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

Soheil Memariansorkhabi

TL;DR

The paper studies hyperbolicity and arithmetic of non-compact complex ball quotients $X=\Gamma\backslash \mathbb{B}^n$ with unipotent cusps, establishing a universal lower bound for subvariety volumes in terms of the systole $\operatorname{sys}(X)$ and a thick-thin decomposition that links geometry to a uniform cusp depth. It then derives effective positivity results for multiples of the toroidal canonical bundle $K_{\overline{X}}$, via Seshadri constants and jet separation, under large $\operatorname{sys}(X)$, and applies Bakker–Tsimerman’s ampleness criteria to obtain global generation and very ampleness modulo the boundary. The work further connects these geometric bounds to arithmetic by bounding the growth of rational points through the determinant method, with exponents that decay as the systole grows. Overall, the results demonstrate that the systole governs both hyperbolicity and Diophantine properties of ball quotients, yielding uniform, arithmetic consequences in addition to refined geometric positivity statements.

Abstract

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.

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

TL;DR

The paper studies hyperbolicity and arithmetic of non-compact complex ball quotients with unipotent cusps, establishing a universal lower bound for subvariety volumes in terms of the systole and a thick-thin decomposition that links geometry to a uniform cusp depth. It then derives effective positivity results for multiples of the toroidal canonical bundle , via Seshadri constants and jet separation, under large , and applies Bakker–Tsimerman’s ampleness criteria to obtain global generation and very ampleness modulo the boundary. The work further connects these geometric bounds to arithmetic by bounding the growth of rational points through the determinant method, with exponents that decay as the systole grows. Overall, the results demonstrate that the systole governs both hyperbolicity and Diophantine properties of ball quotients, yielding uniform, arithmetic consequences in addition to refined geometric positivity statements.

Abstract

Let be an -dimensional complex ball quotient by a torsion-free non-uniform lattice whose parabolic subgroups are unipotent. We prove that the volumes of subvarieties of are controlled by the systole of which is the length of the shortest closed geodesic of . There are a number of arithmetic and geometric consequences: the systole of controls the growth rate of rational points on uniformly in the field of definition. Also, we obtain effective global generation and very ampleness results for multiples of the canonical bundle where is the toroidal compactification of These results follow from the bound we find for the Seshadri constant of in terms of the systole.
Paper Structure (16 sections, 56 theorems, 116 equations)

This paper contains 16 sections, 56 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Geometry of complex ball quotients
  4. Siegel domain model
  5. Bergman metric
  6. Toroidal compactification
  7. Stabilizer of cusps
  8. Neighborhood of cusps
  9. Systole and depth of cusps
  10. Thick-thin decomposition
  11. Volume estimate of subvarieties
  12. Effective global generation and very ampleness
  13. Base-point freeness and very ampleness on $D$
  14. Global generation and very ampleness on $\overline{X}$
  15. Seshadri Constant
  16. ...and 1 more sections

Key Result

Theorem A

(Theorem Bound Hyperbolic Volume) Let $X=\Gamma \backslash \mathbb{B}^{n}$ be a complex ball quotient by a torsion-free non-uniform lattice $\Gamma$ whose parabolic stabilizers are unipotent. Let $V\subset X$ be an irreducible subvariety of dimension $m>0.$ Then, where $\operatorname{vol}_{X}(V)$ is the volume of $V$ induced by the Bergman metric on $V$ and $\operatorname{sys}(X)$ is the length o

Theorems & Definitions (109)

  • Theorem A
  • Theorem B
  • Corollary C
  • Corollary D
  • Corollary E
  • Corollary F
  • Remark A
  • Lemma B
  • Lemma C
  • proof
  • ...and 99 more