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

TL;DR

This paper proves that holomorphic curves exiting a bounded symmetric domain are asymptotically totally geodesic and applies this to characterize totally geodesic subvarieties in uniformized algebraic subsets, advancing understanding of geometric structures in complex domains.

Contribution

It establishes asymptotic total geodesy of boundary-exiting holomorphic curves and characterizes totally geodesic algebraic subsets in uniformizations of bounded symmetric domains.

Findings

Holomorphic curves exiting the boundary are asymptotically totally geodesic.

Equivariant holomorphic embeddings between bounded symmetric domains are totally geodesic.

Totally geodesic subvarieties are characterized by bi-algebraicity in uniformized algebraic subsets.

Abstract

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