Zariski closures of images of algebraic subsets under the uniformization map on finite-volume quotients of the complex unit ball. (19th September 2019)

• Record Type: Journal Article
• Authors: Mok, Ngaiming
• Is Part Of: Compositio mathematica. Volume 155:Number 11(2019)
• Page Start: 2129
• Page End: 2149
• Publication Date: 2019-09-19
• Subjects: 11J81, -- 14C05, -- 32F32, -- 53C55 (primary); bounded symmetric domain, -- uniformization, -- Chow space, -- foliation, -- Kähler metric, -- asymptotic geometry, -- strictly pseudoconvex domains; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X19007577
• Languages: English
• ISSNs: 0010-437X

Abstract

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 $\mathbb{B}^{n}$ using methods of several complex variables, algebraic geometry and Kähler geometry. Consider a torsion-free lattice $\unicode[STIX]{x1D6E4}\, \subset \, \text{Aut}(\mathbb{B}^{n})$ and the associated uniformization map $\unicode[STIX]{x1D70B}:\mathbb{B}^{n}\rightarrow \mathbb{B}^{n}/\unicode[STIX]{x1D6E4}=:X_{\unicode[STIX]{x1D6E4}}$ . Given an algebraic subset $S\, \subset \, \mathbb{B}^{n}$ and writing $Z$ for the Zariski closure of $\unicode[STIX]{x1D70B}(S)$ in $X_{\unicode[STIX]{x1D6E4}}$ (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 $\widetilde{Z}$ of $\unicode[STIX]{x1D70B}^{-1}(Z)$ as $\widetilde{Z}$ exits the boundary $\unicode[STIX]{x2202}\mathbb{B}^{n}$ by exploiting the strict pseudoconvexity of $\mathbb{B}^{n}$, culminating in the proof that $\widetilde{Z}\, \subset \, \mathbb{B}^{n}$ is totally geodesic. Our methodology sets the stage for tackling problems in functional transcendence theory for arbitrary lattices of $\text{ Aut}(\unicode[STIX]{x1D6FA})$ for (possibly reducible) bounded symmetric domains $\unicode[STIX]{x1D6FA}$ .