A complete normal form for everywhere Levi–degenerate hypersurfaces in C³

Abstract

2-nondegenerate real hypersurfaces in complex manifolds play an important role in CR-geometry and the theory of Hermitian Symmetric Domains. In this paper, we obtain a complete convergent normal form for everywhere 2-nondegenerate real-analytic hypersurfaces in complex 3-space. We do so by entirely reproducing the Chern-Moser theory in the 2-nondegenerate setting. This seems to be the first such construction for hypersurfaces of infinite Catlin multitype. We in particular discover chains in an everywhere 2-nondegenerate hypersurface, the tangent lines to which at a point form the so-called canonical cone. Our approach is based on using a rational (nonpolynomial) model for everywhere 2-nondegenerate hypersurfaces, which is the local realization due to Fels-Kaup of the well known tube over the light cone. For the convergence of the normal form, we use an argument due to Zaitsev, based on building a canonical direction field in an appropriate bundle over a hypersurface. As an application, we obtain, in the spirit of Chern-Moser theory, a criterion for the local sphericity (i.e. local equivalence to the model) for a 2-nondegenerate hypersurface in terms of its normal form. As another application, we obtain an explicit description of the moduli space of everywhere 2-nondegenerate hypersurfaces.