Let A ⊂ C N be an irreducible real algebraic set. Assume that there exists p 0 ∈ A such that A is a minimal, generic, holomorphically nondegenerate submanifold at p 0 . We show here that if H is a germ at p 1 ∈ A of a holomorphic mapping from C N into itself, with Jacobian H not identically 0, and H(A) contained in a real algebraic set of the same dimension as A, then H must extend to all of C N (minus a complex algebraic set) as an algebraic mapping. Conversely, we show that for any "model case" (i.e., A given by quasi-homogeneous real polynomials), the conditions on A are actually necessary for the conclusion to hold.
We consider local CR-immersions of a strictly pseudoconvex real hypersurface M ⊂ C n+1 , near a point p ∈ M , into the unit sphere S ⊂ C n+d+1 with d > 0. Our main result is that if there is such an immersion f : (M, p) → S and d < n/2, then f is rigid in the sense that any other immersion of (M, p) into S is of the form φ • f , where φ is a biholomorphic automorphism of the unit ball B ⊂ C n+d+1 . As an application of this result, we show that an isolated singularity of an irreducible analytic variety of codimension d in C n+d+1 is uniquely determined up to affine linear transformations by the local CR geometry at a point of its Milnor link.
We prove here new results about transversality and related geometric properties of a holomorphic, formal, or CR mapping, sending one generic submanifold of C N into another. One of our main results is that a finite mapping is transversal to the target manifold provided this manifold is of finite type. For the case of hypersurfaces, transversality in this context was proved by Baouendi and the second author in 1990. The general case of generic manifolds of higher codimension, which we treat in this paper, had remained an open problem since then. Applications of this result include a sufficient condition for a finite mapping to be a local diffeomorphism.
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