Transverse Analytic Envelopes and Semiglobal Dynamics

Abstract

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.