Weak Whitney regularity implies equimultiplicity for families of complex hypersurfaces. (English. French summary) Zbl 1354.32019
The notion of weakly Whitney stratified sets was introduced by K. Bekka and the first author [in: Real and complex singularities. Proceedings of the 5th workshop. Boca Raton, FL: Chapman & Hall/CRC. 1–15 (2000; Zbl 0942.58010)], where they give examples of real algebraic varieties with weakly Whitney regular stratifications which are not Whitney regular. As an evidence that weak Whitney regularity and Whitney regularity might be equivalent notions for stratifications of complex analytic hypersurfaces, the authors show in this paper that equimultiplicity of a family of hypersurfaces follows from weak Whitney regularity of the family over the parameter space. Equimultiplicity follows from Whitney regularity as was proved for general complex analytic spaces by H. Hironaka [Publ. Math., Inst. Hautes Étud. Sci. 36, 127–138 (1969; Zbl 0219.57022)].
Reviewer: Maria Aparecida Soares Ruas (São Carlos)
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