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The Lê-Ramanujam problem for hypersurfaces with one-dimensional singular sets. (English) Zbl 0657.32005
This paper is a very nice generalization of the excellent result of Lê Dũng Tráng and C. P. Ramanujam [Am. J. Math. 98, 67-78 (1976; Zbl 0351.32009)]. The author investigates the case of a family f: (\({\mathbb{C}}^{n+1}\times D,0\times D)\to ({\mathbb{C}},0)\) such that the singular set of each \(f_ t\) is one dimensional at the origin. First some notations: If \(\alpha\) is an ideal in \({\mathcal O}_ 0^{n+1}\) and \(f\in {\mathcal O}_ 0^{n+1}\) such that \(\dim_ 0(V(\alpha)\cap V(f))=0,\) then \((V(\alpha)\cdot V(f))_ 0\) denotes the scheme-theoretic intersection number. If \(f\in {\mathcal O}_ 0^{n+1}\) and L: \({\mathbb{C}}^{n+1}\to {\mathbb{C}}\) is a linear form, then f can be assumed to be a polynomial in the coordinates \((x_ 1,...,x_ n,L)\). Let \(\nu\) denote the ideal in \({\mathcal O}^{n+1}\) generated by \(\frac{\partial f}{\partial x_ 1},...,\frac{\partial f}{\partial x_ n}\); \(\nu_ f\) the ideal generated by \(\nu\) in the ring of fractions obtained from \({\mathcal O}^{n+1}\) by inverting \(\{f^ n\}_{n\geq 0}\); \(\gamma_{f,L}\) the ideal \(\nu_ f\cap {\mathcal O}^{n+1}\) in \({\mathcal O}^{n+1}\) and \(\Gamma_{f,L}\) denote the scheme Spec \({\mathcal O}^{n+1}/\gamma_{f,L}\). Let \(\sigma_{f,L}\) denote the ideal which is the intersection of the remaining isolated primary components of \(\nu\) and \(\Sigma_{f,L}\) denote the scheme Spec \({\mathcal O}^{n+1}/\sigma_{f,L}.\)
The main tool of the paper is a Iomdine-Lê type formula for families of polynomials. If \(f:(\mathbb{C}^{n+1} \times \overset\circ D\), \(0\times \overset\circ D)\;to (\mathbb{C},0)\) is a polynomial such that for all \(t\in \overset\circ D\), \(f_ t=f_ t(x,s)\) defines a hypersurface in \({\mathbb{C}}^{n+1}\times \{t\}\) with \(\dim_ 0\Sigma V(f_ t)=1\) and the coordinate s has been chosen so that \(f_ 0|_{V(s)}\) has an isolated singularity at 0 and \(\overset\circ D\) is chosen so small that \(f_ t|_{V(s)}\) has an isolated singularity at 0 for all \(t\in \overset\circ D\), then the author proves the “Uniform Iomdine- Lê Formula”: For all k sufficiently large, there exists \(\tau >0\) such that for all \(t\in D_{\tau}\), \(f_ t+s^ k\) has an isolated singularity at the origin and \[ \mu (f_ t+s^ k)+\mu (f_ t|_{V(s)})=(\Gamma_{f_ t,s}\cdot V(f_ t))_ 0+k(\Sigma_{f_ t,s}\cdot V(s))_ 0. \] The major application of this formula is the possibility of establishing numerical conditions for \(\Gamma_{f,L}=\emptyset\). If \(\lambda_ t=(\Sigma_{f_ t,s}\cdot V(s))_ 0\) and \(\beta_ t=(\Gamma_{f_ t,s}\cdot V(\frac{\partial f_ t}{\partial s}))_ 0\) are constant for all small t, then the polar curve of f at the origin is empty. In this case the Milnor fibre of f at the origin is diffeomorphic to the cross-product of a disc and the Milnor fiber of a generic hyperplane section of f.
In the special case when \(f_ t\) is a family of line singularities the author proves the “Non-splitting Theorem”: If \(\lambda_ t\) and \(\beta_ t\) are constant for all small t, then the singular set of f does not split. In the case of constancy of \(\lambda_ t\) and \(\alpha_ t=(\Gamma_{f,s}\cdot V(f_ t))_ 0\) (which implies the constancy of \(\beta_ t\) too), and \(n\geq 3\), using both the h-cobordism theorem and the pseudo-isotopy result of Cerf, the author establishes the constancy of the ambient topological type of \(V(f_ t)\) at the origin. Moreover, it is proved that the diffeomorphism type of the Milnor fibration of \(f_ t\) at the origin is constant.
