The equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics

Abstract

Our main objective in this paper is to study the class of real hypersurfaces M ? Cn+1 which admit holomorphic (or formal) embeddings into the unit sphere (or, more generally, Levi-nondegenerate hyperquadrics) in CN+1 where the codimension k := N ? n is small compared to n. Such hypersurfaces play an important role e.g. in deformation theory of singularities where they arise as links of singularities (see e.g. [BM97]). Another source is complex representations of compact groups, where the orbits are always embeddable into spheres due to the existence of invariant scalar products. One of our main results is a complete normal form for hypersurfaces in this class with a rather explicit solution to the equivalence problem in the following form (Theorem 1.3): Two hypersurfaces in normal form are locally biholomorphically equivalent if and only if they coincide up to an automorphism of the associated hyperquadric. Our normal form here is different from the classical one by Chern?Moser [CM74] (which, on the other hand, is valid for the whole class of Levi nondegenerate hypersurfaces), where, in order to verify equivalence of two hypersurfaces, one needs to apply a general automorphism of the associated hyperquadric to one of the hypersurfaces, possibly loosing its normal form, and then perform an algebraically complicated procedure of putting the transformed hypersurface back in normal form. Another advantage of our normal form, comparing with the classical one, is that it can be directly produced from an embedding into a hyperquadric and hence does not need any normalization procedure.