Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

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

1 Answer

up vote 3 down vote accepted

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 $d^c$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.