Zariski closures of images of algebraic subsets under the uniformization map on finite-volume quotients of the complex unit ball

Abstract

We prove the analogue of the Ax-Lindemann-Weierstrass Theorem for not necessarily arithmetic lattices of the automorphism group of the complex unit ball Bn using methods of several complex variables, algebraic geometry and K¨ahler geometry. Consider a torsion-free lattice Γ ⊂ Aut(Bn) and the associated uniformization map π : Bn → Bn/Γ =: XΓ. Given an algebraic subset S ⊂ Bn and writing Z for the Zariski closure of π(S) in XΓ (which is equipped with a canonical quasi-projective structure), in some precise sense we realize Z as a variety uniruled by images of algebraic subsets under the uniformization map, and study the asymptotic geometry of an irreducible component Ze of π−1(Z) as Ze exits the boundary ∂Bn by exploiting the strict pseudoconvexity of Bn, culminating in the proof that Ze ⊂ Bn is totally geodesic. Our methodology sets the stage for tackling problems in functional transcendence theory for arbitrary lattices of Aut(Ω) for (possibly reducible) bounded symmetric domainsΩ.