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

Abstract

Let $X=\Gamma \backslash \mathbb{B}^{n} $ be an $n$-dimensional complex ball quotient by a torsion-free non-uniform lattice $\Gamma$ 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 $K_{\overline{X}},$ where $\overline{X}$ is the toroidal compactification of $X.$ These results follow from the bound we find for the Seshadri constant of $K_{\overline{X}}$ in terms of the systole.