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Blow-$ADE$ singularities and $μ^*$-constant deformations

Christophe Eyral, Mutsuo Oka

Abstract

We introduce a class of complex surface singularities - the blow-$ADE$ singularities - which are likely to be stable with respect to $μ^*$-constant deformations. We prove such a stability property in several special cases. Here, we emphasize that we are not just considering deformation families for small values of the deformation parameter but families connecting any two elements in the $μ^*$-constant stratum.

Blow-$ADE$ singularities and $μ^*$-constant deformations

Abstract

We introduce a class of complex surface singularities - the blow- singularities - which are likely to be stable with respect to -constant deformations. We prove such a stability property in several special cases. Here, we emphasize that we are not just considering deformation families for small values of the deformation parameter but families connecting any two elements in the -constant stratum.
Paper Structure (8 sections, 8 theorems, 45 equations)

This paper contains 8 sections, 8 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Definition of blow-$ADE$ singularities
  3. $\mu^*$-constant deformations of blow-$ADE$ singularities
  4. Proof of Theorem \ref{['mt1']}
  5. Proof of Theorem \ref{['mt1']} -- case (1)
  6. Proof of Theorem \ref{['mt1']} -- case (2)
  7. Proof of Theorem \ref{['mt1']} -- case (3)
  8. Zeta-multi-pli-cities and zeta-multi-pli-city factors

Key Result

Theorem 3.1

Assume that $f$ has, at $\mathbf{0}$, a blow-$ADE$ singularity of one of the following three specific types: In each case, if $\{f_s\}$, $0\leq s\leq 1$, is any $\mu^*$-constant piecewise complex-analytic family starting at $f$ (i.e., $f_0=f$), then each member $f_s$ of the family defines, at $\mathbf{0}$, a blow-$ADE$ singularity of the same type as that of $f$.

Theorems & Definitions (36)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 3.1
  • Conjecture 3.2
  • Corollary 3.3
  • proof
  • Lemma 4.1: see EO
  • ...and 26 more