Algebraicity of holomorphic mappings between real algebraic sets in ${\bold C}^n$

Content Provider	Semantic Scholar
Author	Baouendi, Mohamed Salah Ebenfelt, Peter Rothschild, Linda Preiss
Copyright Year	1995
Abstract	We give conditions under which a germ of a holomorphic mapping in $\Bbb C^N$, mapping an irreducible real algebraic set into another of the same dimension, is actually algebraic. Let $A\subset \bC^N$ be an irreducible real algebraic set. Assume that there exists $\po \in A$ such that $A$ is a minimal, generic, holomorphically nondegenerate submanifold at $\po$. We show here that if $H$ is a germ at $p_1 \in A$ of a holomorphic mapping from $\bC^N$ 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 $\bC^N$ (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.
File Format	PDF HTM / HTML
Alternate Webpage(s)	https://arxiv.org/pdf/math/9510201v1.pdf
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article