Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

TL;DR

This paper establishes conditions under which holomorphic vector fields with nilpotent linear parts can be normalized holomorphically, extending classical results and providing new criteria related to Bruno's condition in higher dimensions.

Contribution

It introduces a nilpotent version of Bruno's condition ensuring holomorphic normalization of nonlinear perturbations of nilpotent vector fields.

Findings

Provides a sufficient condition for holomorphic normalization based on Bruno's condition (A).

Shows in dimension 2, no additional condition is needed for holomorphic conjugacy.

Uses Newton's method and $rak{sl}_2(C)$-representations in the proof.

Abstract

We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension $n \geq 3$. Based on Belitskii's work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensure that the normalizing transformation is holomorphic at the fixed point. We shall show that this sufficient condition is a nilpotent version of Bruno's condition (A). In dimension 2, no condition is required since, according to Str{\'o}zyna- Zoladek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton's method and $\frak{sl}\_2(\Bbb C)$-representations.