Punctured spheres in complex hyperbolic surfaces and bielliptic ball quotient compactifications

TL;DR

This paper investigates punctured spheres in complex hyperbolic surfaces, establishing non-existence results, ampleness criteria for certain line bundles, and constructing new bielliptic ball quotient examples based on Gaussian integers.

Contribution

It proves that smooth toroidal compactifications cannot contain certain punctured spheres, establishes ampleness of line bundles for specific parameters, and introduces the first bielliptic ball quotient examples with Gaussian integer structure.

Findings

Smooth toroidal compactifications lack properly embedded 3-punctured spheres.

Ampleness of $K_X + eta D$ for $eta ext{ in } (rac{1}{4}, 1)$ is established.

First bielliptic ball quotient examples based on Gaussian integers are constructed.

Abstract

In this paper, we study punctured spheres in two dimensional ball quotient compactifications $(X, D)$. For example, we show that smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded $3$-punctured spheres. We also use totally geodesic punctured spheres to prove ampleness of $K_X + \alpha D$ for $\alpha \in (\frac{1}{4}, 1)$, giving a sharp version of a theorem of the first author with G. Di Cerbo. Finally, we produce the first examples of bielliptic ball quotient compactifications modeled on the Gaussian integers.