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Arc-analytic subanalytic functions on complex manifolds

Janusz Adamus, Rasul Shafikov

Abstract

We show that an arc-analytic subanalytic function on a complex manifold M, which is holomorphic near one point, is a holomorphic function on M. More generally, an arc-analytic subanalytic function on a real analytic CR-manifold M, which is CR on a nonempty open subset of M, is a CR-function on the whole M.

Arc-analytic subanalytic functions on complex manifolds

Abstract

We show that an arc-analytic subanalytic function on a complex manifold M, which is holomorphic near one point, is a holomorphic function on M. More generally, an arc-analytic subanalytic function on a real analytic CR-manifold M, which is CR on a nonempty open subset of M, is a CR-function on the whole M.

Paper Structure

This paper contains 5 sections, 8 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Proof of Theorem \ref{['thm:glob-holo-map']}
  4. Proof of Theorem \ref{['thm:glob-CR-map']}
  5. Examples of arc-analytic subanalytic functions

Key Result

Theorem 1.1

Let $M$ be a connected complex manifold, let $N$ be a complex manifold, and let $f:M\to N$ be an arc-analytic subanalytic mapping. Suppose that $f$ has a holomorphic germ $f_p$, for some $p\in M$. Then, $f$ is a holomorphic map on $M$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Lemma 2.5: Sh
  • Theorem 2.6: cf. AS
  • ...and 8 more