Transcendence Theory over Function Fields on Quotients of Bounded Symmetric Domains

Finite-volume quotients of bounded symmetric domains $\Omega$, 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 revolves around functional transcendence in relation to universal covering maps of such varieties. While a lot has already been achieved in the case of Shimura varieties by means of methods of Model Theory, Hodge Theory and Complex Differential Geometry, techniques for the general case of not necessarily arithmetic quotients $\Omega / \Gamma =: X\Gamma$ have just begun to be developed. For instance, Ax-type problems for subvarieties of products of arbitrary compact Riemann surfaces of genus $\ge 2$ have hitherto been intractable by existing methods. We will explain how uniformization theorems for bi-algebraic varieties can be proven by analytic methods involving the Poincaré-Lelong equation in the cocompact case (joint work with S.-T. Chan), generalizing in the absence of the emisimplicity theorem of André-Deligne for monodromy groups (proven for arithmetic lattices). Klingler-Ullmo-Yafaev (2016) resolved the hyperbolic Ax-Lindemann Conjecture for Shimura varieties in the affirmative ascertaining that the Zariski closure of the image $\pi(S)$ of an algebraic subset $S\subset\Omega$ under the universal covering map $\pi : \Omega \to X\Gamma$ is totally geodesic. I will explain how the arithmeticity condition can be dropped in the cocompact case by a completely different proof using foliation theory, Chow schemes, partial Cayley transforms and Kähler geometry.