Applications of principal isogenies to constructions of ball quotient surfaces

TL;DR

This paper explores how principal isogenies can be used to construct and analyze infinite series of non-birational and birational models of ball quotient surfaces, advancing the understanding of their geometric structures.

Contribution

It introduces a method using principal isogeny pull-backs to generate infinite series of non-birational co-abelian covers of ball quotient surfaces.

Findings

Constructed infinite series of non-birational torsion free Galois covers.

Produced infinite series of birational models of ball quotient surfaces.

Extended the understanding of the geometric structures of ball quotient surfaces.

Abstract

Let $(({\mathbb B} / \Gamma_1)', T(1))$ be \'a torsion free toroidal compactification with abelian minimal model $(A_1, D(1))$. An arbitrary principal isogeny $\mu_a : A_2 \rightarrow A_1$, $a \in {\mathbb C}$ pulls-back $(A_1, D(1))$ to the abelian minimal model $(A_2, D(2))$ of a torsion free toroidal compactification $(({\mathbb B} / \Gamma_2)', D(2))$. The present work makes use of the isogeny pull-backs of abelian ball quotient models, towards the construction of infinite series of mutually non-birational co-abelian torsion free Galois covers $(({\mathbb B} / \Gamma_n)', T(n))$ of a ball quotient compactification $\bar{{\mathbb B} / \Gamma_H}$ of Kodaira dimension $\kappa (\bar{{\mathbb B} / \Gamma_H}) \leq 0$. It provides also infinite series of birational models of certain ${\mathbb B} / \Gamma_H$.