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

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