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Using the homogeneous coordinate on $CP^n$, we consider the open set $U_0:=\{[1, \ldots, z_n]\}$. Then the standard Kähler form of $CP^n$ can be written as

$$ \omega_0=\frac{\sqrt{-1}}{2}\partial\bar{\partial}\log(1+|z_1|^2+\cdots+|z_n|^2) $$

The compatability of this form can be easily checked for other chart $U_i$.

My question is, if I want to deform this Kähler form, an easy way to do this is introducing a function say $\rho: \mathbb R\to \mathbb R$ and write the new Kähler form on $U_0$ as

$$ \omega_\rho=\frac{\sqrt{-1}}{2}\partial\bar{\partial}\log(1+\rho(|z_1|^2+\cdots+|z_n|^2)) $$

Then what are the restrictions on $\rho$ and how to write the form $\omega_\rho$ in other coordinate charts, say $U_1$?

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    Either Kähler or Kaehler is a correct spelling; Kahler is not. If it were, it would be pronounced differently. I changed it to Kähler in the posting.2011-11-07

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