On the Beloshapka's rigidity conjecture for real submanifolds in complex space

TL;DR

This paper proves Beloshapka's rigidity conjecture for all polynomial models with length greater than or equal to three, confirming their automorphism groups are determined by their differential at a fixed point.

Contribution

It extends the proof of Beloshapka's conjecture to all lengths l ≥ 3 and constructs polynomial models with large automorphism groups that are not totally nondegenerate.

Findings

Proved Beloshapka's conjecture for all lengths l ≥ 3.

Constructed polynomial models with large automorphism groups.

Identified models that are not totally nondegenerate with significant automorphisms.

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 {\em 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.