-constant morphisms from a normal variety to a simple abelian variety under specific tangency conditions, using geometric analysis of the associated Hom scheme.

Contribution

It establishes a finiteness result for morphisms into simple abelian varieties satisfying tangency conditions, extending understanding of their mapping properties.

Findings

Finiteness of certain morphisms into simple abelian varieties.

Construction of a quasi-finite non-dominant morphism from the Hom scheme to the abelian variety.

Application of geometric techniques to study the scheme parametrizing these morphisms.

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 $\Delta$ on $A$ is finite. To do so, we study the geometry of the scheme $\underline{\mathrm{Hom}}^{\mathrm{nc}}(C, (A, \Delta))$ parametrizing such morphisms from a smooth curve $C$ and show that it admits a quasi-finite non-dominant morphism to $A$.