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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

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  • $\begingroup$ 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? $\endgroup$
    – Robert Bryant
    Commented Apr 17, 2014 at 11:35
  • $\begingroup$ @Robert Thanks, yes I noticed that but didn't know what the resulting Kaehler potential is. $\endgroup$
    – user43696
    Commented Apr 17, 2014 at 14:56

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Weighted projective spaces are the simplest projective toric varieties and you can find the Kahler potential of singular toric varieties here

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See page 21 of "Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics" by Ross-Thomas for a definition, they also discuss some other definitions.

http://arxiv.org/abs/0907.5214

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