Multiple realizations of varieties as ball quotient compactifications

TL;DR

This paper explores the diversity of smooth projective surfaces that can serve as compactifications of ball quotients, revealing the existence of infinitely many such quotients with identical compactifications.

Contribution

It constructs arbitrarily large families of distinct ball quotients sharing the same smooth toroidal compactification, extending Hirzebruch's earlier results.

Findings

Existence of infinitely many ball quotients with the same compactification.

Construction of arbitrarily large families of such quotients.

Extension of Hirzebruch's work on ball quotient compactifications.

Abstract

We study the number of distinct ways in which a smooth projective surface $X$ can be realized as a smooth toroidal compactification of a ball quotient. It follows from work of Hirzebruch that there are infinitely many distinct ball quotients with birational smooth toroidal compactifications. We take this to its natural extreme by constructing arbitrarily large families of distinct ball quotients with biholomorphic smooth toroidal compactifications.