Holomorphic Normal Form of Nonlinear Perturbations of Nilpotent Vector Fields

We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension $n\geqslant 3$. 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 ensures 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 Stróżyna–Żoładek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton's method and $\mathfrak{sl}_2(\mathbb C)$-representations.