The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincare 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 Poincare disk with ${\rm Aut}(\Omega')$-equivalent tangent spaces into a tube domain $\Omega' \subset \Omega$ and derive a contradiction by means of the Poincare-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 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 Andre-Deligne on subvarieties of Shimura varieties, and as such our proof applies to not necessarily arithmetic cocompact lattices.

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退出有界对称域的局部全纯曲线的渐近全大地测量及其在代数子集均匀化问题中的应用

当前的文章源于我们对 Poincare 圆盘的全纯等距嵌入到有界对称域的渐近行为的研究。作为第一个结果，我们证明任何离开有界对称域 $\Omega$ 边界的全纯曲线必然是渐近完全测地线的。假设否则，我们通过将具有 ${\rm Aut}(\Omega')$ 等效切空间的 Poincare 圆盘的假设全纯等距嵌入重新缩放到管域 $\Omega' \subset \Omega$ 的方法推导出Poincare-Lelong 方程的矛盾。我们推断有界对称域之间的等变全纯嵌入必须是完全测地线的。此外，我们解决了代数子集 $Z \subset \Omega$ 的统一化问题。更确切地说，如果$\check \Gamma\subset {\rm Aut}(\Omega)$ 是一个无扭离散子群，$Z$ 不变使得$Z/\check \Gamma$ 是紧致的，我们证明$Z \subset \Omega$ 是完全测地线的。特别地，令 $\Gamma \subset{\rm Aut}(\Omega)$ 为无扭晶格，$\pi: \Omega \to \Omega/\Gamma =: X_\Gamma$ 为均匀化图，当 $\pi^{-1}(Y)$ 的某些（以及因此任何）不可约分量 $Z$ 是 $\Omega$ 的代数子集时，子变体 $Y \subset X_\Gamma$ 必须是完全测地线的。对于协紧晶格，这通过双代数性产生了 $X_\Gamma$ 的完全测地线子集的特征，而无需求助于 Andre-Deligne 对 Shimura 变体的子变体的著名单调结果，因此我们的证明不一定适用于算术协致晶格。