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A characterization of quasi-homogeneity in terms of liftable vector fields

Ignacio Breva Ribes, Raúl Oset Sinha

TL;DR

This work develops a framework linking quasi-homogeneity of map-germs to liftable vector fields and the notion of substantial unfoldings. It introduces and exploits the concepts of substantial and weakly substantial unfoldings, and uses stable unfoldings and their Euler-type vector fields to diagnose weighted-homogeneous structure. A key contribution is a necessary condition for weighted-homogeneity under stable unfoldings, and a concrete converse in the equidimensional, corank-1, $\mathscr{A}$-finite setting for minimal stable unfoldings or multiplicity 3, along with a general conjecture: a finitely determined map-germ is quasi-homogeneous if and only if it admits a substantial unfolding. These results provide a practical pathway to certify quasi-homogeneity via unfoldings, with implications for calculating invariants and understanding augmentations in singularity theory.

Abstract

We prove under certain conditions that any stable unfolding of a quasi-homogeneous map-germ with finite singularity type is substantial. We then prove that if an equidimensional map-germ is finitely determined, of corank 1, and either it admits a minimal stable unfolding or it is of multipliticy 3, then it admits a substantial unfolding if and only if it is quasi-homogeneous. Based on this we pose the following conjecture: a finitely determined map-germ is quasi-homogeneous if and only if it admits a substantial unfolding.

A characterization of quasi-homogeneity in terms of liftable vector fields

TL;DR

This work develops a framework linking quasi-homogeneity of map-germs to liftable vector fields and the notion of substantial unfoldings. It introduces and exploits the concepts of substantial and weakly substantial unfoldings, and uses stable unfoldings and their Euler-type vector fields to diagnose weighted-homogeneous structure. A key contribution is a necessary condition for weighted-homogeneity under stable unfoldings, and a concrete converse in the equidimensional, corank-1, -finite setting for minimal stable unfoldings or multiplicity 3, along with a general conjecture: a finitely determined map-germ is quasi-homogeneous if and only if it admits a substantial unfolding. These results provide a practical pathway to certify quasi-homogeneity via unfoldings, with implications for calculating invariants and understanding augmentations in singularity theory.

Abstract

We prove under certain conditions that any stable unfolding of a quasi-homogeneous map-germ with finite singularity type is substantial. We then prove that if an equidimensional map-germ is finitely determined, of corank 1, and either it admits a minimal stable unfolding or it is of multipliticy 3, then it admits a substantial unfolding if and only if it is quasi-homogeneous. Based on this we pose the following conjecture: a finitely determined map-germ is quasi-homogeneous if and only if it admits a substantial unfolding.

Paper Structure

This paper contains 6 sections, 17 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Necessary condition for weighted-homogeneity
  4. Equidimensional, corank 1, $\mathscr{A}$-finite case
  5. Minimal stable unfolding case
  6. Multiplicity 3 case

Key Result

Lemma 2.2

Let $f,g\colon(\mathbb{K}^n,0)\to(\mathbb{K}^p,0)$ be smooth map-germs and assume there are some germs of diffeomorphism $\phi\colon(\mathbb{K}^n,0)\to(\mathbb{K}^n,0)$ and $\psi\colon(\mathbb{K}^p,0)\to(\mathbb{K}^p,0)$ such that $\psi\circ f\circ \phi = g$. Then, the map is a bijection. In particular, it is an isomorphism of $\mathcal{O}_p$-modules via $\psi^{-1}$.

Theorems & Definitions (54)

  • Conjecture 1.1
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Lemma 2.7
  • proof
  • Definition 2.8
  • ...and 44 more