3
$\begingroup$

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.

Thanks

$\endgroup$
4
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ 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 '13 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 $\endgroup$ – Francois Ziegler Sep 24 '13 at 1:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.