Volumes of Subvarieties of Complex Ball Quotients and Effective Very Ampleness

Soheil Memarian

Published 2023-10-12Version 1

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 $X$. As applications, 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.