Soheil Memariansorkhabi

arXiv2025

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

Let $X=Γ\backslash \mathbb{B}^{n} $ be an $n$-dimensional complex ball quotient by a torsion-free non-uniform lattice $Γ$ 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.

math.AG↗

arXiv2024

Positivity of the Cotangent Bundle of Complex Hyperbolic Manifolds with Cusps

Let $\overline{X}$ be the toroidal compactification of a cusped complex hyperbolic manifold $X=\mathbb{B}^n/Γ$ with the boundary divisor $D=\overline{X}\setminus X$. The main goal of this paper is to find the positivity properties of $Ω^{1}_{\overline{X}}$ and $Ω^{1}_{\overline{X}}\big(\log(D)\big)$ depending intrinsically on $X$. We prove that $Ω^{1}_{\overline{X}}\big(\log(D)\big) \langle -r D \rangle$ is ample for all sufficiently small rational numbers $r >0$, and $Ω^{1}_{\overline{X}}\big(\log(D)\big)$ is ample modulo $D.$ Further, we conclude that if the cusps of $X$ have uniform depth greater than $4π$, then $Ω^{1}_{\overline{X}}$ is semi-ample and is ample modulo $D$, all subvarieties of $X$ are of general type, and every smooth subvariety $V\subset \overline{X}$ intersecting $\overline{X}$ has ample $K_{V}$. Finally, we show that the minimum volume of subvarieties of $\overline{X}$ intersecting both $X$ and $D$ tends to infinity in towers of normal covering of $X.$

math.AG↗