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

DOI: https://dx.doi.org/10.4310/MRL.2022.v29.n2.a4

Authors

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)

Abstract

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.

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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