Curves on Compact Arithmetic Quotients of Hyperbolic 2-ball

We study the geometry of the simplest type of compact arithmetic quotients of the hyperbolic 2-ball ${\mathbb{B}}^2$, which has a moduli interpretation for certain types of abelian varieties of dimension 6 with ${\mathcal{O}}_F$-endomorphism, where $F$ is a CM extension of a real quadratic field $\mathbb{Q}(\sqrt{D})$. Under mild assumption, we prove that for any fixed $g$, when the defining discriminant $D$ is large, there will be no complex curves of genus $g$ on this type of arithmetic quotients. The proof uses the technique of volume estimates, which requires us to understand the distribution of special subvarieties and the geometry near quotient and cusp singularities.