Universal Covering Maps onto Finite-Volume Quotients of Bounded Symmetric Domains from the Perspective of Complex Differential Geometry

Abstract

By the Uniformization Theorem a compact Riemann surface other than the Riemann Sphere or an elliptic curve is uniformized by the unit disk and equivalently by the upper half plane. The upper half plane is also the universal covering space of the moduli space of elliptic curves equipped with a suitable level structure. In Several Complex Variables, the Siegel upper half plane (which is biholomorphic to a bounded symmetric domain) is an analogue of the upper half plane, and it is the universal covering space of moduli spaces of polarized Abelian varieties with level structures. In general, finite-volume quotients of bounded symmetric domains, which are naturally quasi-projective varieties, are of immense interest to Several Complex Variables, Algebraic Geometry and Number Theory, and an important object of study is the universal covering map πΓ : Ω → XΓ from a bounded symmetric domain Ω onto its quotient XΓ := Ω/Γ by a torsion-free discrete lattice Γ ⊂ Aut(Ω). We will explain a differential-geometric approach to the study of the universal covering map revolving around the notion of asymptotic curvature behavior, rescaling arguments and the use of meromorphic foliations, and illustrate how this approach using transcendental techniques leads to various results concerning totally geodesic subvarieties of finite-volume quotients without the assumption of arithmeticity.