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 $\Omega$ 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 $\operatorname{Aut}(\Omega^\prime)$-equivalent tangent spaces into a tube domain $\Omega^\prime \subset \Omega$ 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 $Z \subset \Omega$. More precisely, if $\check{\Gamma} \subset \operatorname{Aut} (\Omega)$ 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 $\Gamma \subset \operatorname{Aut} (\Omega)$ 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 $\pi-(Y)$ 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 André–Deligne on subvarieties of Shimura varieties, and as such our proof applies to not necessarily arithmetic cocompact lattices. In place of the monodromy result of André–Deligne we exploit the existence theorem of Aubin and Yau on Kähler–Einstein metrics for projective manifolds $Y$ satisfying $c_1 (Y) \lt 0$ and make use of Nadel’s semisimplicity theorem on automorphism groups of noncompact Galois covers of such manifolds, together with the total geodesy of equivariant holomorphic isometric embeddings between bounded symmetric domains.