Curvature and Uniformization on Bounded Symmetric Domains

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 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. The Siegel upper half plane belongs, up to biholomorphic equivalence, to the set of bounded symmetric domains, on which a great deal of mathematical research is taking place. Especially, finite-volume quotients of bounded symmetric domains, which are naturally quasi-projective varieties, are objects of immense interest to Several Complex Variables, Algebraic Geometry and Number Theory, and an important topic is the study of uniformizations of algebraic subsets of such quasi-projective varieties. While a lot has already been achieved from methods of Diophantine Geometry, Model Theory, Hodge Theory and Algebraic Geometry for Shimura varieties, techniques for the general case of not necessarily arithmetic quotients have just begun to be developed. We will explain a differential-geometric approach to the study of such algebraic subsets revolving around the notion of asymptotic curvature behavior and the use of rescaling arguments, and illustrate how this approach using transcendental techniques leads to various characterization results for totally geodesic subvarieties of finite-volume quotients without the assumption of arithmeticity.