The Kaehler potential for the standard Fubini-Study Kaehler form in projective space $\mathbb{C} P^n$ is given by:


$$\log(\sum\_{i=0}^n |z\_i|^2)).$$


What is the analogous formula for a Kaehler potential in a weighted projective space $\mathbb{C} P(m\_0, \ldots, m\_n)$ with weights $m\_0, \ldots, m\_n \in \mathbb{Z}\_{\geq 0}$? That is, $\mathbb{C} P(m\_0, \ldots, m\_n) = (\mathbb{C}^{n+1} \setminus \{0\})/\sim$ where $(z\_0, \ldots, z\_n) \sim (\lambda^{m\_0}z\_0, \ldots, \lambda^{m\_n}z\_n)$ 
for any $0 \neq \lambda \in \mathbb{C}$.


Of course the weighted projective space is usually singular, and we are only asking for Kaehler potential in its smooth locus.


Thanks



Kaehler form on weighted projective space
Weighted projective spaces are the simplest projective toric varieties and you can find the Kahler potential of singular toric varieties [here](http://msp.org/pjm/2008/238-1/pjm-v238-n1-p03-p.pdf) 


