In the question here Simplifying the Kahler form, user290605 asked a question about how is that when we take the differential of Kahler form:$$\mathcal{K}=\frac{\sqrt{-1}}{2\pi}g_{i\bar{j}}dz^i\wedge d\bar{z}^{\bar{j}},$$ we get$$\partial_ig_{j\bar{k}}=\partial_jg_{i\bar{k}} \hspace{1cm} \text{and} \hspace{1cm} \partial_{\bar{i}}g_{j\bar{k}}=\partial_{\bar{k}}g_{j\bar{i}}.\hspace{1cm} (2)$$ Then, was my question, that how starting from (2)
(for example here p.45)] do we get $$g_{i\bar{k}}=∂_i∂_\bar{k}K(z,\bar{z})$$ where $K(z,\bar{z})$ is the Kahler potential?
As you can see, John, answered me there by saying that
That follows from the fact that $\mathcal{K}$ is closed and of type
$(1,1)$. This is called the $\partial\bar{\partial}$-lemma and the
proof can be found in (e.g.) p.14 of this note. Note that
$\bar{\partial}$-Poincare lemma is needed.
However, as a physicist I am not familiar with this lemma and would really appreciate if anyone can elaborate on this more.
Best Answer
The $\partial\overline{\partial}$-lemma says that a closed (1,1)-form $\omega$ locally arises as $\partial\overline{\partial}f$ for some smooth function $f$. This is in the same spirit as the regular Poincare lemma (which says closed differential forms are locally exact) but instead of exactness meaning "equal to d(something)", as in the Poincare lemma, it here means $\partial\overline{\partial}$(something). It is worth remembering that the Poincare lemma is true on all smooth manifolds, the $\overline{\partial}$-lemma (in which 'exactness' means $\overline{\partial}$(something)) is true on all complex manifolds, but we only have a general proof of the $\partial\overline{\partial}$-lemma on Kahler manifolds, which is one of the reasons they are distinctly important.