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Morse numbers of function germs with isolated singularities

Laurentiu Maxim, Mihai Tibăr

Abstract

A set of Morse numbers is associated to a holomorphic function germ with stratified isolated singularity, extending the classical Milnor number to the setting of a singular base space.

Morse numbers of function germs with isolated singularities

Abstract

A set of Morse numbers is associated to a holomorphic function germ with stratified isolated singularity, extending the classical Milnor number to the setting of a singular base space.
Paper Structure (7 sections, 3 theorems, 18 equations)

This paper contains 7 sections, 3 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Brieskorn principle and Morse numbers
  3. Polar locus and general linear functions
  4. The Brieskorn principle on Morsifications
  5. The set of Morse pairs
  6. The polar formula for the Morse numbers
  7. The Comparison Argument

Key Result

Lemma 2.1

There is a Zariski open dense subset $\Omega' :=\Omega'_{f} \subset \check \mathbb{P}^{N-1}$ such that $\Gamma(l,f)$ is either a reduced curve germ for all $l \in \Omega'$, or is empty for all $l \in \Omega'$, and that the hyperplane $\{ l=0\}$ is transverse to all strata of $X \setminus \{ 0\}$. I

Theorems & Definitions (9)

  • Lemma 2.1: Polar Curve Lemma
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Example 3.4
  • Remark 3.5