Abstract
We study how to minimize the number of invariants that is sufficient for the Whitney equisingularity of a one parameter deformation of any finitely determined holomorphic germ f: (ℂn, 0) → (ℂ3, 0), with n > 3. Gaffney showed in [3] that the invariants for the Whitney equisingularity are the 0-stable invariants and the polar multiplicities of the stable types of the germ. First we describe all stable types which appear in these dimensions. Then we find relationships between the polar multiplicities of the stable types in the singular set and also in the discriminant. When n > 3, for any germ f there is an hypersurface in ℂn, which is of special interest, the closure of the inverse image of the discriminant by f, which possibly is with non isolated singularities. For this hypersurface we apply results of Gaffney and Gassler [6], and Gaffney and Massey [7], to show how the Lê numbers control the polar invariants of the strata in this hypersurface. Gaffney shows that the number of invariants needed is 4n+10. In the corank one case we reduce this number to 2n+2. The polar multiplicities are also an interesting tool to compute the local Euler obstruction of a singular variety, see [12]. Here we apply this result to obtain explicit algebraic formulae to compute the local Euler obstruction of the stable types which appear in the singular set and also for the stable types which appear in the discriminant, of corank one map germs from ℂn to ℂ3 with n ≥ 3.
The first named author was partially supported by CAPES - Grant 3131/04-3.
The third named author is partially supported by CNPq - Grant 300880/03-6.
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References
J.P. Brasselet and M.H. Schwartz, Sur les classes de Chern d’un ensemble analytique complexe, Asterisque 82–83, 93–146, 1981.
J. Damon, Nonlinear sections of nonisolated complete intersections, Kluwer Academic Publishers, 2001.
T. Gaffney, Polar Multiplicities and Equisingularity of Map Germs, Topology, vol.32, No.1, 185–223, 1993.
T. Gaffney, Properties of Finitely Determined Germs, Ph.D. Thesis, Brandeis University, 1975.
T. Gaffney, Multiplicities and Equisingularities of I.C.I.S Germs, Invent. Math. 123, 200–220, 1996.
T. Gaffney and R. Gassler, Segre Numbers and Hypersurfaces Singularities, Jour. Algebraic Geometry 08, 695–736, 1999.
T. Gaffney and D. Massey, Trends in Equisingularity Theory in Singularity Theory, edited by Bruce, J.W. and Mond. D., Cambridge University Press, 1999.
T. Gaffney and R. Vohra, A numerical characterization of equisingularity for map germs from n-space, (n ≥ 3), to the plane. J. Dyn. Syst. Geom. Theor. 2, 43–55, 2004.
G. Gonzalez-Sprinberg, L’obstruction locale d’Euler et le théorème de Mac-Pherson, Astérisque 82 et 83, 7–33, ed. J.L. Verdier, Soc. Math. France, Paris, 1981.
V.V. Goryunov, Semi-simplicial resolutions and homology of images and discriminants of mappings, Proc. London Math. Soc. 3,70, 363–385, 1995.
Lê D.T., Calculation of Milnor number of isolated singularity of complete intersection, Funktsional’ny: Analizi Ego Prilozheniya, vol. 8, no. 2, 45–49, 1974.
Lê D.T. and B. Teissier, Variétés polaires locales et classes de Chern de variétés singulières. Annals of Math., vol. 114, 457–491, 1981.
E.J.N. Looijenga, Isolated singular points on complete intersections, London Mathematical Soc., Lecture Note series, 77, 1984.
R.D. MacPherson, Chern class for singular algebraic varieties, Ann. of Math., 100, 423–432, 1974.
D. Massey, Le Cycles and Hypersurface Singularities, Springer Lecture Notes in Mathematics, No. 1615, 1995.
D. Massey, Numerical invariants of perverse sheaves, Duke Math. Journal, 73, #2 307–369, 1994.
V.H. Jorge Pérez, Polar multiplicities and equisingularity of map germs from ℂ 3 to ℂ 3, Houston Journal of Mathematics. 29 n: 04, 901–923, 2003.
V.H. Jorge Pérez, D. Levcovitz and M.J. Saia, Invariants, equisingularity and Euler obstruction of map germs from ℂ n to ℂ n, Journal fur die Reine und angewandte Mathematik, Crelle’s Journal, 587 2005, 145–167.
V.H.J. Jorge Pérez and M.J. Saia. Euler Obstruction, polar multiplicities and equisingularity of map germs in O(n, p), n < p. Int. Jour. Maths. 17, n. 8, 887–903, 2006.
B. Tessier, Variétés Polaires 2: Multiplicités Polaires, Sections Planes, et Conditions de Whitney, Actes de la conference de géometrie algébrique à la Rábida, Springer Lecture Notes, 961, 314–491, 1981.
B. Tessier, Cycles Evanescents, Sections Planes et Conditions de Whitney II. Proc. Symposia Pure Math., vol. 40, part 2, Amer. Math. Soc., 65–104, 1983.
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Jorge Pérez, V.H., Rizziolli, E.C., Saia, M.J. (2006). Whitney Equisingularity, Euler Obstruction and Invariants of Map Germs from ℂn to ℂ3, n > 3. In: Brasselet, JP., Ruas, M.A.S. (eds) Real and Complex Singularities. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7776-2_19
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