Deforming the weighted-homogeneous foliation, and trivializing families of semi-weighted homogeneous ICIS

TL;DR

This paper investigates how weighted-homogeneous foliations of complete intersection singularities deform under higher order perturbations, identifying obstructions and constructing trivializations with controlled regularity properties.

Contribution

It introduces the concept of an obstruction locus for compatible foliation deformation and constructs a real-analytic or Nash trivialization with controlled Lipschitz and differentiability properties.

Findings

Identification of the obstruction locus  in the singularity

Construction of a contact trivialization with controlled regularity

Deformation of foliations compatible with higher order perturbations

Abstract

Let X_o be a weighted-homogeneous complete intersection germ in (R^N,o) or (C^N,o), with arbitrary singularities, possibly non-reduced. Take the foliation of the ambient space by weighted-homogeneous real arcs, \ga_s. Take a deformation of X_o by higher order terms, X_t. Does the foliation \ga_s deform compatibly with X_t? We identify the ``obstruction locus", \Sigma in X_o, outside of which such a deformation does exist, and possesses exceptionally nice properties. Using this deformed foliation we construct a contact trivialization of the family of defining equations by a homeomorphism that is real analytic (resp. Nash) off the origin, differentiable at the origin, whose presentation in weighted-polar coordinates is globally real-analytic (resp. globally Nash), and with controlled Lipschitz/C^1-properties.