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Embedding nonisolated singularities into isolated singularities. (English) Zbl 0910.32037
Arnold, V. I. (ed.) et al., Singularities. The Brieskorn anniversary volume. Proceedings of the conference dedicated to Egbert Brieskorn on his 60th birthday, Oberwolfach, Germany, July 1996. Basel: Birkhäuser. Prog. Math. 162, 103-115 (1998).
This paper answers positively a question raised by Malgrange, whether it is possible to embed the Milnor fibre of a non-isolated singular function germ \(f\) into the Milnor fibre of some isolated singularity \(g\). If \(f\) is defined on a singular germ \(X\) and the dimension of the singular locus (with respect to a given stratification of \(X\)) is \(k\) then \(g\) can be taken in the form \(g=f+\varepsilon_1x_1^{N_1}+\cdots+\varepsilon_kx_k^{N_k}\) for general coordinate functions and large enough \(N_i\). The construction proceeds by induction on \(k\) and uses the description of the Milnor fibre based on the polar curve with respect to a general linear function. The construction gives an embedding of the monodromy fibration.
As a corollary in the case \(k=1\) precise results are obtained on the homotopy type of the Milnor fibre of \(f_N:=f+\varepsilon x^N\) in terms of attaching cell to the Milnor fibre of \(f\). Also formulas for the zeta-function of the monodromy are given.
For the entire collection see [Zbl 0890.00033].

32S25 Complex surface and hypersurface singularities
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)