Structure of leaves and the complex Kupka-Smale property
Annales de l'Institut Fourier, Volume 63 (2013) no. 5, p. 1849-1879

We study topology of leaves of $1$-dimensional singular holomorphic foliations of Stein manifolds. We prove that for a generic foliation all leaves, except for at most countably many, are contractible, the rest are topological cylinders. We show that a generic foliation is complex Kupka-Smale.

Nous étudions la topologie des feuilles d’un feuilletage holomorphe singulier de dimension $1$ sur des variétés de Stein. Nous prouvons que pour un feuilletage générique, toutes les feuilles, sauf au plus un nombre dénombrable, sont contractiles, les autres étant topologiquement des cylindres. Nous montrons aussi qu’un feuilletage générique est Kupka-Smale complexe.

DOI : https://doi.org/10.5802/aif.2816
Classification:  37F75,  32M25,  32E10
Keywords: holomorphic foliations, complex differential equations, Stein manifolds, Kupka-Smale property, generic properties
@article{AIF_2013__63_5_1849_0,
author = {Firsova, Tanya},
title = {Structure of leaves and the complex Kupka-Smale property},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {63},
number = {5},
year = {2013},
pages = {1849-1879},
doi = {10.5802/aif.2816},
mrnumber = {3186510},
zbl = {1294.37020},
language = {en},
url = {http://www.numdam.org/item/AIF_2013__63_5_1849_0}
}

Firsova, Tanya. Structure of leaves and the complex Kupka-Smale property. Annales de l'Institut Fourier, Volume 63 (2013) no. 5, pp. 1849-1879. doi : 10.5802/aif.2816. http://www.numdam.org/item/AIF_2013__63_5_1849_0/

[1] Arnol’D, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Vol.II. The classification of critical points, caustics and wave fronts, Birkhauser Boston, Inc., Boston, MA, Monographs in Mathematics, Tome 82 (1985) | MR 777682 | Zbl 0659.58002

[2] Buzzard, G.T. Kupka-Smale Theorem for Automorphisms of ${ℂ}^{n}$, Duke Math. J., Tome 93 (1998), pp. 487-503 | Article | MR 1626731 | Zbl 0946.32012

[3] Candel, A.; Gomez-Mont, X. Uniformization of the leaves of a rational vector field, Annales de l’Institute Fourier, Tome 45(4) (1995), pp. 1123-1133 | Article | Numdam | MR 1359843 | Zbl 0832.32017

[4] Chaperon, Marc ${C}^{k}$-conjugacy of holomorphic flows near a singularity, Inst. Haute Études Sci. Publ. Math., Tome 64 (1986), pp. 143-183 | Numdam | MR 876162 | Zbl 0625.57011

[5] Chaperon, Marc Generic complex flows, Complex Geometry II: Contemporary Aspects of Mathematics and Physics, Hermann (2004), pp. 71-79 | MR 2493571 | Zbl 1145.37305

[6] Chirka, E. M. Complex analytic sets, Kluwer, Dordrecht (1989) | MR 1111477 | Zbl 0683.32002

[7] Firsova, T.S. Topology of analytic foliations in ${ℂ}^{2}.$ Kupka-Smale property, Proceedings of the Steklov Institute of Mathematics, Tome 254 (2006), pp. 152-168 | Article | MR 2301003

[8] Glutsyuk, A. Hyperbolicity of phase curves of a general polynomial vector field in ${ℂ}^{n}$, Func. Anal. Appl., Tome 28(2) (1994), pp. 1-11 | MR 1283247 | Zbl 0848.32032

[9] Golenishcheva-Kutuzova, T.; Kleptsyn, V. Minimality and ergodicity of a generic foliation of ${ℂ}^{2}$, Ergod. Th. & Dynam. Sys, Tome 28 (2008), pp. 1533-1544 | Article | MR 2449542 | Zbl 1169.37007

[10] Golenishcheva-Kutuzova, T. I. A generic analytic foliation in ${ℂ}^{2}$ has infinitely many cylindrical leaves, Proc. Steklov Inst. Math., Tome 254 (2006), pp. 180-183 | Article | MR 2301005

[11] Hörmander, L. An Introduction to complex analysis in several variables, North Holland, Neitherlands (1990) | MR 1045639 | Zbl 0685.32001

[12] Ilyashenko, Yu Selected topics in differential equations with real and complex time. Normal forms, bifurcations and finiteness problems in differential equations, NATO Sci. Ser. II Math. Phys. Chem., Tome 137 (2004), pp. 317-354 | MR 2083252 | Zbl 0884.00026

[13] Ilyashenko, Yu Some open problems in real and complex dynamical systems, Nonlinearity, Tome 21(7) (2008), pp. 101-107 | Article | MR 2425322 | Zbl 1183.37016

[14] Ilyashenko, Yu.; Yakovenko, S. Lectures on analytic differential equations, Amer. Math. Soc., Graduate Studies in Mathematics, Tome 86 (2008) | MR 2363178 | Zbl 1186.34001

[15] Landis, E.; Petrovskii, I. On the number of limit cycles of the equation $\frac{dy}{dx}=\frac{P\left(x,y\right)}{Q\left(x,y\right)},$ where $P$ and $Q$ of second degree (Russian), Mat. sbornik, Tome 37(79), 2 (1955) | MR 73004

[16] Lins Neto, A. Simultaneous uniformization for the leaves of projective foliations by curves, Bol. Soc. Brasil. Mat. (N.S.), Tome 25 (1994) no. 2, pp. 181-206 | Article | MR 1306560 | Zbl 0821.32027

[17] Moldavskis, V. New generic properties of complex and real dynamical systems, PhD thesis, Cornell University (2007)

[18] Siu, Yum-Tong Every Stein subvariety admits a Stein Neighborhood, Inventiones Math., Tome 38 (1976), pp. 89-100 | Article | MR 435447 | Zbl 0343.32014

[19] Stolzenberg, G. Uniform approximation on smooth curves, Acta. Math, Tome 115, 3-4 (1966), pp. 185-198 | Article | MR 192080 | Zbl 0143.30005

[20] Volk, D.S. The density of separatrix connections in the space of polynomial foliations in $ℂ{\mathrm{P}}^{2}$, Proc. Steklov Inst. Math., Tome 3(254) (2006), pp. 169-179 | Article | MR 2301004

[21] Wermer, J. The hull of curve in ${ℂ}^{n}$, Annals of Mathematics, Tome 68, 3 (1958) | MR 100102 | Zbl 0084.33402