Mathematical Research Letters

Volume 29 (2022)

Number 2

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

Pages: 399 – 420



Jan Gregorovič (Faculty of Science, University of Hradec Králové, Czech Republic; and Faculty of Mathematics, University of Vienna, Austria)

Francine Meylan (Department of Mathematics, University of Fribourg, Switzerland)


We first construct a counterexample of a generic quadratic submanifold of codimension $5$ in $\mathbb{C}^9$ 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.

J. G. gratefully acknowledges support via Czech Science Foundation (project no. 19-14466Y) and partial support via Austrian Science Fund (FWF): 13472.

Received 20 October 2020

Accepted 4 April 2021

Published 29 September 2022