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Curvature properties and Shafarevich conjecture for toroidal compactifications of ball quotients

William Sarem

TL;DR

The paper analyzes curvature and Shafarevich-type questions for toroidal compactifications $X_ ext{G}$ of ball quotients $\, abla ackslash \\mathbb{H}^{n}_{\\mathbb{C}}/\, ext{Gamma}$. It shows $X_ ext{G}$ cannot admit a Kähler metric of nonpositive sectional curvature, yet admits a metric with nonpositive holomorphic bisectional curvature, and for deep arithmetic lattices the universal cover $ ilde{X}_ ext{G}$ is Stein. A key part is the control of Albanese maps near the boundary tori, along with a general PSH-exhaustion framework that converts holomorphic convexity into Steinness of the universal cover. The results supply explicit examples solving Diverio–Kobayashi-type questions about quasi-negative bisectional curvature without negative sectional curvature and establish Shafarevich-ness in higher dimensions for a broad class of toroidal compactifications. Altogether, the work advances understanding of curvature and holomorphic convexity phenomena in complex hyperbolic geometry and their arithmetic manifestations.

Abstract

We study toroidal compactifications of finite volume complex hyperbolic manifolds. We obtain results on the existence or nonexistence of Kähler metrics satisfying certain nonpositive curvature properties on these compactifications. Starting from quotients of complex hyperbolic space by deep enough non-uniform arithmetic lattices, we also verify the Shafarevich conjecture for their compactifications, by showing that their universal covers are Stein.

Curvature properties and Shafarevich conjecture for toroidal compactifications of ball quotients

TL;DR

The paper analyzes curvature and Shafarevich-type questions for toroidal compactifications of ball quotients . It shows cannot admit a Kähler metric of nonpositive sectional curvature, yet admits a metric with nonpositive holomorphic bisectional curvature, and for deep arithmetic lattices the universal cover is Stein. A key part is the control of Albanese maps near the boundary tori, along with a general PSH-exhaustion framework that converts holomorphic convexity into Steinness of the universal cover. The results supply explicit examples solving Diverio–Kobayashi-type questions about quasi-negative bisectional curvature without negative sectional curvature and establish Shafarevich-ness in higher dimensions for a broad class of toroidal compactifications. Altogether, the work advances understanding of curvature and holomorphic convexity phenomena in complex hyperbolic geometry and their arithmetic manifestations.

Abstract

We study toroidal compactifications of finite volume complex hyperbolic manifolds. We obtain results on the existence or nonexistence of Kähler metrics satisfying certain nonpositive curvature properties on these compactifications. Starting from quotients of complex hyperbolic space by deep enough non-uniform arithmetic lattices, we also verify the Shafarevich conjecture for their compactifications, by showing that their universal covers are Stein.
Paper Structure (8 sections, 22 theorems, 73 equations)

This paper contains 8 sections, 22 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonpositively curved Kähler manifolds containing submanifolds with $c_1=0$
  4. A metric on $X_{\Gamma}$ with nonpositive holomorphic bisectional curvature
  5. Commensurators, parabolic subgroups and Albanese maps
  6. The Shafarevich conjecture for the manifold $X_\Gamma$
  7. A general result on holomorphic maps whose target have Stein universal cover
  8. A strictly plurisubharmonic exhaustion function on the universal cover of $X_\Gamma$

Key Result

Theorem 1

Let $\Gamma_0$ be a non-uniform torsion-free lattice in $\mathrm{PU}(n,1)$ for which the quotient space $\mathbb{H}^{n}_{\mathbb{C}}/\Gamma_{0}$ admits a toroidal compactification $X_{\Gamma_0}$. Then $X_{\Gamma_0}$ is a Kähler manifold. Moreover, there exists a finite index subgroup $\Gamma'<\Gamma

Theorems & Definitions (38)

  • Theorem : hummelCuspClosingRank1996
  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Proposition 6
  • proof
  • Corollary 7
  • Proposition 8
  • ...and 28 more