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One-cusped complex hyperbolic 2-manifolds

Martin Deraux, Matthew Stover

TL;DR

The authors construct the first explicit geometric family of complete one-cusped complex hyperbolic 2-manifolds by starting from genus-four curves with an A4 symmetry, forming a product quotient, and resolving singularities to obtain a minimal surface Z with an elliptic curve E such that Z\setminus E is a ball quotient. They establish the base case (Z1,E1) and then produce all odd-d versions (Zd,Ed) via étale covers, preserving the ball-quotient structure and yielding manifolds of volume $16\pi^2 d$. A key consequence is that every odd-d 3-dimensional nilmanifold with Euler number $12d$ geometrically bounds a complex hyperbolic 2-manifold, with divisibility constraints on cusp data arising from toroidal compactifications. The work blends explicit group-theoretic constructions, product-quotient surfaces, and Kobayashi’s uniformization criterion to realize new higher-dimensional ball quotients and geometric bounding phenomena.

Abstract

This paper builds one-cusped complex hyperbolic $2$-manifolds by an explicit geometric construction. Specifically, for each odd $d \ge 1$ there is a smooth projective surface $Z_d$ with $c_1^2(Z_d) = c_2(Z_d) = 6d$ and a smooth irreducible curve $E_d$ on $Z_d$ of genus one so that $Z_d \smallsetminus E_d$ admits a finite volume uniformization by the unit ball $\mathbb{B}^2$ in $\mathbb{C}^2$. This produces one-cusped complex hyperbolic $2$-manifolds of arbitrarily large volume. As a consequence, the $3$-dimensional nilmanifold of Euler number $12d$ bounds geometrically for all odd $d \ge 1$.

One-cusped complex hyperbolic 2-manifolds

TL;DR

The authors construct the first explicit geometric family of complete one-cusped complex hyperbolic 2-manifolds by starting from genus-four curves with an A4 symmetry, forming a product quotient, and resolving singularities to obtain a minimal surface Z with an elliptic curve E such that Z\setminus E is a ball quotient. They establish the base case (Z1,E1) and then produce all odd-d versions (Zd,Ed) via étale covers, preserving the ball-quotient structure and yielding manifolds of volume . A key consequence is that every odd-d 3-dimensional nilmanifold with Euler number geometrically bounds a complex hyperbolic 2-manifold, with divisibility constraints on cusp data arising from toroidal compactifications. The work blends explicit group-theoretic constructions, product-quotient surfaces, and Kobayashi’s uniformization criterion to realize new higher-dimensional ball quotients and geometric bounding phenomena.

Abstract

This paper builds one-cusped complex hyperbolic -manifolds by an explicit geometric construction. Specifically, for each odd there is a smooth projective surface with and a smooth irreducible curve on of genus one so that admits a finite volume uniformization by the unit ball in . This produces one-cusped complex hyperbolic -manifolds of arbitrarily large volume. As a consequence, the -dimensional nilmanifold of Euler number bounds geometrically for all odd .
Paper Structure (13 sections, 15 theorems, 33 equations, 5 figures)

This paper contains 13 sections, 15 theorems, 33 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The main result, history, and context
  3. The main technical result
  4. Geometric bounding
  5. Background
  6. Construction of the first example
  7. The curves of genus four
  8. The $(2,3,12)$-triangle group
  9. Two subgroups of index $6$ in $\Gamma_0$
  10. The product action
  11. The elliptic curve $E$
  12. Proof that $(Z, E)$ is a ball quotient pair
  13. The proof of Theorem \ref{['thm:MainTech']}

Key Result

Theorem 1.1

For each odd $d \ge 1$, there is a one-cusped complex hyperbolic $2$-manifold of volume $16 \pi^2 d$.

Figures (5)

  • Figure 1: Fundamental domains for $\Gamma_1$ (left) and $\Gamma_2$ (right), as copies of a fundamental domain for $\Gamma_0$ (shown in bold on both pictures)
  • Figure 2: Coverings of hyperbolic orbifolds
  • Figure 3: Further coverings of hyperbolic orbifolds
  • Figure 4: The $\Gamma_0 \times \Gamma_0$ orbits of $\widetilde{D}$ that meet $\widetilde{D}$
  • Figure 5: The immersion of $D_3$ in $X_3$

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Remark 3.1
  • Proposition 3.2
  • ...and 18 more