A generalization of Milnor's formula
type: Item
instance of: scholarly article
title: A generalization of Milnor's formula
zbMATH Open document ID: 1503.32015
author: Matthias Zach
published in: Mathematische Annalen
publication date: 2022-05-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/2002.04009
review text: In the article a generalization of the classical notion of the Milnor number \(\mu _{f}(0)\) of a holomorphic function \(f :(\mathbb{C}^{n} ,0) \rightarrow (\mathbb{C} ,0)\) with an isolated critical point at \(0\) (= an isolated singularity) is given in the following setting: 1. \((X ,0) \subset (\mathbb{C}^{n} ,0)\) is a singular reduced analytic space, 2. \((\mathcal{S}_{\alpha })\) is the canonical Whitney stratification of \((X ,0) ,\) 3. \(f :(\mathbb{C}^{n} ,0) \rightarrow (\mathbb{C} ,0)\) is a holomorphic function such that \(f\vert (X ,0)\) has an isolated singularity in the stratified sense at 0 (it means \(f\vert \mathcal{S}_{\alpha }\) has only regular points in a small vicinity of \(0 \in \mathcal{S}_{\alpha }\)). The author defines this generalization for any strata \(\mathcal{S}_{\alpha }\). The Milnor number \(\mu _{f}(\alpha ;X ,0)\) is the number of Morse critical points on \(\mathcal{S}_{\alpha \text{}}\) in a morsification of \(f\vert (X ,0)\) (a morsification is an unfolding \((f_{t})\) of \(f\) such that for small \(t\) and any \(\mathcal{S}_{\alpha \text{}}\) the restriction \(f_{t}\vert \mathcal{S}_{\alpha \text{}}\) has only classical Morse critical points and at these points the differential \(df_{t}\) does not annihilate any limiting tangent space from other adjacent strata \(\mathcal{S}_{\beta }\)). He shows that the numbers \(\mu _{f}(\alpha ;X ,0)\) generalize the classical known characterizations of the Milnor number \(\mu _{f}(0)\).
Mathematics Subject Classification ID: 32S05
Mathematics Subject Classification ID: 32S30
Mathematics Subject Classification ID: 58K60
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 7522795
zbMATH Keywords: isolated singularity
zbMATH Keywords: Milnor number
zbMATH Keywords: morsification
zbMATH Keywords: Milnor fiber
zbMATH Keywords: Whitney stratification
Wikidata QID: Q123150663
reviewed by: Tadeusz Krasiński
describes a project that uses: Surfer
describes a project that uses: SINGULAR
MaRDI profile type: Publication
OpenAlex ID: W3186948169
arXiv ID: 2002.04009
cites work: Q3217281
cites work: EULER OBSTRUCTION AND DEFECTS OF FUNCTIONS ON SINGULAR VARIETIES
cites work: Q3924846
cites work: Q4274969
cites work: Local homological properties of analytic sets
cites work: Q5662690
cites work: Radial index and Euler obstruction of a 1-form on a singular variety
cites work: Q3424740
cites work: HOMOLOGICAL INDEX FOR 1-FORMS AND A MILNOR NUMBER FOR ISOLATED SINGULARITIES
cites work: Q4797619
cites work: Q4267942
cites work: The index of a holomorphic flow with an isolated singularity
cites work: Q4039743
cites work: Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen
cites work: Der Gauß-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten
cites work: Q2781760
cites work: Lokale topologische Eigenschaften komplexer Räume
cites work: Q4143433
cites work: Q3858550
cites work: Calculation of Milnor number of isolated singularity of complete intersection
cites work: Q3864794
cites work: Q3803424
cites work: Q5287189
cites work: Variétés polaires locales et classes de Chern des variétés singulieres
cites work: Chern classes for singular algebraic varieties
cites work: A little microlocal Morse theory
cites work: Singular Points of Complex Hypersurfaces. (AM-61)
cites work: Milnor numbers and Euler obstruction
cites work: Q4324884
cites work: Q3710626
cites work: Bouquet decomposition of the Milnor fibre
DOI: 10.1007/S00208-021-02223-5
Recommended article: Q3646860
Recommended article: A generalization of a theorem of Milnor
Recommended article: A generalization of the Milnor number
Recommended article: Q4725343
Recommended article: Q4870753
Recommended article: Q4528628
Recommended article: A generalization of a formula due to Kummer†
Recommended article: Q3059147
Recommended article: An extension of Milnor's inequality
Recommended article: Q4860491
description: scientific article