A general calculation of the number of vanishing cycles
type: Item
instance of: scholarly article
title: A general calculation of the number of vanishing cycles
zbMATH Open document ID: 0824.32013
DOI: 10.1016/0166-8641(94)00042-2
author: David B. Massey
published in: Topology and its Applications
publication date: 1995-04-10T00:00:00Z
review text: Let \(X\) be a reduced complex analytic space containing an open neighborhood \(U\) of the origin in \(\mathbb{C}^ n\) and let \(f : (U,0) \to (\mathbb{C})\) be an analytic function. Suppose that \(\mathbb{F}^ \bullet\) is a bounded complex of sheaves on \(X\) which is constructible with respect to a Whitney stratification \(\{S_ \alpha\}\) of \(X\). Let \(H\) be a hypersurface in \(U\) which transversely intersects the strata of \(\{S_ \alpha\}\) except, perhaps, at the origin. The author obtains an algebraic formula for calculating the difference between the Euler characteristics of the stalks cohomology at the origin of the sheaves of vanishing cycles along \(f\) and along the restriction \(f \mid_ H\). Thus, he generalizes a result of \textit{Le Dung Trang} [Ann. Inst. Fourier 23(1973), No. 4, 261- 270 (1974; Zbl 0293.32013)]. In the case where \(\mathbb{F}^ \bullet\) is a perverse sheaf on a complete intersection the formula (5.11a) of \textit{E. J. N. Looijenga} [`Isolated singular points on complete intersections', Lond. Math. Soc. Lectures Note Series 77, Cambridge (1984; Zbl 0552.14002)] is also generalized. As an application it is proved that the complex link of any isolated complete intersection singularity is never contractible.
Mathematics Subject Classification ID: 32S60
Mathematics Subject Classification ID: 32S55
Mathematics Subject Classification ID: 32S30
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14F43
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57Q45
zbMATH DE Number: 735668
zbMATH Keywords: isolated singularities
zbMATH Keywords: polar curve
zbMATH Keywords: vanishing cycles
zbMATH Keywords: perverse shaves
zbMATH Keywords: constructible sheaves
zbMATH Keywords: characteristic cycle
zbMATH Keywords: Euler characteristic
zbMATH Keywords: complete intersections
zbMATH Keywords: complex link of a singularity
reviewed by: Aleksandr G. Aleksandrov
MaRDI profile type: Publication
full work available at URL: 0166-8641(94)00042-2
OpenAlex ID: W2045357041
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description: scientific article; zbMATH DE number 735668