Article
Stabilité structurelle du feuilletage de Jouanolou de degré 2
Publications Mathématiques de l'IHÉS, Volume 141 (2025), pp. 191-247

Nous démontrons que le feuilletage de Jouanolou de degré 2 sur le plan projectif complexe est structurellement stable. De plus, son ensemble de Fatou est une fibration holomorphe sur la quartique de Klein ayant une structure de fibré lisse localement trivial en disques. En particulier, aucune feuille de $\mathcal{J}_{2}$ n’est dense dans $\mathbf{P}^{2}$.

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DOI: 10.1007/s10240-024-00153-x

Aurélien Alvarez 1; Bertrand Deroin 1

1
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     title = {Stabilit\'e structurelle du feuilletage de {Jouanolou} de degr\'e 2},
     journal = {Publications Math\'ematiques de l'IH\'ES},
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Aurélien Alvarez; Bertrand Deroin. Stabilité structurelle du feuilletage de Jouanolou de degré 2. Publications Mathématiques de l'IHÉS, Volume 141 (2025), pp. 191-247. doi: 10.1007/s10240-024-00153-x

[1.] E. Arbarello; M. Cornalba; P. A. Griffiths; J. E. Harris Geometry of Algebraic Curves, Springer, Berlin, 1984

[2.] T. Asuke A Fatou-Julia decomposition of transversally holomorphic foliations, Ann. Inst. Fourier, Volume 60 (2010), pp. 1057-1104 | MR | Zbl | Numdam | DOI

[3.] F. Bogomolov Complex Manifolds and Algebraic Foliations, 1084, 1996

[4.] C. Bonatti; R. Langevin; R. Moussu Feuilletages de 𝐏 n : de l’holonomie hyperbolique pour les minimaux exceptionnels, Publ. Math. IHES, Volume 75 (1992), pp. 123-134 | DOI | Zbl | Numdam

[5.] M. Brunella Birational Geometry of Foliations, 2015 | DOI | Zbl

[6.] C. Camacho; L. H. de Figueiredo The dynamics of the Jouanolou foliation on the complex projective 2-space, Ergod. Theory Dyn. Syst., Volume 21 (2001), pp. 757-766 | MR | Zbl | DOI

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

[8.] D. Cerveau Densité des feuilles de certaines équations de Pfaff à 2 variables, Ann. Inst. Fourier, Volume 33 (1983), pp. 185-194 | MR | DOI | Zbl | Numdam

[9.] B. Deroin; A. Guillot Foliated affine and projective structures, Compositio Math., Volume 159 (2023), pp. 1053-1187 | MR | Zbl | DOI

[10.] B. Deroin; V. Kleptsyn Random conformal dynamical systems, Geom. Funct. Anal., Volume 17 (2007), pp. 1043-1105 | MR | Zbl | DOI

[11.] T. Fisher; B. Hasselblatt Hyperbolic Flows, 25, 2020

[12.] É. Ghys Sur les groupes engendrés par des difféomorphismes proches de l’identité, Bol. Soc. Bras. Mat., Volume 24 (1993), pp. 137-178 | DOI | Zbl

[13.] É. Ghys; X. Gomez-Mont; J. Saludes Fatou and Julia components of transversely holomorphic foliations. Essays on geometry and related topics, Monogr. Enseign. Math., Volume 38 (2001), pp. 287-319 | Zbl

[14.] A. Haefliger Foliations and Compactly Generated Pseudogroups Foliations : Geometry and Dynamics, World Scientific, Singapore, 2000, pp. 275-295 | Zbl

[15.] M. O. Hudaj-Verenov A property of the solutions of a differential equation, Math. USSR Sb., Volume 56 (1962), pp. 301-308 (en russe) | MR | Zbl

[16.] Y. Ilyashenko Global and local aspects of the theory of complex differential equations, Proc. Int. Cong. Math. Helsinski, 1978, pp. 821-828 | Zbl

[17.] J.-P. Jouanolou Équations de Pfaff algébriques, 708, Springer, Berlin, 1979 | DOI | Zbl

[18.] A. Lins Neto Algebraic solutions of polynomial differential equations and foliations in dimension two, Holomorphic Dynamics (1986), 1345, Springer, Berlin, 1988, pp. 192-232 | Zbl | DOI

[19.] F. Loray; J. Rebelo Minimal, rigid foliations by curves on 𝐏 n , J. Eur. Math. Soc., Volume 5 (2003), pp. 147-201 | DOI | Zbl

[20.] B. Mjuller On the density of solutions of an equation in 𝐂𝐏 2 , Mat. Sb., Volume 98 (1975), pp. 325-338 | MR | Zbl | DOI

[21.] I. Nakai Separatrices for nonsolvable dynamics on C 0, Ann. Inst. Fourier, Volume 44 (1994), pp. 569-599 | MR | DOI | Zbl | Numdam

[22.] M. Oka On mixed projective curves, Singularities in Geometry and Topology (2009), 20, 2012, pp. 133-147 | Zbl | DOI

[23.] C. Robinson Structural stability manifolds with boundary, J. Differ. Equ., Volume 37 (1980), pp. 1-11 | MR | DOI | Zbl

[24.] A. A. Scherbakov On the density of an orbit of a pseudogroup of conformal mappings and a generalization of the Hudai-Verenov theorem, Vestn. Mosk. Univ. Math., Volume 31 (1982), pp. 10-15

[25.] A. Takeuchi Domaines Pseudo-Convexes dans les variétés Kählériennes, J. Math. Kyoto Univ., Volume 6 (1967), pp. 323-357 | MR | Zbl

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