On the Zariski closure of an infinite number of totally geodesic subvarieties of Ω/Γ

Abstract

Let $Omega$ be a bounded symmetric domain, $Gamma subset ext{Aut}(Omega)$ be a torsion-free lattice, $X := Omega/Gamma$. Let $Z subset X$ be an irreducible quasi-projective variety such that $Z$ is the Zariski closure of the union of an infinite family of totally-geodesic complex subvarieties $S_alpha subset Z, , alpha in A$. Under a non-degeneracy condition one expects $Z$ to be also totally geodesic, so that $Z$ is again uniformized by a bounded symmetric domain. This set-up is related to a well-known problem on Shimura varieties $X = Omega/Gamma$ in which one tries to characterize the Zariski closure of an infinite number of `special' subvarieties $S_alpha$. Here special subvarieties are defined by arithmetic conditions, but it is known that they are always totally geodesic. While the case where $S_alpha$ are 0-dimensional, for which the problem is called the André-Oort Conjecture, cannot be dealt with directly using methods of complex geometry, the case where $S_alpha$ are of positive dimension is naturally a problem in complex geometry. From our complex-analytic perspective, no arithmeticity assumption is placed on $Gamma subset ext{Aut}(Omega)$, and the `distinguished' subvarieties are simply the totally-geodesic subvarieties of $X$.

Using methods of Kähler geometry, we solve the afore-mentioned problem in the rank-1 case. A generalization of the argument to bounded symmetric domains $Omega$ leads to the study of holomorphic isometries from complex unit balls $B^m$ to $Omega$. We explain a method along this line of thoughts for solving the general problem, and illustrate how the problem is solved when $Z$ is a complex surface and $S_alpha subset Z$ are totally-geodesic holomorphic curves using our recent results on holomorphic isometries with respect to the Bergman metric.