Infinitesimal symmetries of weakly pseudoconvex manifolds

Abstract

© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.We consider weakly pseudoconvex hypersurfaces with polynomial models in CN and their symmetry algebras. In the most prominent case of special models, given by sums of squares of polynomials, we give their complete classification. In particular, we prove that such manifolds do not admit any nonlinear symmetries, depending only on complex tangential variables, nor do they admit real or nilpotent linear symmetries. This leads to a sharp 2-jet determination result for local automorphisms. We also give partial results in the general case and a more detailed description of the graded components in complex dimension three. The results also provide an important necessary step for solving the local equivalence problem on such manifolds.