The Lê varieties. II (Q803342)

Let \(f: ({\mathbb{C}}^{n+1},0)\to ({\mathbb{C}},0)\) be a polynomial with singular locus of dimension s. In part I of this paper [ibid. 99, 357-376 (1990; Zbl 0712.32020)] the author defines a sequence of varieties \(\Lambda_ f^{(k)}\) using the polar varieties. The aim is to obtain similar informations in case \(s>0\) as in the case of isolated singularities from the Jacobian ideal. Especially the multiplicities of \(\Lambda_ f^{(k)}\) (the so called Lê-numbers) generalize the Milnor number. In case of an isolated singularity \(\Lambda_ f^{(k)}=\emptyset\) if \(k>0\) and \(\Lambda_ f^{(0)}\) is the variety defined by the Jacobian ideal \((\frac{\partial f}{\partial z_ 0},...,\frac{\partial f}{\partial z_ n})\). The author investigates the Milnor fibration of f using his Lê varieties to extend the result of Lê and Ramanujan concerning the constancy of the Milnor fibration in a family of isolated singularities to nonisolated singularities: The constancy of the Lê-numbers in a family of singularities implies the constancy of the fibre-homotopy type of the Milnor-fibration.