Construction of counterexamples to the $2-$jet determination Chern-Moser Theorem in higher codimension

TL;DR

This paper constructs counterexamples to the $2$-jet determination in higher codimension CR geometry, resolving a long-standing open question and providing conditions for generating more such examples with varying jet determination orders.

Contribution

It presents the first known counterexample to the $2$-jet determination in higher codimension and offers criteria to produce additional counterexamples with different jet determination properties.

Findings

Constructed a counterexample in $C^9$ with a real analytic infinitesimal CR automorphism.

Resolved a 50-year-old open question in Tanaka prolongation theory.

Provided conditions to generate counterexamples with arbitrarily high jet determination.

Abstract

We first construct a counterexample of a generic quadratic submanifold of codimension $5$ in $\Bbb C^9$ which admits a real analytic infinitesimal CR automorphism with homogeneous polynomial coefficients of degree $4.$ This example also resolves a question in the Tanaka prolongation theory that was open for more than 50 years. Then we give sufficient conditions to generate more counterexamples to the $2-$jet determination Chern-Moser Theorem in higher codimension. In particular, we construct examples of generic quadratic submanifolds with jet determination of arbitrarily high order.