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• W2015028055 abstract "In this paper, we investigate the Milnor fibrations [27] of complex polynomials with non-isolated singularities using a new tool the L~ varieties. We began this investigation in [22] where we defined the L8 varieties and used them to give a handle decomposition of the Milnor fibre which generalizes the result of Milnor for isolated singularities and the result of Vannier [28], [39] for one-dimensional singularities. The primary new result of this paper is to extend the result of L~ and Ramanujam 1,17] on the constancy of the Milnor fibrations in families of isolated singularities (and another result of Vannier for families of one-dimensional singularities) to families of singularities of arbitrary dimension (however, the singularity must be of codimension at least 2 inside the hypersurface). We accomplish this essentially using the same method that we used in 5.1 and 5.2 of [21] together with a trick. This trick gives one very little insight into why the theorem should be true. A sketch of the proof is as follows. Suppose that we have a one-parameter family of polynomials f : (C 1, 0) --* (C, 0) in the coordinates ( Z o , . . . , z,). Let s denote dim0 S f o. We would like to know what numerical data we should require to be constant in the family in order to guarantee that that the diffeomorphism type (or, at least, the fibre-homotopy type) of the Milnor fibrations at the origin remains constant. Now, it is certainly reasonable to expect that the Milnor fibrations in the family f ' are constant if and only if the Milnor fibrations are constant in the family f ' + w i, where w is a variable disjoint from (z o . . . . . z,) and j > 2. (This seems reasonable since, up to homotopy, the Milnor fibre off, + w i is the one-point union of j l copies of the suspension of the Milnor fibre of f . See [28], 1-31], [32], and 4.2 of this paper.) But why should the family f + w J be any easier to study than f itself? It is easier because we have very nice hyperplanes of the form L = w Zo such that when we take the sections f ' + WJlwL~ we get the family f ' + z~ which, for" @default.
• W2015028055 created "2016-06-24" @default.
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• W2015028055 date "2020-11-24" @default.
• W2015028055 modified "2023-09-25" @default.
• W2015028055 title "Varieties II:" @default.
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• W2015028055 doi "https://doi.org/10.2307/j.ctv18zhcs4.15" @default.
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