Algebraic approximation in CR geometry

TL;DR

This paper establishes a CR analogue of Artin's approximation theorem, showing that holomorphic maps with algebraic constraints on real-algebraic CR manifolds can be approximated by algebraic maps up to any order.

Contribution

It extends Artin's approximation theorem to the setting of CR geometry, providing algebraic approximation results for holomorphic mappings between real-algebraic CR manifolds.

Findings

Existence of algebraic approximations of holomorphic maps up to any order

CR version of Artin's approximation theorem proved

Applicable to real-algebraic CR submanifolds with uniform CR orbit dimension

Abstract

We prove the following CR version of Artin's approximation theorem for holomorphic mappings between real-algebraic sets in complex space. Let $M\subset \C^N$ be a real-algebraic CR submanifold whose CR orbits are all of the same dimension. Then for every point $p\in M$, for every real-algebraic subset $S'\subset \C^N\times\C^{N'}$ and every positive integer $\ell$, if $f\colon (\C^N,p)\to \C^{N'}$ is a germ of a holomorphic map such that ${\rm Graph}\, f \cap (M\times \C^{N'})\subset S'$, then there exists a germ of a complex-algebraic map $f^\ell \colon (\C^N,p)\to \C^{N'}$ such that ${\rm Graph}\, f^\ell \cap (M\times \C^{N'})\subset S'$ and that agrees with $f$ at $p$ up to order $\ell$.