# Kaehler form on weighted projective space

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

• I think you can get this just by following the Marsden-Weinstein reduction procedure using the obvious circle generated by the quotient action you have, but setting $|\lambda|=1$. Have you tried this? – Robert Bryant Apr 17 '14 at 11:35
• @Robert Thanks, yes I noticed that but didn't know what the resulting Kaehler potential is. – user43696 Apr 17 '14 at 14:56