Hyperbolicity and Rational Points on Complex Ball Quotients

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. Let $\overline{X}$ be the unique toroidal complication of $X.$ In the first part of this thesis we prove positivity properties of $\Omega^{1}_{\overline{X}}$ and $\Omega^{1}_{\Xb}\big(\log(D)\big)$ depending intrinsically on $X$. We prove that $\Omega^{1}_{\Xb}\big(\log(D)\big) \langle} -r D \rangle}$ is ample for all sufficiently small rational numbers $r >0$, and $\Omega^{1}_{\Xb}\big(\log(D)\big)$ is ample modulo $D.$ Further, we conclude that if the cusps of $X$ have uniform depth greater than $4\pi$, then $\Omega^{1}_{\Xb}$ is semi-ample and is ample modulo $D$, and all subvarieties of $X$ are of general type. In the second part of this thesis 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 of $\overline{X}.$ Also, we obtain effective global generation and very ampleness results for multiples of the canonical bundle $K_{\overline{X}}.$ These results follow from the bound we find for the Seshadri constant of $K_{\overline{X}}$ in terms of the systole.

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