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

Laurent Stolovitch

• English

  We consider a commutative family of holomorphic vector fields in a neighbourhood of a common singular point, say 0 ¡ô Cn. Let g be a commutative complex Lie algebra of dimension l. Let ¥ë1, . . . , ¥ën ¡ô g. and let us set S(g) =
  n i=1 ¥ëi(g)xi ¡Ó ¡Óxi . We assume that this Lie morphism is diophantine in some sense. Let X1 be a holomorphic vector fields in a neighbourhood of 0 ¡ô Cn. We assume that its linear part s is regular relative to S, that it belongs to S(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 (Cn, 0) such that the family of vector fields are in normal form in these formal coordinates. This means that each element of the family commutes with s. We assume that the normal forms of the Xi¡¯s belong to OS n . S(g) ( OS n is the ring of formal first integral of S). We also assume that the family of lowest order parts of the normal forms is free over OS n. Then, we show that there exists a holomorphic diffeomorphism of (Cn, 0) which transforms the family into a normal form. The elements of the family, but one, may not have a nonzero linear part.

• français

  Nous montrons un r¿esultat de normalisation holomorphe d¿une famille commutative de champs de vecteurs holomorphes au voisinage de leur point singulier commun. Pour ce faire, nous supposons qu¿une condition diophantienne portant sur une alg`ebre de Lie commutative de champs lin¿eaires associ¿ee est satisfaite. D¿autre part, nous imposons certaines conditions alg¿ebriques sur leur forme normale formelle. Les champs d¿une telle famille, sauf un, peuvent ne pas avoir de partie lin¿eaire `a l¿origine.