Chern–Moser operators and polynomial models in CR geometry

Abstract

We consider the fundamental invariant of a real hypersurface in CN – its holomorphic symmetry group – and analyze its structure at a point of degenerate Levi form. Generalizing the Chern–Moser operator to hypersurfaces of finite multitype, we compute the Lie algebra of infinitesimal symmetries of the model and provide explicit description for each graded component. Compared with a hyperquadric, it may contain additional components consisting of nonlinear vector fields defined in terms of complex tangential variables.As a consequence, we obtain exact results on jet determination for hypersurfaces with such models. The results generalize directly the fundamental result of Chern and Moser from quadratic models to polynomials of higher degree.