Algebraicity of holomorphic mappings between real algebraic sets in Cn

Abstract

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.

Holomorphic correspondences between CR manifolds

“…Our main construction is based on the technique of these papers. A somewhat different approach was developed in [BJT], [BR1] and [BER1]. We rely on the characterization of minimality in terms of Segre sets proved in [BER1].…”

“…This follows from the minimality of M and the property of Segre sets proved in [BER1]. Furthermore, for compact minimal manifolds such neighborhoods can be chosen of uniform size for all points on M , which allows us to extend f to any simply-connected relatively compact subset of M .…”

“…According to [BER1] (see also [BER2] and [Me3] for a short proof of this fact), there exists an integer j 0 , 0 < j 0 < ∞, such that ∪ j≤j 0 Q j 0 contains a neighborhood of the origin in C n , provided that M is minimal at 0. We define…”

“…By the construction, from minimality of M and from [BER1], for some j 0 > 1, the set A j 0 defines a holomorphic correspondence F j 0 in a neighborhood Ω 0 ⊂ U j 0 of the origin. Note that the size of this neighborhood depends only on the geometry of M and is independent of U 0 , where f was originally defined.…”

“…Under certain conditions, it then turns out that a locally defined holomorphic map between such objects must necessarily be algebraic. See Baouendi, Ebenfelt and Rothschild [BER1], Huang [Hu], Sharipov and Sukhov [SS], Baouendi, Huang and Rothschild [BHR], Coupet, Meylan and Sukhov [CMS], Zaitsev [Za], Merker [Me1] and many additional references contained therein. The special case of hyperquadrics was also considered in Tumanov [Tu], Forstnerič [Fo], Sukhov [Su] and other papers.…”