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Generic rigidity of singular foliations. (Rigidité générique des feuilletages singuliers.) (French) Zbl 0934.32023
The paper deals with differential equations $$\omega=0$$, where $$\omega=A(x,y)dx+B(x,y)dy$$ is a holomorphic $$1$$-form on some neighborhood $$U$$ of $$(0,0)\in{\mathbb C}^2$$ such that $$(A,B)=1$$, $$A(0,0)=B(0,0)=0$$ and $$\omega\wedge d\omega=0$$; thus $$\omega=0$$ defines a singular foliation on $$U$$.
In the first part of the paper, the author considers families of $$1$$-forms $$\omega_{\varepsilon}$$ as above, depending holomorphically on $$\varepsilon\in V$$, where $$V$$ is a neighborhood of $$0\in{\mathbb C}$$. The first main result shows that, when the germ of $$\omega_0$$ at $$(0,0)$$ is non-exceptional, such a family $$\omega_{\varepsilon}$$ is analytically conjugate to the constant family $$\omega_0$$ if it is formally conjugate to $$\omega_0$$. Here, the condition “being non-exceptional” can be explained as follows. Let $$\text{Diff}$$ denote the group of germs at $$0\in{\mathbb C}$$ of analytic diffeomorphisms that fix $$0$$. A subgroup $$H\subset\text{Diff}$$ is called super-rigid if every formal conjugate of $$H$$ is an analytic conjugate. D. Cerveau and R. Moussu have proved that non-exceptional subgroups $$H\subset\text{Diff}$$ are just those which are non-exceptional, where exceptional subgroups $$H$$ are those whose commutator $$[H,H]$$ is a non-trivial abelian group [Bull. Soc. Math. Fr. 116, No. 4, 459-488 (1988; Zbl 0696.58011)]. Then the germ of $$\omega_0$$ at $$(0,0)$$ is non-exceptional if there is a non-dicritical irreducible component of its desingularization tree whose holonomy group is nonabelian and non-exceptional.
The second part of the paper is devoted to the study how general is the hypothesis “being non-exceptional” of the first main result. The second main result shows that this property is generic in the class of the so called generalized curves, which are those germs whose desingularization has no trivial eigenvalue in the linear part at singular points.

##### MSC:
 32S65 Singularities of holomorphic vector fields and foliations
##### Keywords:
holomorphic differential forms; formal type; analytic type
Full Text:
##### References:
 [1] M. ARTIN , On the solutions of analytic equations , (Invent. Math., vol. 5, 1968 , p. 277-291). MR 38 #344 | Zbl 0172.05301 · Zbl 0172.05301 [2] C. BANICA et O. STANASILA , Algebraic methods in the global theory of complex spaces , John Wiley and sons, 1976 . MR 57 #3420 | Zbl 0334.32001 · Zbl 0334.32001 [3] M. BERTHIER et R. MOUSSU , Réversibilité et classification des centres nilpotents , (Ann. Inst. Fourier, vol. 44, 2, 1994 , p. 465-494). Numdam | MR 95h:58103 | Zbl 0803.34005 · Zbl 0803.34005 [4] F. CANO et D. CERVEAU , Desingularization of non dicritical holomorphic foliations and existence of separatrices , (Acta Math., vol. 169, 1992 , p. 1-103). MR 93k:32069 | Zbl 0771.32018 · Zbl 0771.32018 [5] F. CANO et J.-F. MATTEI , Hypersurfaces intégrales des feuilletages holomorphes , (Ann. Inst. Fourier, vol. 42, 1-2, 1992 , p. 49-72). Numdam | MR 93h:32043 | Zbl 0762.32018 · Zbl 0762.32018 [6] C. CAMACHO et P. SAD , Invariant varieties through singularities of holomorphic vector fields , (Annals Math., vol. 115, 1982 , p. 579-595). MR 83m:58062 | Zbl 0503.32007 · Zbl 0503.32007 [7] C. CAMACHO , A. LINS NETO et P. SAD , Topological invariants and equidesingularization for holomorphic vector fields , (J. Diff. Geom., vol. 20, N^\circ 1, 1984 , p. 143-174). MR 86d:58080 | Zbl 0576.32020 · Zbl 0576.32020 [8] D. CERVEAU et J.-F. MATTEI , Formes intégrales holomorphes singulières , (Astérisque, vol. 97, 1982 ). MR 86f:58006 | Zbl 0545.32006 · Zbl 0545.32006 [9] D. CERVEAU et R. MOUSSU , Groupes d’automorphismes de \Bbb C,0 et équations différentielles y dy + ... = 0 , (Bull. Soc. Math. France, vol. 116, 1988 , p. 459-488). Numdam | MR 90m:58192 | Zbl 0696.58011 · Zbl 0696.58011 [10] L. LE FLOCH , Un problème de rigidité pour une famille à un paramètre de 1-formes holomorphes , Thèse de l’Université de Rennes I, 1995 . [11] F. LORAY , Feuilletages holomorphes à holonomie résoluble , Thèse de l’Université de Rennes I, 1994 . [12] J.-F. MATTEI , Modules de feuilletages holomorphes singuliers. I : équisingularité , (Ivent. Math., vol. 103, N^\circ 2, 1991 , p. 297-325). MR 92f:32056 | Zbl 0709.32025 · Zbl 0709.32025 [13] J.-F. MATTEI et E. SALEM , Complete system of topological and analytical invariants for a generic foliation of \Bbb C$$^{2}$$,0 , Preprint de l’Université Paul Sabatier, 1997 . MR 98a:32044 [14] J. MARTINET et J.-P. RAMIS , Classification analytique des équations différentielles non linéaires résonnantes du premier ordre , (Ann. Sc. É. Norm. Sup., série 4, t 16, 1983 , p. 571-621). Numdam | MR 86k:34034 | Zbl 0534.34011 · Zbl 0534.34011 [15] J.-F. MATTEI et R. MOUSSU , Holonomie et intégrales premières , (Ann. Sc. É. Norm. Sup., série 4, t 13, 1980 , p. 469-523). Numdam | MR 83b:58005 | Zbl 0458.32005 · Zbl 0458.32005
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