Polynomial models with few infinitesimal symmetries

One possible approach to study local CR equivalence of real hypersurfaces in C^{n+1} is the “Taylor expansion” approach: it involves the assignment of different weights to different directions. When the hypersurface is of finite type m (in the sense of Kohn and Bloom-Graham) at a given point, the complex normal coordinate is assigned the weight 1 while the n complex tangential coordinates are assigned the weight 1/m . They allow to define a homogeneous model (hypersurface) for which the Lie algebra of infinitesimal symmetries admits a natural grading. Since the kernel of the generalized Chern–Moser operator corresponds to the above Lie algebra, this gives a tool for addressing the equivalence problem. We point out that full classification of such Lie algebras seems still out of reach, due in particular to the presence of a component containing nonlinear rigid vector fields, with arbitrarily high degree coefficients. In this talk, we will focus on a class of Levi degenerate hypersurfaces for which the structure of the Lie algebra of infinitesimal CR automorphisms of their models is very simple, leading to second jet determination for their automorphisms.