On Beloshapka’s rigidity conjecture for real submanifolds in complex space

Abstract

A well known Conjecture due to Beloshapka asserts that all totally nondegenerate polynomial models with the length $l \geq 3$ of their Levi–Tanaka algebra are rigid, that is, any point preserving automorphism of them is completely determined by the restriction of its differential at the fixed point onto the complex tangent space. For the length $l = 3$, Beloshapka’s Conjecture was proved by Gammel and Kossovskiy in 2006. In this paper, we prove the Conjecture for arbitrary length $l \geq 3$. As another application of our method, we construct polynomial models of length $l \geq 3$, which are not totally nondegenerate and admit large groups of point preserving nonlinear automorphisms.