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I am trying to prove the local $\partial \bar{\partial}$ lemma. This says that for a polydisc in $\mathbb{C}^{n}$, a form in $A^{p,q}(U)$ being $d$-closed implies that it is $\partial \bar{\partial}$-exact. I've tried to use the $\partial$ and $\bar{\partial}$ Poincare lemma's to no avail.. I'd like a hint as to where I should get started, I'd greatly appreciate it.

EDIT:

Due to my lack of conceptual tools and methods of approach.. I guess I would like to try going down this avenue:

Given that my form (say, $\alpha$) is $d$-closed, it is $\partial$-closed. By Poincare's lemma, I can find a form $\gamma$ in $A^{p-1,q}(U)$ such that $\partial \gamma = \alpha$. As a stab in the dark, I would like to attempt to decompose $\gamma$ as $P + \bar{\partial}Q$ for a $\partial$-closed form $P$. If I can accomplish this, then $\partial \gamma = \partial (P + \bar{\partial}Q) = \partial \bar{\partial}Q$ and I'll be done.

Do you think that this is a long shot? I'm just trying to get my hands dirty and actually expanding and applying $\partial$ to the terms to see if I can fuddle around enough to arise at such a decomposition..

This arises as an exercise in Huybrecht's book "Complex geometry". It is exercise 1.3.4/1.3.3

Thank you!

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1 Answer 1

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First, you need to assume that $p$ and $q$ are both positive; otherwise there are no nontrivial $(p-1,q-1)$-forms. You'll need to use both the ordinary Poincaré lemma and the $\partial$- and $\overline\partial$-Poincaré lemmas.

First, if $\alpha$ is a $d$-closed $(p,q)$ form on $U$, then the ordinary Poincaré lemma implies that $\alpha=d\eta$ for some complex $p+q-1$-form $\eta$. Since the only parts of $\eta$ that can contribute to the $(p,q)$-part of $d\eta$ are the $(p,q-1)$ and $(p-1,q)$ parts, we may as well assume that $\eta$ decomposes as $\eta =\eta^{(p,q-1)}+\eta^{(p-1,q)}$. Using the fact that $d=\partial+\overline\partial$, we can decompose the equation $d\eta=\alpha$ as follows: \begin{align*} \partial\eta ^{(p,q-1)}&= 0 &&\text{($(p+1,q-1)$-part)}\\ \overline\partial\eta^{(p,q-1)} + \partial\eta^{(p-1,q)} &= \alpha&&\text{($(p,q)$-part})\\ \overline\partial\eta ^{(p-1,q)}&= 0 &&\text{($(p-1,q+1)$-part)}. \end{align*} Now apply the $\partial$- and $\overline\partial$-Poincaré lemmas to conclude that there exist $(p-1,q-1)$-forms $\beta$ and $\gamma$ such that $\partial \beta=\eta^{(p,q-1)}$ and $\overline\partial \gamma=\eta^{(p-1,q)}$. You can probably take it from there.

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  • $\begingroup$ Perfect! I forgot about the ordinary Poincare lemma.. it turned out to be quite easy, when you put it that way. Thanks! $\endgroup$
    – Elliot
    Sep 14, 2015 at 6:46

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