发表机构
Università degli Studi di Bari; Universidad Nacional Autónoma de México（巴里大学; 墨西哥国立自治大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对复向量群对角全纯作用的奇异轨道，定义了典范横向解析包络，揭示其对剩余线性作用的恢复机制，结合相关指标与留数可重构环境权配置，为分析横向动力学与全纯首次积分的信息差异提供了框架。
AI 中文摘要
我们将复向量群的对角全纯作用的每个奇异轨道与一个典范横向解析包络关联起来，该包络由局部截面上的全纯首次积分定义，是一个正规仿射环簇芽。即使横向叶的局部叶空间不满足T₁分离公理，该包络也能实现横向叶的全纯分离。我们精确确定该包络能恢复剩余线性作用的哪一部分：带有标记包络的剩余作用构成格拉斯曼族，当积分共振关系张成整个复关系空间时，该包络可精确确定剩余作用。对于一维剩余作用，该准则体现为Baum–Bott留数；在二维横向维度中，则体现为Camacho–Sad指标。沿正维奇异轨道，包络带有典范运输和平坦联络，可视为正则情形下横向全纯性的奇异对应；当支撑权重独立时，该联络是对数的，其留数与全纯性编码了逐点包络未包含的半全局信息。在全共振秩下，带标记包络与对数留数可重构环境权配置至线性等价。因此，横向解析包络及其典范平坦联络提供了一个框架，用于精确衡量从横向动力学到全纯首次积分过程中丢失的信息，并确定何时该信息足以重构原始线性作用。
英文摘要
We associate with every singular orbit of a diagonal holomorphic action by a complex vector group a canonical transverse analytic envelope, defined by the holomorphic first integrals on a local transversal. This envelope is a normal affine toric germ. It provides a holomorphic separation of the transverse leaves even when their local leaf space fails the $T_1$ separation axiom. We determine exactly which part of the residual linear action is recovered by this envelope. Residual actions with a fixed labelled envelope form Grassmannian families, and the envelope determines the residual action precisely when the integral resonance relations span the full complex relation space. For one-dimensional residual actions, this criterion is reflected in Baum--Bott residues and, in transverse dimension two, in the Camacho--Sad indices. Along positive-dimensional singular orbits, the envelopes carry canonical transport and a flat connection, which may be viewed as a singular counterpart of transverse holonomy in the regular setting. The connection is logarithmic when the support weights are independent, and its residues and holonomy encode semiglobal information not contained in the pointwise envelope. At full resonance rank, the labelled envelope together with the logarithmic residues reconstructs the ambient weight configuration up to linear equivalence. Thus, the transverse analytic envelope and its canonical flat connection provide a framework for measuring exactly the information lost in passing from the transverse dynamics to holomorphic first integrals, and for determining when this information suffices to reconstruct the original linear action.
CommentsPdf has 39 pages. No figures
