Kähler form on a complex projective space This is what I found in:
H.B. Lawson, Lectures on Minimal Submanifolds, Vol.1,
Publish or Perish, pp.34-36.
On a complex projective space Kähler form looks like this 
\begin{align}
\omega_{0}=\frac{\sqrt{-1}}{2}\sum_{i,j}g_{ij}dz^{i}\wedge d\overline{z}^{j}.
\end{align}
On the other hand 
\begin{align}
\omega_{0} = 4\partial\overline{\partial}\log |z|^{2}.
\end{align}
$z=(z^{0},..,z^{n})$.
When I calculate the second equation I can not get the same expression as in the first equation. The problem is $\sqrt{-1}$, I can not get it. What would be the problem here?
Thank you.

The reason why we have $\sqrt{-1}$ is  $K\ddot{a}hler$ form is a real valued (1,1) form. E.g. for complex 1-dim, locally, $\frac{\sqrt{-1}}{2}dz\wedge d\bar z=dx\wedge dy$. Also, $\sqrt{-1}\partial\bar\partial=kdd^c$, where k is a constant depending on how you define $d^c$. 
