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Differentiable maps on links of complex isolated hypersurface singularities

Osamu Saeki, Shuntaro Sakurai

Abstract

We consider links of complex isolated hypersurface singularities in $\mathbb{C}^{n+1}$ and study differentiable maps defined by restricting holomorphic functions to the links. We give an explicit example in which such a restriction gives a fold map into the plane $\mathbb{C} = \mathbb{R}^2$ whose singular value set consists of concentric circles.

Differentiable maps on links of complex isolated hypersurface singularities

Abstract

We consider links of complex isolated hypersurface singularities in and study differentiable maps defined by restricting holomorphic functions to the links. We give an explicit example in which such a restriction gives a fold map into the plane whose singular value set consists of concentric circles.
Paper Structure (7 sections, 7 theorems, 36 equations, 3 figures)

This paper contains 7 sections, 7 theorems, 36 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem
  4. Round fold maps
  5. Inner products
  6. Singular point set
  7. Explicit construction

Key Result

Lemma 2.4

We always have ${(\mathbf{u}, \mathbf{v})_\mathbb{R} = \mathrm{Re} \langle \mathbf{u}, \mathbf{v} \rangle.}$

Figures (3)

  • Figure 1: Link of an isolated hypersurface singularity
  • Figure 2: Singular value set $h(S(h))$ of $h$
  • Figure 3: $S^1$--action

Theorems & Definitions (16)

  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Proposition 2.8
  • Remark 3.1
  • ...and 6 more