Foliations on the complex projective plane with many parabolic leaves

Marco Brunella

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 4, page 1237-1242
  • ISSN: 0373-0956

Abstract

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We prove that a foliation on C P 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.

How to cite

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Brunella, Marco. "Foliations on the complex projective plane with many parabolic leaves." Annales de l'institut Fourier 44.4 (1994): 1237-1242. <http://eudml.org/doc/75095>.

@article{Brunella1994,
abstract = {We prove that a foliation on $\{\bf C\}P^2$ 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.},
author = {Brunella, Marco},
journal = {Annales de l'institut Fourier},
keywords = {holomorphic foliations; harmonic measures; parabolic Riemann surfaces},
language = {eng},
number = {4},
pages = {1237-1242},
publisher = {Association des Annales de l'Institut Fourier},
title = {Foliations on the complex projective plane with many parabolic leaves},
url = {http://eudml.org/doc/75095},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Brunella, Marco
TI - Foliations on the complex projective plane with many parabolic leaves
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 4
SP - 1237
EP - 1242
AB - We prove that a foliation on ${\bf C}P^2$ 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.
LA - eng
KW - holomorphic foliations; harmonic measures; parabolic Riemann surfaces
UR - http://eudml.org/doc/75095
ER -

References

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  1. [Arn] V. I. ARNOL'D, Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Mir, Moscou (1980). Zbl0956.34501MR83a:34003
  2. [CLS1] C. CAMACHO, A. LINS NETO, P. SAD, Minimal sets of foliations on complex projective spaces, Publ. IHES, 68 (1988), 187-203. Zbl0682.57012MR90e:58129
  3. [CLS2] C. CAMACHO, A. LINS NETO, P. SAD, Foliations with algebraic limit sets, Ann. of Math., 136 (1992), 429-446. Zbl0769.57017MR93i:32035
  4. [Gar] L. GARNETT, Foliations, the ergodic theorem and brownian motion, Jour. of Funct. Anal., 51 (1983), 285-311. Zbl0524.58026MR84j:58099
  5. [Ghy] E. GHYS, Topologie des feuilles génériques, preprint ENS de Lyon (1993). Zbl0843.57026
  6. [KM] Y. KUSUNOKI, S. MORI, On the harmonic boundary of an open Riemann surface, I, Jap. Jour. of Math., 29 (1959), 52-56. Zbl0098.28002MR22 #11125a
  7. [Suz] M. SUZUKI, Sur les intégrales premières de certains feuilletages analytiques complexes, Séminaire Norguet, Springer Lect. Notes, 670 (1977), 53-79. Zbl0391.32017MR80h:57038
  8. [Tsu] M. TSUJI, Potential theory in modern function theory, Maruzen, Tokyo (1959). Zbl0087.28401MR22 #5712
  9. [Var] N. Th. VAROPOULOS, Random walks on groups. Applications to Fuchsian groups, Ark. för Math., 23 (1985), 171-176. Zbl0596.60073MR87b:60107

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