Asymptotic total geodesy of local holomorphic curves exiting a bounded symmetric domain and applications to a uniformization problem for algebraic subsets

Abstract

The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincaré disk into bounded symmetric domains. As a first result we prove that any holomorphic curve exiting the boundary of a bounded symmetric domain $Ω$ must necessarily be asymptotically totally geodesic. Assuming otherwise we derive by the method of rescaling a hypothetical holomorphic isometric embedding of the Poincaré disk with ${\rm Aut}(Ω')$-equivalent tangent spaces into a tube domain $Ω' \subset Ω$ and derive a contradiction by means of the Poincaré-Lelong equation. We deduce that equivariant holomorphic embeddings between bounded symmetric domains must be totally geodesic. Furthermore, we solve a uniformization problem on algebraic subsets $Z \subset Ω$. More precisely, if $\check Γ\subset {\rm Aut}(Ω)$ is a torsion-free discrete subgroup leaving $Z$ invariant such that $Z/\check Γ$ is compact, we prove that $Z \subset Ω$ is totally geodesic. In particular, letting $Γ\subset{\rm Aut}(Ω)$ be a torsion-free cocompact lattice, and $π: Ω\to Ω/Γ=: X_Γ$ be the uniformization map, a subvariety $Y \subset X_Γ$ must be totally geodesic whenever some (and hence any) irreducible component $Z$ of $π^{-1}(Y)$ is an algebraic subset of $Ω$. For cocompact lattices this yields a characterization of totally geodesic subsets of $X_Γ$ by means of bi-algebraicity without recourse to the celebrated monodromy result of André-Deligne on subvarieties of Shimura varieties, and as such our proof applies to not necessarily arithmetic cocompact lattices.