Holomorphic Linear $\C^k$-Actions, Trace Foliations, and Higher-Rank Poincar\'e Dynamics
We study the orbit decomposition on $\C^n$ generated by diagonal holomorphic $\C^k$-actions in the higher-rank setting of the classical Poincar\'e--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 Poincar\'e 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.