Convergence of the Chern–Moser–Beloshapka normal forms

Abstract

Abstract In this article, we give a normal form for real-analytic, Levi-nondegenerate submanifolds of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℂ</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> {\mathbb{C}^{N}} of codimension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {d\geq 1} under the action of formal biholomorphisms. We find a very general sufficient condition on the formal normal form that ensures that the normalizing transformation to this normal form is holomorphic. In the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {d=1} our methods in particular allow us to obtain a new and direct proof of the convergence of the Chern–Moser normal form.

Locations

• arXiv (Cornell University) - View - PDF
• HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF
• Journal für die reine und angewandte Mathematik (Crelles Journal) - View