I am struggling trying to understand an statement in a paper I am reading:

Let $M$ be a complex manifold of dimension $2n$. Let's consider a function $\xi$: $M$ $\rightarrow$ $\mathbb{C}$ whose real and imaginary parts are real analytic functions. Let $diag$($M$,$M$) be the diagonal of $M\times M$. Then there is a holomorphic extension $\Xi$ of $\xi$ where $\Xi$ is defined in a open neighbourhood of $diag$($M$,$M$).

I already checked some references about real analytic functions but I could not find anything useful about holomorphic extensions of real analytic function in several complex variables. Even in the case $M$ = $\Omega$ a domain of $\mathbb{C}^n$ I could not figure out how to extend $\xi$($z_1$,.....,$z_n$) to a holomorphic function with the double of complex variables $\Xi$($z_1$,..,$z_n$,$\lambda_1$,..,$\lambda_n$) when we are off-diagonal.



1 Answer 1


The statement you require (IIUC a simple consequence of analytic continuation) is e.g. in Bourbaki, Variétés différentielles et analytiques, Fascicule de résultats, 5.14.7, page 60. It requires the manifold to be paracompact.

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    $\begingroup$ Thanks for the reference! I found the result is proved in: Whitney, H.; Bruhat, F. Quelques propriétés fondamentales des ensembles analytiques-réels. (French) Comment. Math. Helv. 33 1959 132–160. In English the result (and a very nice sketch of the proof) can be found in Kai Cieliebak, Yakov Eliashberg, From Stein to Weinstein and Back Symp. Geometry.. page 102 books.google.ca/… $\endgroup$
    – Coffee
    Sep 23, 2013 at 18:19
  • $\begingroup$ @Josh: Nice! These sound like much better references than mine -- you should not hesitate to make them an answer and accept it. The Whitney-Bruhat paper is freely available here: dx.doi.org/10.5169/seals-26014 $\endgroup$ Sep 24, 2013 at 1:02

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