# Convergence of the Chern-Moser-Beloshapka normal forms

## TL;DR

This paper develops a formal normal form for certain real-analytic submanifolds in complex space, providing conditions for holomorphic normalization and offering a new proof of the Chern-Moser theorem.

## Contribution

It introduces a normal form for Levi-nondegenerate submanifolds under formal biholomorphisms and establishes conditions for holomorphic normalization, extending and simplifying existing results.

## Key findings

- Provided a sufficient condition for holomorphic normalization.
- Extended techniques to the case d=1 for a new proof of Chern-Moser form.
- Described a normal form for perturbations of Levi-nondegenerate hyperquadrics.

## Abstract

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.