从有界对称域穿出的局部全纯曲线的渐近全测地性及其在代数子集单值化问题中的应用
Asymptotic total geodesy of local holomorphic curves exiting a bounded symmetric domain and applications to a uniformization problem for algebraic subsets
AI总结：
本文研究全纯曲线与嵌入的渐近全测地性，证明从有界对称域穿出的全纯曲线渐近全测地，推导等变全纯嵌入的全测地性，并解决域内代数子集的单值化问题，给出余紧格下全测地子集的双代数性刻画。
摘要：
本文源于我们对庞加莱圆盘到有界对称域的全纯等距嵌入渐近行为的研究。作为首个结果，我们证明：任何从有界对称域$\Omega$的边界穿出的全纯曲线，必然是渐近全测地的。假设该结论不成立，我们通过重标度方法，构造出一个将庞加莱圆盘嵌入管域$\Omega' \subset \Omega$的假想全纯等距嵌入，且该嵌入具有与${\rm Aut}(\Omega')$等价的切空间，再借助庞加莱-勒龙方程导出矛盾。\n我们由此推导出：有界对称域之间的等变全纯嵌入必定是全测地的。此外，我们解决了有界对称域内代数子集$Z \subset \Omega$的单值化问题。更确切地说，若$\check{\Gamma}\subset {\rm Aut}(\Omega)$是一个无挠离散子群，且保持$Z$不变，使得商空间$Z/\check{\Gamma}$是紧的，则我们证明$Z \subset \Omega$是全测地的。\n特别地，设$\Gamma\subset{\rm Aut}(\Omega)$为无挠余紧格，$\pi: \Omega\to \Omega/\Gamma=: X_\Gamma$为单值化映射，则只要$\pi^{-1}(Y)$的某个（进而是任意一个）不可约分支$Z$是$\Omega$的代数子集，子簇$Y \subset X_\Gamma$就必定是全测地的。对于余紧格，这一结果通过双代数性给出了$X_\Gamma$的全测地子集的刻画，无需借助André-Deligne关于志村簇子簇的著名单值性结果，因此我们的证明适用于不一定是算术的余紧格。
英文摘要：
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.
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