NORMAL FORM OF HOLOMORPHIC VECTOR FIELDS WITH AN INVARIANT TORUS UNDER BRJUNO'S A CONDITION

This article proves the existence of an analytic normal form for some holomorphic differential systems in the neighborhood of a fixed point and of an invariant torus. Once a formal normal form is constructed, one shows that the initial system with quasilinear part S can be holomorphically conjugated to a normal form, i.e. a vector field which commutes with S, under two arithmetical conditions known as Brjuno's gamma and omega conditions, and an algebraic condition known as Brjuno's A-condition, which requires the formal normal form to be proportional to S.