On Effective Existence of Symmetric Differentials of Complex Hyperbolic Space Forms

TL;DR

This paper establishes conditions under which symmetric differentials exist on noncompact complex hyperbolic space forms, linking geometric measurements of cusps to the ampleness of the cotangent bundle.

Contribution

It introduces the canonical radius as a geometric measure and proves the existence of symmetric differentials vanishing at infinity under certain cusp conditions.

Findings

Existence of symmetric differentials when cusp radii exceed a dimension-dependent threshold.

Ampleness of the cotangent bundle modulo infinity under interior injectivity radius conditions.

A new geometric criterion for producing symmetric differentials on complex hyperbolic space forms.

Abstract

For a noncompact complex hyperbolic space form of finite volume $X=\mathbb{B}^n/\Gamma$, we consider the problem of producing symmetric differentials vanishing at infinity on the Mumford compactification $\overline{X}$ of $X$ similar to the case of producing cusp forms on hyperbolic Riemann surfaces. We introduce a natural geometric measurement which measures the size of the infinity $\overline{X}-X$ called `canonical radius' of a cusp of $\Gamma$. The main result in the article is that there is a constant $r^*=r^*(n)$ depending only on the dimension, so that if the canonical radii of all cusps of $\Gamma$ are larger than $r^*$, then there exist symmetric differentials of $\overline{X}$ vanishing at infinity. As a corollary, we show that the cotangent bundle $T_{\overline{X}}$ is ample modulo the infinity if moreover the injectivity radius in the interior of $\overline{X}$ is larger than some constant $d^*=d^*(n)$ which depends only on the dimension.