Volumes of Subvarieties of Complex Ball Quotients and Sparsity of Rational Points

Let $X=\mathrm{\Gamma }\mathrm{\backslash }{\mathbb{B}}^n$ be an $n$-dimensional complex ball quotient by a torsion-free non-uniform lattice $\mathrm{\Gamma }$ whose parabolic subgroups are unipotent. We prove that the volumes of subvarieties of $X$ are controlled by the systole of $X,$ which is the length of the shortest closed geodesic of $X$. There are a number of arithmetic and geometric consequences: the systole of $X$ controls the growth rate of rational points on $X,$ uniformly in the field of definition. Also, we obtain effective global generation and very ampleness results for multiples of the canonical bundle $K_{\bar{X}},$ where $\bar{X}$ is the toroidal compactification of $X\mathrm{.}$ These results follow from the bound we find for the Seshadri constant of $K_{\bar{X}}$ in terms of the systole.