Zariski closures of images of algebraic subsets under the uniformization map on finite-volume quotients of the complex unit ball. (November 2019) - Mok, Ngaiming
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 B^n using methods of several complex variables, algebraic geometry and Kähler geometry. Consider a torsion-free lattice [STIX]x1D6E4 Aut(B^n) and the associated uniformization map [STIX]x1D70B:B^n→ B^n/[STIX]x1D6E4=:X_[STIX]x1D6E4 . Given an algebraic subset S B^n and writing Z for the Zariski closure of [STIX]x1D70B(S) in X_[STIX]x1D6E4 (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 Z of [STIX]x1D70B^-1(Z) as Z exits the boundary [STIX]x2202B^n by exploiting the strict pseudoconvexity of B^n, culminating in the proof that Z B^n is totally geodesic. Our methodology sets the stage for tackling problems in functional transcendence theory for arbitrary lattices of Aut([STIX]x1D6FA) for (possibly reducible) bounded symmetric domains [STIX]x1D6FA .