New Normal Forms for Levi-nondegenerate Hypersurfaces

In this paper we construct a large class of new normal forms for Levi-nondegenerate real hypersurfaces in complex spaces. We adopt a general approach illustrating why these normal forms are natural and which role is played by the celebrated Chern-Moser normal form [CM74]. The latter appears in our class as the one with the ”maximum normalization” in the lowest degree. However, there are other natural normal forms, even with normalization conditions for the terms of the same degree. Some of these forms do not involve the cube of the trace operator and, in that sense, are simplier than the one by Chern-Moser. We have attempted to give a complete and self-contained exposition (including proofs of well-known results about trace decompositions) that should be accessible to graduate students. All normal forms here are formal, i.e. at the level of formal power series. This is sufficient for most purposes such as constructing invariants or solving the local equivalence problem for real-analytic hypersurfaces. In fact, a formal equivalence map between Levi-nondegenerate realanalytic hypersurfaces is automatically convergent and is therefore a local biholomorphic map. This is a special case of an important result of Baouendi-Ebenfelt-Rothschild [BER00a] and can also be obtained from the Chern-Moser theory [CM74]. The reader is referred for more details to an excellent survey [BER00b]. We also refer to normal forms for Levi-degenerate hypersurfaces [E98a, E98b, Ko05], for real CR submanifolds of higher codimension [ES95, SS03], for submanifolds with CR singularities [MW83, HY08], and for non-integrable Levi-nondegenerate hypersurface type CR structures [Z08]. Throughout the paper we consider a real-analytic hypersurface M in C passing through 0 and locally given by an equation