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

Zbl 0696.58011
Full Text:

References:

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