AI 中文总结
本文研究高阶框架下对角全纯$\boldsymbol{\rm C}^k$-作用生成的轨道分解，通过坐标分层分析叶的结构，揭示其微分同胚类型的阈值现象，并证明奇异叶状结构具有典范局部横向全纯结构。
AI 中文摘要
我们研究$\boldsymbol{\rm C}^n$上由对角全纯$\boldsymbol{\rm C}^k$-作用生成的轨道分解，该研究处于线性向量场经典庞加莱-西格尔二分法的高阶框架下。坐标分层确定了叶的维数和迷向群，在极大秩条件下，可精确描述每个坐标分层上的轨道结构。对于庞加莱域内的构型，一条分离实方向提供了穿孔轨道叶状结构的全局锥模型，该模型由其在欧氏球面上的迹构成。我们在球面上构造了对应径向重参数化作用，将其叶描述为齐性空间，并证明轨道闭包受坐标支撑约束：极限点无法获得新的非零坐标，尽管非闭迹叶也可能在固定支撑分层内积累。我们证明权构型的几何决定叶的复维数，而其有效迷向群的算术决定叶的微分同胚类型。这在坐标分层上产生阈值现象：叶的微分同胚类型在低维和高维区域是刚性的，但在中间范围$k<|I|<2k$内具有算术不稳定性。最后，我们证明这些奇异叶状结构具有典范的局部横向全纯结构，其精神与Haefliger针对正则叶状结构的横向几何一致，这些结构确定了内在的横向伪群，为轨道叶状结构提供了良定义的局部横向全纯几何。
英文摘要
We study the orbit decomposition on $\C^n$ generated by diagonal holomorphic $\C^k$-actions in the higher-rank setting of the classical Poincaré--Siegel dichotomy for linear vector fields. The coordinate stratification determines the dimensions and isotropy groups of the leaves and, under a maximal-rank condition, gives a precise description of the orbit structure on every coordinate stratum. For configurations in the Poincaré domain, a separating real direction provides a global conical model of the punctured orbit foliation by its traces on Euclidean spheres. We construct the corresponding radially reparametrized action on a sphere, describe its leaves as homogeneous spaces, and prove that orbit closures are constrained by coordinate supports. In particular, a limit point cannot acquire a new nonzero coordinate, although nonclosed trace leaves may also accumulate within a fixed support stratum. We show that the geometry of the weight configuration determines the complex dimensions of the leaves, whereas the arithmetic of their effective isotropy groups determines their diffeomorphism types. This gives rise to a threshold phenomenon across the coordinate stratification: the diffeomorphism type of the leaves is rigid in the low- and high-dimensional regimes, but becomes arithmetically unstable in the intermediate range $k<|I|<2k$. Finally, we show that these singular foliations admit canonical local transverse holomorphic structures in the spirit of Haefliger's transverse geometry for regular foliations. These structures determine intrinsic transverse pseudogroups, yielding a well-defined local transverse holomorphic geometry for the orbit foliation.
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