The C 1 invariance of the algebraic multiplicity of a holomorphic vector field
[La C 1 -invariance de la multiplicité algébrique d’un champ de vecteurs holomorpe]
Rosas, Rudy1, 2
1 Pontificia Universidad Católica del Perú Av Universitaria 1801 San Miguel, Lima 32 (Perú)
2 Instituto de Matemática y Ciencias Afines (IMCA) Jr. los Biólogos 245 La Molina, Lima (Perú)
Annales de l'Institut Fourier, Tome 60 (2010) no. 6, pp. 2115-2135.

On démontre que la multiplicité algébrique d’une singularité d’un champ de vecteurs holomorphe est invariante par C 1 -equivalences.

We prove that the algebraic multiplicity of a holomorphic vector field at an isolated singularity is invariant by C 1 equivalences.

DOI : 10.5802/aif.2578
Classification : 37F75
Keywords: Algebraic multiplicity, holomorphic vector field, holomorphic foliation
Mot clés : multiplicité algébrique, champ de vecteurs holomorphique, feuilletage holomorphique
Rosas, Rudy 1, 2

1 Pontificia Universidad Católica del Perú Av Universitaria 1801 San Miguel, Lima 32 (Perú)
2 Instituto de Matemática y Ciencias Afines (IMCA) Jr. los Biólogos 245 La Molina, Lima (Perú) 
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Rosas, Rudy. The $C^1$ invariance of the algebraic multiplicity of a holomorphic vector field. Annales de l'Institut Fourier, Tome 60 (2010) no. 6, pp. 2115-2135. doi : 10.5802/aif.2578. http://www.numdam.org/articles/10.5802/aif.2578/

[1] Bers, L. Riemann surfaces, New York University, 1958 (Notes by Richard Pollack and James Radlow)

[2] Burau, W. Kennzeichnung der schlauchknoten, Abh. Math. Sem. Ham. Univ., Volume 9 (1932), pp. 125-133 | DOI | Zbl

[3] Camacho, C. and Sad, P. Invariant varieties through singularities of holomorphic vector fields, Ann. Math., Volume 115(3) (1982), pp. 579-595 | DOI | MR | Zbl

[4] Camacho, C. and Sad, P. and Lins, A. Topological invariants and equidesingularization for holomorphic vector fields, J. Differential Geometry, Volume 20 (1984), pp. 143-174 | MR | Zbl

[5] Dieudonné, J. Foundations of modern analysis, enlarged and corrected printing, Academic Press, New York, 1969 | MR | Zbl

[6] Dold, A. Lecture notes on algebraic topology, Springer, Berlin, 1972

[7] Gomez Mont, X. and Seade, J. and Verjovsky, A. The index of a holomorphic flow with an isolated singularity, Math. Ann., Volume 291(4) (1991), pp. 737-751 | MR | Zbl

[8] Mattei, J.F. and Cerveau, D. Formes intégrables holomorphes singuliéres, Astérisque 97, Société Mathématique, Paris, 1982 | Numdam | MR | Zbl

[9] Mattei, J.F. and Moussu, R. Holonomie et intégrales premiéres, Ann. Sci. École Norm. Sup.(4), Volume 13(4) (1980), pp. 469-523 | Numdam | MR | Zbl

[10] Milnor, J. Lecture notes on algebraic topology, University Press of Virginia, Charlottesville, 1965

[11] Pommerenke, Ch. Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften 299, A Series of Comprehensive Studies in Mathematics, Springer-Verlag, 1992 | MR | Zbl

[12] Rosas, R. On the topological invariance of the algebraic multiplicity of a holomorphic vector field, IMPA, Rio de Janeiro (2005) (Ph. D. Thesis)

[13] Rudin, W. Real and complex analysis, Tata McGraw-Hill, New Delhi, 1979 | Zbl

[14] Taylor, M.E. Partial Differential Equations I, Springer, New York, 1996 | MR | Zbl

[15] Zariski, O. On the topology of algebroid singularities, Amer. Journ. of Math., Volume 54 (1932), pp. 453-465 | DOI | JFM | MR | Zbl

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