Lefschetz number of the monodromy and resolution of singularities
type: Item
instance of: scholarly article
title: Lefschetz number of the monodromy and resolution of singularities
zbMATH Open document ID: 0815.32018
DOI: 10.1016/0019-3577(94)90038-8
published in: Indagationes Mathematicae. New Series
publication date: 1994-08-11T00:00:00Z
review text: Let \(f(X,x) \to (\mathbb{C}, 0)\) be a germ of holomorphic function on an analytic space germ \(X\), not supposed to be smooth. We say that \(f\) is a smoothing if the singular locus of \(X\) is contained in \(H\), the zero locus of \(f\). Then the generic fibre of \(f\) is smooth and the monodromy acts on the cohomology of the Milnor fiber and the Lefschetz number \(\Lambda (f)\) is the alternating sum of the trace of these monodromies. The author proves that \(\Lambda (f)\) depends only on the residue class of \(f\) in \(m/{\mathcal I}\) where \(m\) is the maximal ideal and \({\mathcal I}\) is an ideal defined by using some resolution of \(X\). In fact \({\mathcal I}\) doesn't depend on this resolution. In particular, if the singularity of \(X\) is isolated, \({\mathcal I}\) contains \(m^ 2\). The Lefschetz number is computed when \(X\) is the cone over a smooth projective hypersurface, or the core over a hypersurface with isolated singularities.
Mathematics Subject Classification ID: 32S30
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 32S45
zbMATH DE Number: 604898
zbMATH Keywords: smoothing of analytic space germ
zbMATH Keywords: monodromy
zbMATH Keywords: Milnor fiber
zbMATH Keywords: Lefschetz number
zbMATH Keywords: singularities
author: Mihai-Marius Tibar
reviewed by: J.-Y. Mérindol
MaRDI profile type: Publication
full work available at URL: 0019-3577(94)90038-8
OpenAlex ID: W2155682815
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description: scientific article; zbMATH DE number 604898