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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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[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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