Reviewer: A.Némethi

32S05 Local complex singularities
Full Text: DOI EuDML
[1] Briançon, J., Speder, J.P.: La trivialité topologique n’implique pas les conditions de Whitney. C.R. Acad. Sci. Paris280, 365-367 (1975) · Zbl 0331.32010
[2] Cerf, J.: La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie. I.H.E.S.39, 187-353 (1970)
[3] Fulton, W.: Intersection theory. Ergebnisse der Math. Berlin, Heidelberg, New York: Springer 1984 · Zbl 0541.14005
[4] Gibson, C. et al.: Topological stability of smooth mappings. Lect. Notes in Math. 552. Berlin, Heidelberg, New York: Springer 1976 · Zbl 0377.58006
[5] Husemoller, D.: Fibre bundles. Graduate Texts in Math. 20. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0144.44804
[6] Hamm, H., Lê D?ng Tráng: Un théorème de Zariski du type de Lefschetz. Ann. Sci. L’Ecole Norm. Sup.6, 317-366 (1973)
[7] Kato, M., Matsumoto, Y.: On the connectivity of the Milnor fiber of a holomorphic function at a critical point. Proc. of 1973 Tokyo Manifolds Conference, 131-136 · Zbl 0309.32008
[8] Lê D?ng Tráng: Topologie des singularities des hypersurfaces complexes. Singularités à Cargése. Asterique7, 171-182 (1973)
[9] Lê D?ng Tráng: Calcul du nombre de cycles évanouissants d’une hypersurface complexe. Ann. Inst. Fourier, Grenoble23, 261-270 (1973) · Zbl 0293.32013
[10] Lê D?ng Tráng: Ensembles analytiques complexes avec lieu singulier de dimesnion un (d’après I.N. Iomdine) Seminar on Singularities (Paris, 1976/1977) pp. 87-95, Publ. Math. Univ. Paris VII 7, Univ. Paris VII, Paris, 1980
[11] Lê D?ng Tráng, Ramanujam, C.P.: The invariance of Milnor’s number implies the invariance of the topological type. Am. J. Math.98, 67-78 (1976) · Zbl 0351.32009 · doi:10.2307/2373614
[12] Lê D?ng Tráng, Saito, K.: La constance du nombre e Milnor donne des bonnes stratifications. C.R. Acad. Sci. Paris277, 793-795 (1973)
[13] Massey, D.: Families of hypersurfaces with one-dimensional singular sets. Dissertation, Duke University, 1986
[14] Mather, J.: Notes on topological stability. Harvard University, 1970 · Zbl 0207.54303
[15] Milnor, J.: Singular points of complex hypersurfaces. Ann. Math. Stud.61, (1968) · Zbl 0184.48405
[16] Perron, B.: ? constant implique type topologique constant en dimension complex trois. C.R. Acad. Sci. Paris295, 735-738 (1982) · Zbl 0526.32009
[17] Randell, R.: On the topology of non-isolated singularities.Geometric topology (Proc. Georgia Top. Conf., Athens, Ga., 1977) 445-473. New York: Academic Press 1979
[18] Teissier, B.: Résolution simultanée-II. Résolution simultanée et cycles evanescents. Lectures Notes in Math. 777, 82-146 (Séminaire sur les Singularités des Surfaces, Palaiseau, France 1976/1977). Berlin, Heidelberg, New York: Springer 1980
[19] Timourian, J.G.: The invariance of Milnor’s number implies topological triviality. Am. J. Math.99, 437-446 · Zbl 0373.32003
[20] Vannier, J.P.: Sur les fibres de Milnor d’une famille à un paramètre de fonctions analytiques à singularité isolée, C.R. Acad. Sci. Paris, 302, Serie I (1986) · Zbl 0586.32007
[21] Vannier, J.P.: Familles à un paramètre de fonctions analytiques à lieu singulier de dimension un. C.R. Acad. Sci. Paris 303, Serie I (1986) · Zbl 0596.32014
[22] Vannier, J.P.: Vannier, J.P.: Familles a paramètres de fonctions holomorphes a ensembles singulier de dimension zero ou un. These, Université de Bourgogne, 1987 · Zbl 0687.32014
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