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Invariants, equisingularity and Euler obstruction of map germs from \(\mathbb C^n\) to \(\mathbb C^n\). (English) Zbl 1083.32023

By a result of T. Gaffney [Topology 32, No. 1, 185–223 (1993; Zbl 0790.57020)] a family of finitely determined map germs \(f_t\colon {\mathbb C}^n\to {\mathbb C}^p\) of discrete stable type is Whitney equisingular if finitely many invariants are constant in the family, namely all the \(0\)-stable invariants and all polar multiplicities which appear in the stable types of a stable deformation. But there are many of these.
The paper under review gives, for the case \(p=n\), a minimal set of invariants which, if constant, guarantee that \(\{f_t\}\) is Whitney equisingular. More precisely, the authors study \(1\)-parameter deformations of a corank \(1\) finitely determined holomorphic germ \(f\colon {\mathbb C}^n\to{\mathbb C}^n\). The authors explain their procedure well in the abstract: “first we describe all stable types, then we show how the invariants in the source and target are related and reduce the number using these relations. We also investigate the relationship between the local Euler obstruction and the polar multiplicities of the stable types.”

MSC:

32S15 Equisingularity (topological and analytic)

Citations:

Zbl 0790.57020
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References:

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