Modules de feuilletages holomorphes singuliers. I: équisingularité. (Modules of singular holomorphic foliations. I: Equisingularity). (French) Zbl 0709.32025

We prove that there exists a semi-universal unfolding in the equisingular unfolding class, for a germ in \(0\in ({\mathbb{C}}^ 2)\) of holomorphic singular foliation. We give an explicit formula for the dimension of the universal parameter space, with respect to multiplicities of the infinitely near points. This result applied to a function gives a formula for the dimension of the \(\mu\)-constant strata.
Reviewer: J.F.Mattei


32S15 Equisingularity (topological and analytic)
32S65 Singularities of holomorphic vector fields and foliations
Full Text: DOI EuDML


[1] [C-L-S] Camacho, C., Lins, A., Sad, P.: Topological invariant and equisingularisation for holomorphic vector fields. Differ. Geom.20, 143-174 (1984) · Zbl 0576.32020
[2] [C-M] Cerveau, D., Mattei, J.F.: Formes intégrables holomorphes singulières. Astérisque vol. 97 (1982) · Zbl 0545.32006
[3] [F] Frankel, J.: Cohomologie non abélienne et espaces fibrés. Bull. Soc. Math. France85, 135-220 (1957)
[4] [G] Godement, R.: Théorie des faisceaux. Acta Sci. Ind. Hermann, Paris (1964)
[5] [H] Hironaka, H.: Introduction to the theory of infinitely near singular points. Mem. Mat. Inst. Jorge Juan vol. 28 (1974) · Zbl 0366.32007
[6] [Ka] Kabila, A.: Formes intégrables à singularités lisses, Thèse, Université de Dijon, 1983
[7] [Ki] King, H.C.: Topological type of isolated critical points. Ann. Math.107, 385-397 (1978) · doi:10.2307/1971121
[8] [Kl] Klughertz, M.: Feuilletages holomorphes à singularité isolée ayant une infinité de courbes intègrales, Thèse Université de Toulouse III, 1988
[9] [K-M] Klughertz, M., Mattei, J.F.: Champs holomorphes à courbes intégrales analytiques. (En préparation)
[10] [Ko] Kodaira, K.: Complex Manifolds and deformation of complex Structure. (Grundlehren der math. Wiss., vol. 283). Berlin-Heidelberg-New York: Springer 1986
[11] [Ku] Kupka, I.: Singulatities of integrable Pfaffian fams. Proc. Natl. Acad. Sci. USA52, 1431-1432 (1964) · Zbl 0137.41404 · doi:10.1073/pnas.52.6.1431
[12] [L-R] Le Dung Trang, Ramanuyan, 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
[13] [Ma] Martinet, J.: Déploiements versels des applications différentiables et classification des application stables (Lect. Notes in Math., vol. 535). Berlin-Heidelberg-New York: Springer 1976
[14] [M-M] Mattei, J-F., Moussu, R.: Holonomie et intégrales premières. Ann. Sci. Ec. Norm. Super., IV. Ser.13, 469-523 (1980) · Zbl 0458.32005
[15] [Pn] Pnevmatikos, S.: Une remarque sur les déformations d’un germe d’hypersurface à type topologique constant. Can. Math. Bull.28, 455-462 (1985) · Zbl 0574.32009 · doi:10.4153/CMB-1985-054-4
[16] [Se] Seidenberg, A.: Reduction of singularities of the differentiable equation AdY=BdX. Am. J. Math. 90, 248-269 (1968) · Zbl 0159.33303 · doi:10.2307/2373435
[17] [Si] Siu, Y.T.: Every Stein subvariety admits a Stein neighborhood. Invent. Math.38, 89-100 (1976) · Zbl 0343.32014 · doi:10.1007/BF01390170
[18] [Te] Tessier, B., Appendice à O. Zariski le problème des modules pour les bandes planes, publ. école polytechnique, 1973
[19] [VdE] Essen, ven den A.: Reduction of singularities of the differentiable equation Adx+Bdy (Lect. Notes in Math., vol. 712, pp. 44-59). Berlin-Heidelberg-New York: Springer 1979
[20] [W] Wahl, J.: Equisingular deformations of plane Algebrad Curves. Trans. Am. Mat. Soc.193, 143-170 (1974) · Zbl 0294.14007 · doi:10.1090/S0002-9947-1974-0419439-2
[21] [Z1] Zariski, O.: Studies in equisingularity I; Equivalent singularities of plane algebroïde curves. Am. J. Math.87, 507-536 (1965) · Zbl 0132.41601 · doi:10.2307/2373019
[22] [Z2] Zariski, O.: Studies in equisingularity II; Equisingularity in codimension 1 (and caracteristic zero). Am. J. Math.87, 972-1006 (1965) · Zbl 0146.42502 · doi:10.2307/2373257
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.