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Fibred links and a construction of real singularities via complex geometry. (English) Zbl 0881.32019
Let \(F, X\) be continuous vector fields at the origin in \({\mathbb{C}}^n.\) Similary to the earlier work [X. Gomez-Mont, J. Seade and A. Verjovsky, Funct. Anal. Appl. 27, No. 2, 97-103 (1993); translation from Funkts. Anal. Prilozh. 27, No. 2, 22-31 (1993; Zbl 0812.32019)] the author defines a continuous map \(\varphi : {\mathbb{C}}^n \cong {\mathbb{R}}^{2n} \longrightarrow {\mathbb{C}} \cong {\mathbb{R}}^2\) as follows: \(\varphi(z) = \sum_{i=1}^n F_i(z) \cdot \overline{X}_i(z).\) In fact, the map \(\varphi\) is not complex analytic but real analytic only even in the case where \(F\) and \(X\) are holomorphic non-constant fields. In the note the author studies \(\varphi\) when \(F\) is a holomorphic field and \(X\) is the gradient vector field of a real analytic function \(f: {\mathbb{C}}^n \cong {\mathbb{R}}^{2n} \longrightarrow {\mathbb{R}}\) with an isolated critical point at the origin. Some general properties of the polar variety of \(f\) and \(F\) are stated. Then an explicit construction of infinite families of real singularities satisfying the hypothesis of the famous Milnor’s fibration theorem is described. The question about the existence of such a family was posed by J. Milnor [in ‘Singular points of complex hypersurfaces’ (1968; Zbl 0184.48405)] and it was answered positively by E. Looijenga [in Nederl. Akad. Wet., Proc., Ser. A 74, 418-421 (1971; Zbl 0234.57010)].

32S65 Singularities of holomorphic vector fields and foliations
57R30 Foliations in differential topology; geometric theory
32S55 Milnor fibration; relations with knot theory
58C25 Differentiable maps on manifolds
Full Text: DOI
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