Convergence of the Chern-Moser-Beloshapka normal forms

In this article, we first describe a normal form of real-analytic, Levi-nondegenerate submanifolds of $C^N$ of codimension d $\ge$ 1 under the action of formal biholomorphisms, that is, of perturbations of Levi-nondegenerate hyperquadrics. We give a sufficient condition on the formal normal form that ensures that the normalizing transformation to this normal form is holomorphic. We show that our techniques can be adapted in the case d = 1 in order to obtain a new and direct proof of Chern-Moser normal form theorem.