Quadratic functions and smoothing surface singularities
type: Item
instance of: scholarly article
title: Quadratic functions and smoothing surface singularities
zbMATH Open document ID: 0615.32014
DOI: 10.1016/0040-9383(86)90044-3
author: Eduard Looijenga
author: Jonathan Wahl
published in: Topology
publication date: 1986-01-01T00:00:00Z
review text: This paper is, in the reviewers opinion, a major contribution to singularity theory. Based on the remarkable paper by \textit{V. V. Nikulin} [Math. USSR Izv. 14, 103-167 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 111- 177 (1979; Zbl 0408.10011)] the authors attack the problem of characterizing the smoothing components and the intersection forms of the corresponding Milnor fiber, for isolated complex-analytic surface singularities. Let (X,0) be any such singularity, i.e. a Stein space of dimension \(2\) with a smooth \((C^{\infty})\) boundary \(\partial X=L\), and a unique singular point 0. L is usually called the link of the singularity. Let \(\pi: \tilde X\to X\) be a good resolution, i.e. one with an exceptional curve \(E=\pi^{-1}(0)\) consisting of nonsingular irreducible components intersecting transversally. Consider a smoothing \(f: {\mathcal X}\to \Delta,\) where \(\Delta\) is an open disc in \({\mathbb{C}}\) containing 0. Here \({\mathcal X}\) is a Stein space with a partial \(C^{\infty}\)-boundary \(\partial {\mathcal X}\). There is an isomorphism of analytic spaces with boundary \(i: f^{- 1}(0)\overset \sim \rightarrow X\) and \(f| int({\mathcal X}-\{0\})\) and \(f| \partial {\mathcal X}\) are both submersions. The Milnor fiber M, i.e. the generic fiber of f, has a boundary \(\partial M\) which is diffeomorphic to L. Therefore the 3-dimensional manifold L bounds both of the two oriented 4- dimensional manifolds \(\tilde X\) and M. The main result of this paper, (3.7), shows that the classical linking pairing of the link L, \(b: H_ 1(L)_ t\times H_ 1(L)_ t\to Q/Z\) where the subscript t denotes the torsion part, is in an obvious sense the link between the intersection pairings of \(H_ 2(\tilde X)\) and \(H_ 2(M)\). In fact, the complex structures of \(\tilde X\) and M induces an, up to homotopy, unique complex structure on the sum of the trivial bundle R and the tangent bundle \(\tau_ L\) of L. Using the first Chern class of \(-\tau_ L\opl…
Mathematics Subject Classification ID: 32S30
Mathematics Subject Classification ID: 32S05
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 32Sxx
zbMATH DE Number: 3996104
zbMATH Keywords: smoothing components
zbMATH Keywords: intersection forms
zbMATH Keywords: Milnor fiber
zbMATH Keywords: isolated complex- analytic surface singularities
zbMATH Keywords: link of the singularity
zbMATH Keywords: discriminant quadratic function
reviewed by: Olav Arnfinn Laudal
MaRDI profile type: Publication
full work available at URL: 0040-9383(86)90044-3
OpenAlex ID: W1996300040
description: scientific article; zbMATH DE number 3996104