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Singularités des flots holomorphes. II. (Singularities of holomorphic flows. II.). (French) Zbl 0938.32019
Ann. Inst. Fourier 47, No. 4, 1117-1174 (1997); erratum ibid. 50, No. 3, 1019-1020 (2000).
Summary: In part I [Ann. Inst. Fourier 46, No.2, 411-428 (1996; Zbl 0853.34002)], the second author showed, in particular, that a complete holomorphic vector field on a complex surface cannot have an isolated singularity with a vanishing second jet. In this paper, we give a precise description of complete holomorphic vector fields on complex surfaces admitting an isolated singularity with trivial first jet. If the ambient surface is compact, we describe explicitly all possible surfaces and vector fields.

##### MSC:
 32S65 Singularities of holomorphic vector fields and foliations 37F75 Dynamical aspects of holomorphic foliations and vector fields
##### Keywords:
singularities; flow; vector field
Full Text:
##### References:
 [1] D. AKHIEZER, Lie group actions in complex analysis, Vieweg (1994). · Zbl 0845.22001 [2] V. ARNOLD, Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Mir-Moscou, 1980. · Zbl 0956.34501 [3] W. BARTH & C. PETERS & A. VAN de VEN, Compact complex surfaces, Springer, Berlin, 1984. · Zbl 0718.14023 [4] C. CAMACHO & P. SAD, Invariant varieties through singularities of holomorphic vector fields, Annals of Math., 115 (1982), 579-595. · Zbl 0503.32007 [5] J. CARREL, A. HOWARD & C. KOSNIOWSKI, Holomorphic vector fields on complex surfaces, Math. Ann., 204 (1973), 73-81. · Zbl 0253.14005 [6] D. CERVEAU & J.-F. MATTEI, Formes intégrables holomorphes singulières, Astérisque, 97, 1982. · Zbl 0545.32006 [7] D. CERVEAU & B. SCARDUA, On the integration of polynomial vector fields in dimension two, Preprint IRMAR (Rennes), 1996. · Zbl 1118.32022 [8] H. DULAC, Recherches sur LES points singuliers des équations différentielles, J. École Polytechnique, 9 (1904), 1-125. · JFM 35.0331.02 [9] H. HUKUARA, T. KIMURA & T. MATUDA, Équations différentielles ordinaires du premier ordre dans le champ complexe, Publ. Math. Soc. of Japan, 1961. · Zbl 0101.30002 [10] K. KODAIRA, On compact analytic surfaces II, Annals of Math., 77 (1963), 563-626. · Zbl 0118.15802 [11] F. LORAY, Feuilletages holomorphes à holonomie résoluble, Thèse, Univ. Rennes I, 1994. [12] J.-F. MATTEI, Manuscrit, 1996. [13] J.-F. MATTEI & R. MOUSSU, Holonomie et intégrales premières, Ann. Scient. Ec. Norm. Sup., 16 (1983), 469-523. · Zbl 0458.32005 [14] J.-C. REBELO, Singularités des flots holomorphes, Ann. Inst. Fourier, 46-2 (1996), 411-428. · Zbl 0853.34002 [15] A. SEIDENBERG, Reduction of singularities of the differentiable equation A dy = B dx, Amer. J. of Math., 90 (1968), 248-269. · Zbl 0159.33303
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