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New invariant tensors in CR structures and a normal form for real hypersurfaces at a generic Levi degeneracy. (English) Zbl 0945.32020
The author investigates the $$CR$$ geometry of a real hypersurface $$M$$ of a complex manifold at a point $$p\in M$$ where the determinant of its Levi form has a first order zero. He finds normal forms, uniquely defined modulo a second degree polynomial change of holomorphic coordinates, providing a complete set of holomorphic invariants in the real-analytic case. The first terms in the formal power series expansion of the normal form are related to two tensors, the first being the Levi form, and the second being related to the cubic form of S. Webster [Ann. Math. Stud. 137, 327-342 (1995; Zbl 0870.32008)]. The author shows that in general, for a generic $$CR$$ submanifold of a complex manifold, at a point where it is of finite $$k$$-nondegeneracy in the sense of M. S. Baouendi, P. Ebenfelt and L. P. Rothschild [Acta Math. 177, No. 2, 225-273 (1996; Zbl 0890.32005)] there are $$k$$ such tensor invariants.
In the case the Levi form of the hypersurface is semidefinite, he finds for the normal form an easier and more explicit expression involving numerical invariants. This applies in particular to hypersurfaces in $$\mathbb{C}^3$$ that the author considered in [Indiana Univ. Math. J. 42, No. 2, 311-366 (1998)].

##### MSC:
 32V05 CR structures, CR operators, and generalizations 32V40 Real submanifolds in complex manifolds
##### Keywords:
invariant tensors; CR geometry; real hypersurface; normal forms
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