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

The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings Poincaré disk into bounded symmetric domains. As a first result we prove that any curve exiting boundary domain $\Omega$ must necessarily be asymptotically totally geodesic. Assuming otherwise derive by method rescaling hypothetical embedding with $\operatorname{Aut}(\Omega^\prime)$-equivalent tangent spaces tube $\Omega^\prime \subset \Omega$ and contradiction means Poincaré–Lelong equation. We deduce equivariant between domains Furthermore, solve uniformization problem algebraic subsets $Z \Omega$. More precisely, if $\check{\Gamma} \operatorname{Aut} (\Omega)$ is torsion-free discrete subgroup leaving $Z$ invariant such / \check{\Gamma}$ compact, In particular, letting $\Gamma cocompact lattice, $\pi : \Omega \to \Gamma =: X_\Gamma$ map, subvariety $Y geodesic whenever some (and hence any) irreducible component $\pi-(Y)$ an subset $\Omega$. For lattices this yields characterization $X_\Gamma$ bi-algebraicity without recourse to celebrated monodromy André–Deligne subvarieties Shimura varieties, as proof applies not arithmetic lattices. place exploit existence theorem Aubin Yau Kähler–Einstein metrics for projective manifolds $Y$ satisfying $c_1 (Y) \lt 0$ make use Nadel’s semisimplicity automorphism groups noncompact Galois covers manifolds, together total geodesy