Curves on Compact Arithmetic Quotients of Hyperbolic 2-ball
Zhehao Li
TL;DR
The paper proves that compact arithmetic quotients of the complex hyperbolic 2-ball $\mathbb{B}^2$, arising from CM-type constructions over a real quadratic field, contain no nontrivial complex curves of fixed genus once the real discriminant $D$ and the relative discriminant $D'$ are large. The authors develop volume-based techniques to bound curve multiplicities near elliptic loci, classify and bound stabilizers of elliptic points and lines, and obtain injectivity-radius estimates that lead to a contradiction for large $D$. A key outcome is the repulsion of elliptic points and robust tubular-neighborhood multiplicity bounds, which together with Schwarz–Gauss–Bonnet-type arguments yield isotriviality results for families of abelian sixfolds with $\mathcal{O}_F$-endomorphism. The work provides a detailed geometric and arithmetic analysis of singularities in complex hyperbolic quotients and connects these to a moduli interpretation in terms of polarized $\mathcal{O}_F$-type abelian varieties. Overall, it advances our understanding of how discriminant growth constrains the geometry and moduli of higher-dimensional Shimura-type spaces.
Abstract
We study the geometry of the simplest type of compact arithmetic quotients of the hyperbolic 2-ball $\mathbb{B}^2$, which has a moduli interpretation for certain types of abelian varieties of dimension 6 with $\mathcal{O}_F$-endomorphism, where $F$ is a CM extension of a real quadratic field $\mathbb{Q}(\sqrt{D})$. Under mild assumption, we prove that for any fixed $g$, when the defining discriminant $D$ is large, there will be no complex curves of genus $g$ on this type of arithmetic quotients. The proof uses the technique of volume estimates, which requires us to understand the distribution of special subvarieties and the geometry near quotient and cusp singularities.