zbMATH — the first resource for mathematics

Desingularization of non-dicritical holomorphic foliations and existence of separatrices. (English) Zbl 0771.32018
The authors continue their earlier work [cf. the first author, C. R. Acad. Sci., Paris, Sér. I 307, No. 15, 795-798 (1988; Zbl 0669.32007)]; they have completed the reduction of the singularities for non-dicritical holomorphic foliations in order to get only the so-called simple singularities. As a consequence, they have proved R. Thom’s conjecture about the existence of convergent separatrices, in dimension three [cf. the authors, C. R. Acad. Sci., Paris, Sér. I Math. 307, 387-390 (1988; Zbl 0656.57019)].

32S30 Deformations of complex singularities; vanishing cycles
32S45 Modifications; resolution of singularities (complex-analytic aspects)
32S65 Singularities of holomorphic vector fields and foliations
57R30 Foliations in differential topology; geometric theory
Full Text: DOI
[1] Aroca, J. M., Hironaka, H. &Vicente, J. L., Introduction to the theory of infinitely near singular points. The theory of the maximal contact. Desingularization theorems.Mem. Mat. Inst. Jorge Juan, 28, 29, 30. Madrid CSIC, 1977.
[2] Banika, E.,Algebraic Methods in the Global Theory of Complex Spaces. John Wiley & Sons, 1976.
[3] Bauer, M., Feuilletage singulier définie par une distribution presque régulière. Thèse, Université Louis Pasteur, Strasbourg, 1985.
[4] Camacho, C. &Sad, P., Invariant varieties through singularities of holomorphic vector fields.Ann. of Math., 115 (1982), 579–595. · Zbl 0503.32007
[5] Cano, F., Réduction des singularités des feuilletages holomorphes.C. R. Acad. Sci. Paris Sér. I Math., 307 (1988), 795–798. · Zbl 0669.32007
[6] Cano, F., Dicriticalness of a singular foliation, inHolomorphic Dynamics. Proceedings, 1986, X. Gomez-Mont, J. Seade, A. Verjovski (eds.).Lecture Notes in Mathematics, 1345 (1988), 73–95. Springer-Verlag.
[7] Cano, F., Reduction of the singularities of non-dicritical singular foliations, dimension three. Preprint Universidad Valladolid, 61 p., 1988. · Zbl 0786.32023
[8] Cano, F. &Cerveau, D., Le problème de la séparatrice, une conséquence de la réduction des singularités des feuilletages holomorphes.C. R. Acad. Sci. Paris Sér. I Math., 307 (1988), 387–390.
[9] Cerveau, D. & Ecalle, J., Preprint, Rennes-Orsay.
[10] Cerveau, D. &Lins, A., Formes tangentes à des actions commutatives.Ann. Fac. Sci. Toulouse, 6 (1984), 51–85. · Zbl 0539.58004
[11] Cerveau, D. & Mattei, J. F.,Formes intégrables holomorphes singulières. Asterisque, 97 (1982). · Zbl 0545.32006
[12] Ecalle, J., Les fonctions résurgentes I, II, III. Publ. Math. Orsay.
[13] Hironaka, H., Resolution of the singularities of an algebraic variety over a field of characteristic zero, I, II.Ann. of Math., 79 (1964), 109–326. · Zbl 0122.38603
[14] Ince, E. L.,Ordinary Differential Equations. Dover Publications, New York, 1956. · Zbl 0063.02971
[15] Jouanolou, J. P.,Equations de Pfaff algébriques. Lecture Notes in Mathematics, 708. Springer-Verlag, 1979. · Zbl 0477.58002
[16] Martinet, J., Normalisation des champs holomorphes (d’apres Brujno).Séminaire Bourbaki, 1980, exp. 564.Lecture Notes in Mathematics, 901. Springer-Verlag, 1982.
[17] Martinet, J. &Ramis, J. P., Problèmes de modules pour les équations différentielles du premier ordre.Publ. Math. I.H.E.S., 55 (1982), 63–164. · Zbl 0546.58038
[18] –, Classification analytique des équations différentielles de premier ordre.Ann. Sci. École Norm. Sup. (4), 16 (1982), 571–621.
[19] Mattei, J. F. &Moussu, R., Holonomie et intégrales premières.Ann. Sci. École Norm. Sup. (4), 13 (1980), 469–523. · Zbl 0458.32005
[20] Paul, E., Etude topologique des formes logarithmiques fermées.Invent. Math., 95 (1989), 395–420. · Zbl 0641.57013
[21] Seidenberg, A., Reduction of the singularities of the differential equationA dy=B dx.Amer. J. Math., 90 (1968), 248–269. · Zbl 0159.33303
[22] Siegel, C. L., Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung.Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1952, pp. 21–30. · Zbl 0047.32901
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.