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This is $\bar{\partial}$-Poincaré lemma: Given a holomorphic funtion $f:U\subset \mathbb{C} \to \mathbb{C}$ ,locally on $U$ there is a holomorphic function $g$ such that : $\frac{\partial g}{\partial \bar z}=f$

The author says that this is a local statement so we may assume $f$ with compact support and defined on the whole plane $\mathbb{C}$, my question is why she says that... thanks.

*Added*

$f,g$ are suppose to be $C^k$ not holomorphic, by definition $\frac{\partial g}{\partial \bar z}=0$ if $g$ were holomorphic...

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    Is my response still not clear? You can respond under my post to let me know what is still making you uncomfortable.2012-06-09

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I don't have the book, and thus I can't check the statement. However, I believe that the statement holds for smooth $f$.

Basically we want to construct/find $g$ as the following integral:

$g(z) = \frac{1}{2 \pi i}\int_{w\in \mathbb{C}} \frac{f(w)}{z-w} d\overline{w}\wedge dw$

In order to do this, $f$ must be defined over the whole complex plane.

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The statement is defined on a local subset $U$... so we can make $f$ have compact support which vanishes outside $U$, thus trivially extending to the whole complex plane (defined to be zero outside the support).

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    So $f: U\to \mathbb{C}$ and the statement applies to a local subset $V\subset U$... i.e. we don't care about outside $V$ (nor $U$ in $\mathbb{C}$)... so taking a compact set $N\subset U$ containing $V$, we haven't lost any information, and $f$ vanishes outside $N$ (hence can be extended to $\mathbb{C}$ trivially). In other words, the original function $f$ and the "new" function with compact support do not look any different in the local region that you care about.2012-06-09