Normalisation holomorphe d'algèbres de type Cartan de champs de vecteurs holomorphes singuliers

Abstract

We consider a commutative family of holomorphic vector fields in an neighbourhood of a common singular point, say $0\in {\mathbb{C}}^n$. Let $\lie g$ be a commutative complex Lie algebra of dimension $l$. Let $λ_1,...,λ_n\in \lie g^*$ and let us set $S(g)={\sum }_{i=1}^n{\lambda }_i(g)x_i\frac{\partial }{\partial x_i}$. We assume that this Lie morphism is {\bf diophantine} in the sense that a diophantine condition $(\omega (S))$ is satisfied. Let $X_1$ be a holomorphic vector field in a neighbourhood of $0\in {\mathbb{C}}^n$. We assume that its linear part $s$ is regular relatively to $S$, that is belongs to $S(\lie g)$ and has the same formal centralizer as $S$. Let $X_2,...,X_l$ be holomorphic vector fields vanishing at 0 and commuting with $X_1$. Then there exists a formal diffeomorphism of $({\mathbb{C}}^n,0)$ such that the family of vector fields are in {\bf normal form} in these formal coordinates. This means that each element of the family commutes with $s$. We show that, if the normal forms of the $X_i$'s belongs to $\hat {\cal O}_n^S\otimes S(\lie g)$ (${\hat{\mathcal{O}}}_n^S$ is the ring of formal first integrals of $S$) and their junior parts are free over ${\hat{\mathcal{O}}}_n^S$, then there exists a holomorphic diffeomorphism of $({\mathbb{C}}^n,0)$ which transforms the family into a normal form. The elements of the family, but one, may not have a non-zero linear part at the origin.

A shorter version is to appear in Annals of Mathematics