The monodromy groups of critical points of functions. II
type: Item
instance of: scholarly article
title: The monodromy groups of critical points of functions. II
zbMATH Open document ID: 0536.32006
DOI: 10.1007/BF01388441
published in: Inventiones Mathematicae
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143059
review text: [For part I see ibid. 67, 123-131 (1982; Zbl 0468.57024).] This article is devoted to a description of the monodromy group of a critical point of a holomorphic function in an even number of variables. The main result of the first part of this article (cited above) reduces the description of the monodromy group to the description of some finite group of automorphisms of some finite \({\mathbb{Z}}_{2\lambda}\)-module, where this module and the natural number \(\lambda\) are determined by the skew-symmetric form. This article is devoted to the description of this finite group. We shall prove that there are invariant subsets of this finite group in the \({\mathbb{Z}}_{2\lambda}\)-module, and any automorphism of the \({\mathbb{Z}}_{2\lambda}\)-module for which these sets are invariant belongs to this finite group (theorem 1). The definition of invariant sets see in section 5. For critical points in two variables: \(\lambda =1\); the \({\mathbb{Z}}_ 2\)-module is the homology group \(H_ 2\) of the nonsingular local level manifold with coefficients in \({\mathbb{Z}}_ 2\); and the invariant subsets are the following: (1) subset of all vanishing vectors; (2) subset of all nonvanishing vectors not belonging to the kernel of the skew-symmetric form and (3) each vector from the kernel of the skew-symmetric form (see theorem 3). - In the case of critical point of a function in an arbitrary even number of variables these invariant subsets are defined by the skew-symmetric form and the set of vanishing vectors in the homology group \(H_ 2\) with coefficients in \({\mathbb{Z}}_ 2.\) In spite of a uniform formulation of the main result of this article (theorem 1) for all critical points in an even number of variables the proof of this result consists of two, very nonsimilar cases. The first case consists of the critical points \(A_{\mu}\) and \(D_{\mu}\). For these critical points theorem 1 was proved by \textit{A. N. Varchenko} (1979, unpublished). This proof is given in an additio…
Mathematics Subject Classification ID: 32S05
Mathematics Subject Classification ID: 32M05
zbMATH DE Number: 3851495
zbMATH Keywords: homology group of the nonsingular local level manifold
zbMATH Keywords: monodromy group of a critical point of a holomorphic function in an even number of variables
zbMATH Keywords: group of automorphisms
zbMATH Keywords: invariant subsets
zbMATH Keywords: vanishing vectors
zbMATH Keywords: nonvanishing vectors
zbMATH Keywords: Dynkin diagram
zbMATH Keywords: \(E_ 6\)
author: S. Chmutov
MaRDI profile type: Publication
cites work: Q3931134
cites work: Singular Points of Complex Hypersurfaces. (AM-61)
description: scientific article; zbMATH DE number 3851495