Milnor numbers and the topology of polynomial hypersurfaces. (English) Zbl 0658.32005

Let F: \({\mathbb{C}}^{n+1}\to {\mathbb{C}}\) be a polynomial. F is called tame if \(\| \partial F(x)\| >a\) for \(\| x\| >R\), where a and R are some positive real numbers. In this case, F has only a finite number of critical points. Let \(\mu\) (F) (resp. \(\mu^ c(F))\) be the sum of the Milnor numbers at all critical points of F in \({\mathbb{C}}^{n+1}\) (resp. lying on \(F^{-1}(c))\). Then the following theorem holds: For any \(c\in {\mathbb{C}}\), \(F^{-1}(c)\) has the homotopy type of a bouquet of \(\mu (F)- \mu^ c(F)\) spheres of dimension n.
It is also shown that the property for a polynomial to be tame is generic.
In the general case (i.e. without the assumption that F is tame), information on the homology groups of level sets \(F^{-1}(c)\) for generic \(c\in {\mathbb{C}}\) is also obtained in terms of the Milnor numbers of isolated critical points of F.
Reviewer: T.Krasiński


32S05 Local complex singularities
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14F45 Topological properties in algebraic geometry
Full Text: DOI EuDML


[1] [Br1] Broughton, S.A.: On the Topology of Polynomial Hypersurfaces. Proceedings of Symposia in Pure Mathematics, Vol. 40 (Arcata Singularities Conference) Am. Math. Soc., pp. 167-178, 1983
[2] [Br2] Broughton, S.A.: On the Topology of Polynomial Hypersurfaces. Ph.D. Thesis, Queen’s University, Kingston, Ontario, 1982
[3] [B-V] Burghelea, D., Verona, A.: Local Homological Properties of Analytic Sets. Manuscr. Math.7, 55-56 (1972) · Zbl 0246.32007
[4] [D] Dold, A.: Lectures on Algebraic Topology. Berlin Heidelberg New York: Springer 1972 · Zbl 0234.55001
[5] [Du] Durfee, A.: Neighbourhoods of Algebraic Sets. Transactions of Am. Math. Soc.276, 517-530 (1983) · Zbl 0529.14013
[6] [G] Grothendieck, A.: Éléments de Géométrie Algebrique. IV, Étude Local de Schémas et de Morphismes de Schémas, Publications Mathématiques I.H.E.S., No. 28, 1966
[7] [H] Hartshorne, R.: Algebraic Geometry. Graduate Texts in Math., Berlin Heidelberg New York: Springer 1977 · Zbl 0367.14001
[8] [K] Kouchnirenko, A.G.: Polyedres de Newton et Nombres de Milnor. Invent. Math.32, 1-31 (1976) · Zbl 0328.32007
[9] [Lo] Lojasiewicz, S.: Triangulations of Semi-Analytic Sets. Ann. Scuola Norm. Super. Pisa, Sci. Fis. Mat. Ser. 3,18, 339-373 (1964)
[10] [L-T] Lejeune-Jalabert, M., Teissier, B.: Contribution à l’étude des singularités de point de vue du poly gone de Newton. Cycle évanoissants, plis évanouissants et conditions de whitney. Thesis, Université de Paris VII, Paris 1973
[11] [Ma] Malgrange, B.: Methodes de la Phase Stationaire et Sommation Borel, in Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory. (Lect. Notes Phys., vol. 126) Berlin Heidelberg New York: Springer 1980, pp. 170-177
[12] [Mi1] Milnor, J.: Singular Points of Complex Hypersurfaces, Ann. Math. Stud. vol. 61, Princeton University Press, 1968 · Zbl 0184.48405
[13] [Mi2] Milnor, J.: Morse Theory. Ann. Math. Stud. vol. 51, Princeton University Press 1963
[14] [Or] Orlik, P.: The Multiplicity of a Holomorphic Map at an Isolated Critical Point. Nordic Summer School/NAVF, Symposium in Pure Mathematics (1976), pp. 405-474
[15] [Pa] Palamadov, V.P.: Multiplicity of Holomorphic Mapping, Functional Analysis and its Applications, 1967, pp. 218-266
[16] [Ph] Pham, F.: Vanishing Homologies and then-Variable Saddlepoint Method. Proceedings of the Symposia in Pure and Applied Math., Vol. 40, Part 2, Am. Math. Soc. 1983, pp. 319-334
[17] [Pr] Prill, D.: Local Classification of Quotients of Complex Manifolds by Discontinuous Groups. Duke Math. J.34, 375-386 (1967) · Zbl 0179.12301
[18] [S] Shafarevich, I.R.: Basic Algebraic Geometry. Berlin Heidelberg New York: Springer 1977 · Zbl 0362.14001
[19] [T] Timourian, J.: The Invariance of Milnor’s Number Implies topological Triviality. Am. J. Math.99, 437-446 (1977) · Zbl 0373.32003
[20] [V] Verdier, J.L.: Stratification de Whitney et Théorème de Bertini-Sard. Invent. Math.99, 295-312 (1977) · Zbl 0333.32010
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