Arc-analytic subanalytic functions on complex manifolds

TL;DR

The paper proves that arc-analytic subanalytic functions on complex manifolds that are holomorphic near a point are globally holomorphic, and similar results hold for CR-functions on CR-manifolds.

Contribution

It establishes that local holomorphicity or CR conditions imply global holomorphicity or CR-ness for arc-analytic subanalytic functions.

Findings

Arc-analytic subanalytic functions holomorphic near a point are globally holomorphic.

Arc-analytic subanalytic CR-functions on a subset extend to the whole CR-manifold.

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.