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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)].

##### MSC:
 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
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##### References:
 [1] [Ar] V. I. Arnol’d:Remarks on singularities of finite codimension in complex dynamical systems. Funct. Anal. Appl.3: (1969), 1-5. · Zbl 0249.34035 [2] [CKP] C. Camacho, N. Kuiper and J. Palis:The topology of holomorphic flows with singularities. Publ. Math. I.H.E.S.,48: (1978), 5-38. · Zbl 0411.58018 [3] [GSV] X. Gomez-Mont, J. Seade and A. Verjovsky:Topology of a holomorphic vector field around an isolated singularity. Funct. Anal Appl.,27: (1993), 97-103. · Zbl 0812.32019 [4] [Gu] J. Guckenheimer:Hartman’s theorem for complex flows in the Poincaré domain. Comp. Math.,24: (1972), 75-82. [5] [It] T. Ito:Some examples of contact sets of spheres and holomorphic vector fields. The Ryukoku J. Humanities and Sci., #15: (1994), 179-190. [6] [Lm] S. López de Medrano:The space of Siegel leaves of a holomorphic vector field. In ?Holomorphic Dynamics?, Springer Verlag Lecture Notes in Math.,1345: 233-245, edit. X. Gomez-Mont et al. [7] [LV] S. López de Medrano and A. Verjovsky:A new family of comples, compact, non-symplectic manifolds. Preprint. · Zbl 0901.53021 [8] [Lo] E. Looijenga:A note on polynomial isolated singularities. Indag. Math.,33: (1971), 418-421. · Zbl 0234.57010 [9] [Mi] J. Milnor: ?Singular points of complex hypersurfaces?. Annals. of Maths. Study,61: (1968), Princeton Univ. Press. [10] [Pa] L. Paunescu:The topology of the real part of a holomorphic function. Math. Nachr.,174: (1995). · Zbl 0832.32009 [11] [Pe] B. Perron:Le noeud ?huit? est algebrique réel. Inv. Math.,65: (1982), 441-451. · Zbl 0503.57003 [12] [Se] J. Seade: ?Open book decompositions associated to holomorphic vector fields?. To appear in Boletin Soc. Mat. Mex. [13] [Su] D. Sullivan:On the intersection ring of compact three manifolds. Topology,14: (1975), 275-277. · Zbl 0312.57003 [14] [Th] R. Thom:Generalization de la théorie de Morse et varietés feuilletées. Ann. Inst. Fourier,14: (1964), 175-190. [15] [Wa] C.T.C. Wall:Stability, pencils and polytopes. Bull. London Math. Soc.,12: (1980), 401-421. · Zbl 0451.58008
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