Uniformization of Subvarieties of Finite-Volume Quotient Spaces of Bounded Symmetric Domains

Abstract

The Siegel upper half plane Hg belongs, up to biholomorphic equivalence, to the set of bounded symmetricdomains, 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, Arithmetic Geometry and Number Theory, and an important topic is the study of covering spaces of algebraic subsets of such quasi-projective varieties. While a lot has already been achieved in the case of Shimura varieties by means of methods of Diophantine Geometry, Model Theory, Hodge Theory and Complex Differential Geometry, techniques for the general case of not necessarily arithmetic quotients =ΩΓ⁄have just begun to be developed. For instance, uniformization problems for subvarieties of products of arbitrary compact Riemann surfaces of genus ≥2 have hitherto been untractable by existing methods. We will explain a differential-geometric approach leading to various characterization results for totally geodesic subvarieties of finite-volume quotients =ΩΓ⁄. Especially, we will explain how the study of holomorphic isometric embeddings of the Poincare disk and more generally complex unit balls into bounded symmetric domains can be further developed to derive uniformization theorems for bi-algebraic varieties and more generally for the Zariski closure of images of algebraic sets under the universal covering map.