Foliations on the complex projective plane with many parabolic leaves
Annales de l'Institut Fourier, Volume 44 (1994) no. 4, pp. 1237-1242.

We prove that a foliation on CP 2 with hyperbolic singularities and with “many" parabolic leaves (i.e. leaves without Green functions) is in fact a linear foliation. This is done in two steps: first we prove that there exists an algebraic leaf, using the technique of harmonic measures, then we show that the holonomy of this leaf is linearizable, from which the result follows easily.

On démontre qu’un feuilletage sur CP 2 avec singularités hyperboliques et ayant “beaucoup" de feuilles paraboliques (i.e. sans fonctions de Green) est en fait un feuilletage linéaire. La preuve est faite en deux temps : d’abord on montre l’existence d’une feuille algébrique, en utilisant la notion de mesure harmonique, puis on montre que l’holonomie de cette feuille est linéarisable, ce qui implique aisément le résultat final.

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     title = {Foliations on the complex projective plane with many parabolic leaves},
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Brunella, Marco. Foliations on the complex projective plane with many parabolic leaves. Annales de l'Institut Fourier, Volume 44 (1994) no. 4, pp. 1237-1242. doi : 10.5802/aif.1432. http://archive.numdam.org/articles/10.5802/aif.1432/

[Arn] V. I. Arnol'D, Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Mir, Moscou (1980). | MR | Zbl

[CLS1] C. Camacho, A. Lins Neto, P. Sad, Minimal sets of foliations on complex projective spaces, Publ. IHES, 68 (1988), 187-203. | Numdam | MR | Zbl

[CLS2] C. Camacho, A. Lins Neto, P. Sad, Foliations with algebraic limit sets, Ann. of Math., 136 (1992), 429-446. | MR | Zbl

[Gar] L. Garnett, Foliations, the ergodic theorem and brownian motion, Jour. of Funct. Anal., 51 (1983), 285-311. | MR | Zbl

[Ghy] E. Ghys, Topologie des feuilles génériques, preprint ENS de Lyon (1993). | Zbl

[KM] Y. Kusunoki, S. Mori, On the harmonic boundary of an open Riemann surface, I, Jap. Jour. of Math., 29 (1959), 52-56. | MR | Zbl

[Suz] M. Suzuki, Sur les intégrales premières de certains feuilletages analytiques complexes, Séminaire Norguet, Springer Lect. Notes, 670 (1977), 53-79. | MR | Zbl

[Tsu] M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo (1959). | MR | Zbl

[Var] N. Th. Varopoulos, Random walks on groups. Applications to Fuchsian groups, Ark. för Math., 23 (1985), 171-176. | MR | Zbl

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