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

Abstract

We first construct a counterexample of a generic quadratic submanifold of codimension 5 in C9 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.