Normal forms of analytic perturbations of quasihomogeneous vector fields

We will first present a survey about normal forms of holomorphic vector fields. Then, we present a more recent result about germs of holomorphic vector fields which are "higher order" perturbations of a quasihomogeneous vector field in a neighborhood of the origin of $Bbb C^n$, fixed point of the vector fields. We define a "diophantine condition'' associated to the quasihomogeneous initial part $S$ which ensures that if such a perturbation of $S$ is formally conjugate to $S$ then it is also holomorphically conjugate to it. We give a condition on $S$ that ensure that there always exists a holomorphic transformation to a normal form. If this condition is not satisfied, we also show, that under some reasonable assumptions, each perturbation of $S$ admits a Gevrey formal normalizing transformation.