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New examples of non-locally embeddable $C R$ structures (with no non-constant $C R$ distributions)

Jean-Pierre Rosay (1989)

Annales de l'institut Fourier

We construct examples of non-locally embeddable $C R$ structures. These examples may show some improvement on previous examples by Nirenberg, and Jacobowitz and Trèves. They are based on a simple construction which consists in gluing two embedded structures. And (this is our main point) we believe that these examples are very transparent, therefore easy to work with.

Non embeddable CR-structures

François Trèves (1981)

Journées équations aux dérivées partielles

Nondegeneracy conditions and normal forms for real hypersurfaces in complex space

Peter Ebenfelt (1997)

Journées équations aux dérivées partielles

Non-Realizable CR Structures.

Howard Jacobowitz, Francois Treves (1982)

Inventiones mathematicae

Non-solvability of the tangential ${\bar{\partial}}_{M}$ -systems

Giuseppe Zampieri (1998)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We prove that for a real analytic generic submanifold $M$ of $C^{n}$ whose Levi-form has constant rank, the tangential ${\bar{\partial}}_{M}$ -system is non-solvable in degrees equal to the numbers of positive and $M$ negative Levi-eigenvalues. This was already proved in [1] in case the Levi-form is non-degenerate (with $M$ non-necessarily real analytic). We refer to our forthcoming paper [7] for more extensive proofs.

Note on double reflection and algebraicity of holomorphic mappings

Joël Merker (2000)

Annales de la Faculté des sciences de Toulouse : Mathématiques