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

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 $A$.