zbMATH — the first resource for mathematics

The Lê varieties. II. (English) Zbl 0727.32015
Let \(f: ({\mathbb{C}}^{n+1},0)\to ({\mathbb{C}},0)\) be a polynomial with singular locus of dimension s. In part I of this paper [ibid. 99, 357-376 (1990; Zbl 0712.32020)] the author defines a sequence of varieties \(\Lambda_ f^{(k)}\) using the polar varieties. The aim is to obtain similar informations in case \(s>0\) as in the case of isolated singularities from the Jacobian ideal. Especially the multiplicities of \(\Lambda_ f^{(k)}\) (the so called Lê-numbers) generalize the Milnor number.
In case of an isolated singularity \(\Lambda_ f^{(k)}=\emptyset\) if \(k>0\) and \(\Lambda_ f^{(0)}\) is the variety defined by the Jacobian ideal \((\frac{\partial f}{\partial z_ 0},...,\frac{\partial f}{\partial z_ n})\). The author investigates the Milnor fibration of f using his Lê varieties to extend the result of Lê and Ramanujan concerning the constancy of the Milnor fibration in a family of isolated singularities to nonisolated singularities: The constancy of the Lê-numbers in a family of singularities implies the constancy of the fibre-homotopy type of the Milnor-fibration.
Reviewer: G.Pfister (Berlin)

32S05 Local complex singularities
32S55 Milnor fibration; relations with knot theory
14B05 Singularities in algebraic geometry
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. (Ergebn. Math.) Berlin Heidelberg New York: Springer 1984 · Zbl 0541.14005
[4] Gibson, C., et al.: Topological stability of smooth mappings. (Lect. Notes Math., vol. 552) Berlin Heidelberg New York: Springer 1976 · Zbl 0377.58006
[5] Goresky, M., MacPherson, R.: Stratified morse theory. (Ergebn. Math. Vol. 14) Berlin Heidelberg New York: Springer 1988
[6] Hamm, H., Lê, D?ng Tráng: Un théorème de Zariski du type de Lefschetz. Ann. Sci. Éc. Norm. Supér.6, 317-366 (1973) · Zbl 0276.14003
[7] Henry, J.-P., Merle, M.: Conditions de régularité et éclatements. Ann. Inst. Fourier37, 159-190 (1987) · Zbl 0596.32018
[8] Henry, J.-P., Merle, M., Sabbah, C.: Sur la condition de Thom stricte pour un morphisme analytique complexe. Ann. E.N.S.17, 227-268 (1984) · Zbl 0551.32012
[9] Husemoller, D.: Fibre bundles. (Grad. Texts in Math. 20) Berlin Heidelberg New York: Springer 1966 · Zbl 0144.44804
[10] Kato, M., Matsumoto, Y.: On the connectivity of the Milnor fibre of a holomorphic function at a critical point. Proc. of 1973 Tokyo manifolds conf., 131-136
[11] Lê, D?ng Tráng: Calcul du Nombre de Cycles Évanouissants D’une Hypersurface Complexe. Ann. Inst. Fourier23, 261-270 (1973) · Zbl 0293.32013
[12] Lê, D?ng Tráng: Ensembles Analytiques Complexes avec Lieu Singulier de Dimension Un (D’Après I. Iomdine). Seminar on Singularities (Paris, 1976/1977) pp. 87-95, Publ. Math. Univ. Paris VII, Paris, 1980
[13] Lê, D?ng Tráng: La Monodromie n’a pas de Points Fixes. J. Fac. Sci. Univ. Tokyo, Sec. 1A22, 409-427 (1975) · Zbl 0355.32012
[14] Lê, D?ng Tráng: Some Remarks on Relative Monodromy, Nordic Summer School/NAVF, Symp. Math., Oslo, 1976
[15] Lê, D?ng Tráng: Topological Use of Polar Curves. Proc. Symp. Pure Math.29, 507-512 (1975)
[16] Lê, D?ng Tráng: Topologie des Singularités des Hypersurfaces Complexes. Sing. a Cargese, Asterisque7 and 8, 171-192 (1973)
[17] Lê, D?ng Tráng, Ramanujam, C.: 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
