2005/02/11 by Laurent Stolovitch, Stolovitch, Laurent
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.CV #math.DS
paper · pdf · doi:10.48550/arxiv.math/0502231
A shorter version is to appear in Annals of Mathematics
arxiv created 2005/02/11 · openalex publication_date 2005/02/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a commutative family of holomorphic vector fields in an neighbourhood of a common singular point, say 0∈ \Bbb Cn. Let \lie g be a commutative complex Lie algebra of dimension l. Let λ1,...,λn∈ \lie g^* and let us set S(g)=∑i=1nλi(g)xi(∂)/(∂ xi). We assume that this Lie morphism is \bf diophantine in the sense that a diophantine condition (ω(S)) is satisfied. Let X1 be a holomorphic vector field in a neighbourhood of 0∈ \Bbb Cn. 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 X2,..., Xl be holomorphic vector fields vanishing at 0 and commuting with X1. Then there exists a formal diffeomorphism of (\Bbb Cn,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 Xi's belongs to \cal OnS⊗ S(\lie g) ( \cal OnS is the ring of formal first integrals of S) and their junior parts are free over \cal OnS, then there exists a holomorphic diffeomorphism of (\Bbb Cn,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.