Semicontinuity of the singularity spectrum
type: Item
instance of: scholarly article
title: Semicontinuity of the singularity spectrum
zbMATH Open document ID: 0568.14021
DOI: 10.1007/BF01388523
author: Jozef Steenbrink
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143212
review text: To each isolated hypersurface singularity f: (\({\mathbb{C}}^{n+1},0)\to ({\mathbb{C}},0)\) with Milnor number \(\mu\) one can associate a sequence of \(\mu\) rational numbers: its spectrum. It arises by choosing rationals \(\lambda\) such that exp \(2\pi\) \(i\lambda\) is an eigenvalue of the monodromy operator T, in a way which is determined by the Hodge filtration on the vanishing cohomology of f. In 1980, \textit{V. I. Arnol'd} [cf. Geometry and analysis, Pap. dedic. Mem. V. K. Patodi, 1-9 (1981; Zbl 0492.58006)] conjectured that the spectrum behaves semicontinuously under deformations of the singularity in a certain way. For a deformation \((f_ t)\) of negative weight of a quasihomogeneous function \textit{A. N. Varchenko} [Sov. Math., Dohl. 27, 735-739 (1983); translation from Dokl. Akad. Nauk SSSR 270, 1294-1297 (1983; Zbl 0537.14003)] proved the following: let \(x_ 1,...,x_ r\) be singular points of \(f_ t^{- 1}(s)\) (t\(\neq 0)\). Then for any \(a\in {\mathbb{R}}\), the number of spectrum numbers of \(f_ 0\) which lie in \((a,a+1)\) is not less than the analogous number for \(f_ t\), summed over all \(x_ i\). In the paper under review, this result is extended to arbitrary deformations of arbitrary isolated singularities, with \((a,a+1)\) replaced by the half-open interval \((a,a+1]\), and Arnol'd's original question is answered affirmatively. The proof uses the existence of limit mixed Hodge structures for geometric variations of mixed Hodge structure. A byproduct is the semicontinuity of Hodge numbers for isolated complex intersection singularities and the semicontinuity of the complex singularity index, conjectured by Malgrange.
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14C30
zbMATH DE Number: 3906635
zbMATH Keywords: singularity spectrum
zbMATH Keywords: vanishing cohomology
zbMATH Keywords: isolated hypersurface singularity
zbMATH Keywords: Milnor number
zbMATH Keywords: deformations of the singularity
zbMATH Keywords: limit mixed Hodge structures
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3906635