Projection on Segre varieties and determination of holomorphic mappings between real submanifolds

It is shown that a germ of a holomorphic mapping sending a real-analytic generic submanifold of finite type into another is determined by its projection on the Segre variety of the target manifold. A necessary and sufficient condition is given for a germ of a mapping into the Segre variety of the target manifold to be the projection of a holomorphic mapping sending the source manifold into the target. An application to the biholomorphic equivalence problem is also given.Comment: 16 page

Remarks on the rank properties of formal CR maps

We prove several new transversality results for formal CR maps between formal real hypersurfaces in complex space. Both cases of finite and infinite type hypersurfaces are tackled in this note

Nowhere minimal CR submanifolds and Levi-flat hypersurfaces

A local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition, being contained a possibly singular real-analytic Levi-flat hypersurface is studied and characterized. This question is completely resolved for algebraic submanifolds of codimension 2 and a sufficient condition for noncontainment is given for non algebraic submanifolds. As a consequence, an example of a submanifold of codimension 2, not biholomorphically equivalent to an algebraic one, is given. We also investigate the structure of singularities of Levi-flat hypersurfaces.Comment: 21 pages; conjecture 2.8 was removed in proof; to appear in J. Geom. Ana

Super-rigidity for CR embeddings of real hypersurfaces into hyperquadrics

Let Q^N_l\subset \bC\bP^{N+1} denote the standard real, nondegenerate hyperquadric of signature $l$ and M\subset \bC^{n+1} a real, Levi nondegenerate hypersurface of the same signature $l$. We shall assume that there is a holomorphic mapping H_0\colon U\to \bC\bP^{N_0+1}, where $U$ is some neighborhood of $M$ in \bC^{n+1}, such that $H_0(M)\subset Q^{N_0}_l$ but $H(U)\not\subset Q^{N_0}_l$. We show that if $N_0-n<l$ then, for any $N\geq N_0$, any holomorphic mapping H\colon U\to \bC\bP^{N+1} with $H(M)\subset Q^{N}_l$ and $H(U)\not\subset Q^{N_0}_l$ must be the standard linear embedding of $Q^{N_0}_l$ into $Q^N_l$ up to conjugation by automorphisms of $Q^{N_0}_l$ and $Q^N_l$