Holomorphic normalization of Cartan type algebras of singular holomorphic vector fields. (Normalisation holomorphe d’algèbres de type Cartan de champs de vecteurs holomorphes singuliers.) (French. English summary) Zbl 1080.32019

The author considers a commutative family of holomorphic vector fields in a neighborhood of a common singular point (the origin in \( \mathbb {C}^{n}\)) and gives some conditions under which there exists a holomorphic diffeomorphism of the germ \( (\mathbb {C}^{n}, 0)\) which transforms this family into a normal form. Several previous results of other authors are thus generalized and several interesting applications are discussed.


32M25 Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions
32M05 Complex Lie groups, group actions on complex spaces
32A38 Algebras of holomorphic functions of several complex variables
58A30 Vector distributions (subbundles of the tangent bundles)
58D25 Equations in function spaces; evolution equations
22E60 Lie algebras of Lie groups
37F99 Dynamical systems over complex numbers
32A19 Normal families of holomorphic functions, mappings of several complex variables, and related topics (taut manifolds etc.)
Full Text: DOI