Carath\'eodory hyperbolicity, volume estimates and level structures over function fields

TL;DR

This paper generalizes the nonexistence of level structures on certain hyperbolic complex manifolds, providing volume estimates and a Schwarz Lemma for manifolds with nonsmooth Finsler metrics, impacting the understanding of maps from Riemann surfaces.

Contribution

It extends previous results on level structures and hyperbolicity to broader classes of manifolds, including those with nonsmooth Finsler metrics, and establishes new volume estimates.

Findings

Nonexistence of high-level structures on certain hyperbolic manifolds.

Volume estimates for curve mappings into these manifolds.

A Schwarz Lemma for nonsmooth Finsler metrics.

Abstract

We give a generalization of the nonexistence of level structures as Nadel, Noguchi, Hwang-To, for quasi-projective manifolds uniformized by strongly Carath\'eodory 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.