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