[18] Lê, D?ng Tráng Kyoji Saito: La constance du nombre de Milnor donne des bonnes stratifications. C.R. Acad. Sci Paris277, 793-795 (1973) · Zbl 0283.32007
[19] Lê, D?ng Tráng and Teissier, B.: Cycles evanescents, sections planes et conditions de Whitney, II. Proc. Symp. Pure. Math.40, Part 2, 65-103 (1983) · Zbl 0532.32003
[20] Lê, D?ng Tráng and Teissier, B.: Varíétés polaires locales et classes de Chern des varíétés singulières. Ann. Math.114, 457-491 (1981) · Zbl 0488.32004 · doi:10.2307/1971299
[21] Massey, D.: The Lê-Ramanujam problem for hypersurfaces with one-dimensional singular sets. Math. Ann.282, 33-49 (1988) · Zbl 0657.32005 · doi:10.1007/BF01457011
[22] Massey, D.: The Lê varieties, I. Invent. Math.99, 357-376 (1990) · Zbl 0712.32020 · doi:10.1007/BF01234423
[23] Massey, D.: Lifting Milnor fibrations in a family. (Prepint) 1988
[24] Massey, D.: The Thom condition along a line. Duke Math. Journ.60, 631-642 (1990) · Zbl 0709.32026 · doi:10.1215/S0012-7094-90-06025-9
[25] Mather, J.: Notes on topological stability. (Xeroxed notes) Harvard University 1970 · Zbl 0207.54303
[26] Milnor, J.: Morse theory, Ann. Math. Stud. 51. Princeton 1963 · Zbl 0108.10401
[27] Milnor, J: Singular points of complex hypersurfaces. Ann. Math. Stud. 61. Princeton 1968 · Zbl 0184.48405
[28] Oka, M.: On the homotopy types of hypersurfaces defined by weighted homogeneous polynomials. Topology12, 19-32 (1973) · Zbl 0263.32005 · doi:10.1016/0040-9383(73)90019-0
[29] Pellikaan, R.: Series of isolated singularities. (To appear in: Proc. of Singularities Conf. Iowa 1986) Contemp. Math. AMS (1988) · Zbl 0621.32019
[30] Randell, R.: On the topology of non-isolated singularities. In: Geometric topology. Proc. Georgia Top. Conf., Athens, Ga., 1977, pp. 445-473. Academic Press: New York 1979
[31] Sakamoto, K.: The Seifert matrices of Milnor fiberings defined by holomorphic functions. J. Math. Soc. Jpn,26 (4), 714-721 (1974) · Zbl 0286.32010 · doi:10.2969/jmsj/02640714
[32] Sebastiani, M., Thom, R.: Un résultat sur la monodromie. Invent. Math.13, 90-96 (1971) · Zbl 0233.32025 · doi:10.1007/BF01390095
[33] Siu, Y.T., Trautmann, G.: Gap-sheaves and extension of coherent analytic subsheaves. (Lect. Notes Math., vol. 172) Berlin Heidelberg New York: Springer 1971 · Zbl 0208.10403
[34] Teissier, B.: Introduction to equisingularity problems. Proc. Symp. Pure Math.29, 593-632 (1975) · Zbl 0322.14008
[35] Teissier, B.: Varíétés polaires, I: Invariants polaires des singularités d’hypersurfaces. Invent. Math.40, 267-292 (1977) · Zbl 0446.32002 · doi:10.1007/BF01425742
[36] Teissier, B.: Varietes polaires, II: Multiplicites polaires, sections planes, et conditions de Whitney. (Lect. Notes Math., vol. 961, pp. 314-491) Berlin Heidelberg New York: Springer 1981
[37] Thom, R.: Ensembles et morphismes stratifiés. Bull. Am. Math. Soc.,75, 240-284 (1969) · Zbl 0197.20502 · doi:10.1090/S0002-9904-1969-12138-5
[38] Vannier, J.-P.: Familles à parametres de fonctions holomorphes à ensemble singulier de dimension zero on un. These, Université de Bourgogne 1987
[39] Vannier, J.-P.: Familles à un paramètre de fonctions analytiques à Lieu singulier de dimension un. C.R. Acad. Sci. Paris,303, Serie I, (8), 367-370 (1986) · Zbl 0596.32014
[40] Whitney, H.: Tangents to an analytic variety. Ann. Math.81, 496-549 (1965) · Zbl 0152.27701 · doi:10.2307/1970400
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.