Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Carathéodory hyperbolicity, volume estimates and level structures over function fields

Kwok-Kin Wong, Sai-Kee Yeung

TL;DR

The paper generalizes the nonexistence of level structures for towers of quasi-projective manifolds uniformized by strongly Carathéodory hyperbolic spaces, including moduli spaces and locally Hermitian symmetric spaces. It develops volume-estimate machinery for curves, combining a Schwarz-type lemma for nonsmooth complex Finsler metrics with a Kähler–Einstein comparison and deformation arguments. The main results show that, for sufficiently high level, any genus-$g_0$ holomorphic map into a compactification must land in the boundary, implying no level structure over function fields in these cases. These methods connect hyperbolicity, volume growth, and ramification/degeneration phenomena, with implications for functional transcendence in period-domain-type settings.

Abstract

We give a generalization of the nonexistence of level structures as Nadel, Noguchi, Hwang-To, for quasi-projective manifolds uniformized by strongly Carathéodory hyperbolic complex manifolds. Examples include moduli space of compact Riemann surfaces with a finite number punctures and locally Hermitian symmetric spaces of finite volume. This leads to the nonexistence of a holomorphic map from a Riemann surface of fixed genus into the compactification of such a quasi-projective manifold when the level structure is sufficiently high. To achieve our goal, we have also established some volume estimates for mapping of curves into these manifolds, extending some earlier result of Hwang-To to a more general setting. A version of Schwarz Lemma applicable to manifolds equipped with nonsmooth complex Finsler metric is also given.

Carathéodory hyperbolicity, volume estimates and level structures over function fields

TL;DR

The paper generalizes the nonexistence of level structures for towers of quasi-projective manifolds uniformized by strongly Carathéodory hyperbolic spaces, including moduli spaces and locally Hermitian symmetric spaces. It develops volume-estimate machinery for curves, combining a Schwarz-type lemma for nonsmooth complex Finsler metrics with a Kähler–Einstein comparison and deformation arguments. The main results show that, for sufficiently high level, any genus- holomorphic map into a compactification must land in the boundary, implying no level structure over function fields in these cases. These methods connect hyperbolicity, volume growth, and ramification/degeneration phenomena, with implications for functional transcendence in period-domain-type settings.

Abstract

We give a generalization of the nonexistence of level structures as Nadel, Noguchi, Hwang-To, for quasi-projective manifolds uniformized by strongly Carathéodory hyperbolic complex manifolds. Examples include moduli space of compact Riemann surfaces with a finite number punctures and locally Hermitian symmetric spaces of finite volume. This leads to the nonexistence of a holomorphic map from a Riemann surface of fixed genus into the compactification of such a quasi-projective manifold when the level structure is sufficiently high. To achieve our goal, we have also established some volume estimates for mapping of curves into these manifolds, extending some earlier result of Hwang-To to a more general setting. A version of Schwarz Lemma applicable to manifolds equipped with nonsmooth complex Finsler metric is also given.
Paper Structure (8 sections, 16 theorems, 67 equations)

This paper contains 8 sections, 16 theorems, 67 equations.

Table of Contents

  1. Compact Riemann surface
  2. Volume estimates
  3. Lower bound in volume
  4. Upper bound in volume
  5. Injectivity radius lower bound
  6. Ramification index lower bound
  7. Proof of Main Theorems
  8. Appendix: Sharp volume lower bounds

Key Result

Theorem 1

Let $M=\overline M- D$ be a quasi-projective manifold uniformized by a strongly Carathéodory hyperbolic manifold $\widetilde{M}$, or a Carathéodory hyperbolic manifold equipped with a smooth bounded plurisubharmonic exhaustion function. Suppose $M$ supports a tower of coverings $\{M_i\}_{i=1}^\infty Let $g_0\geqslant 2$ be a fixed nonnegative integer. Then there exists $i_0\geqslant 0$ such that f

Theorems & Definitions (36)

  • Theorem 1
  • Remark
  • Remark
  • Theorem 2
  • Theorem 3
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Remark
  • Lemma 2.3
  • ...and 26 more