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Constant Milnor number implies constant multiplicity for quasihomogeneous singularities. (English) Zbl 0594.32021

The author studies that the multiplicity does not change for certain topologically trivial deformations of an isolated hypersurface singularity. His work applies to all \(\mu\)-constant first order deformations and to all \(\mu\)-constant deformations of a quasihomogeneous singularity, i.e., an isolated hypersurface singularity with \({\mathbb{C}}^*\)-action. His argument uses a valuation test of Lê and Saito. He can apply this to arbitrary isolated singularities. It is known that constant Milnor number implies constant multiplicity in some special cases. In the case of quasihomogeneous (semi-quasihomogeneous) singularities, the author gives a positive answer to Zariski’s question whether for a hypersurface singularity the multiplicity is an invariant of the topological type. The statement of his result is the following: Let f be a quasi-homogeneous polynomial with isolated singularity and \(F: ({\mathbb{C}}^ n\times {\mathbb{C}},0)\to ({\mathbb{C}},0)\) a \(\mu\)-constant unfolding of f. Then, \(mult(f_ t)=mult(f)\) for small values of t, where \(f_ t(x)=F(x,t)\) for \((x,y)\in {\mathbb{C}}^ n\times {\mathbb{C}}\).
Reviewer: S.Ohyanagi

MSC:

32S30 Deformations of complex singularities; vanishing cycles
32S05 Local complex singularities
14B07 Deformations of singularities
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
14J17 Singularities of surfaces or higher-dimensional varieties
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References:

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