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Non-isolated complete intersection singularities and the \(A_f\) condition. (English) Zbl 1177.32019

Brasselet, Jean-Paul (ed.) et al., Singularities I. Algebraic and analytic aspects. Proceedings of the international conference “School and workshop on the geometry and topology of singularities” in honor of the 60th birthday of Lê Dũng Tráng, Cuernavaca, Mexico, January 8–26, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4458-8/pbk). Contemporary Mathematics 474, 85-93 (2008).
The Milnor fibration is an important tool for studying singular germs. The theorem on the existence of a Milnor fibration is quite general: the domain might have some singularities. For example, Hamm proved it for isolated complete intersection singularities. On the other hand, Lê noticed that the analogous result for complete intersections with non-isolated singularities could not hold: the key stratification condition needed to establish the existence of the Milnor fibration is Thom’s \(A_f\) condition. Lê asked if a complete intersection non-isolated singularity could be embedded in such a way, that for some choice of the generators, the map defined by these generators had the needed Thom’s condition.
The main result of the article shows that the condition for the Thom’s \(A_f\) property to hold puts stringent conditions on the map.
The article contains some interesting examples as well.
For the entire collection see [Zbl 1151.14002].

MSC:

32S55 Milnor fibration; relations with knot theory
32B15 Analytic subsets of affine space
32C35 Analytic sheaves and cohomology groups
32C18 Topology of analytic spaces
32B10 Germs of analytic sets, local parametrization
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