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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.

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  • $\begingroup$ There should be an $i$ in the defintion of the Kähler potential. See for example en.wikipedia.org/wiki/K%C3%A4hler_manifold $\endgroup$ – mrf Sep 19 '13 at 7:04
  • $\begingroup$ Or perhaps Lawson uses $dd^c$ instead of $\partial \bar\partial$. $\endgroup$ – mrf Sep 19 '13 at 7:07
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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 $d^c$.

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