# Fubini-Study form on weighted projective spaces

As it is known, $\mathbb CP^n$ with Fubini-Study symplectic form can be get by the symplectic reduction of $\mathbb C^{n+1}$ with a symplectic form $\sum_{i=0}^n dz_i\wedge d\bar z_i$ by a hamiltonian action of $S^1$:

$$t\cdot (z_0,...z_n) = (e^{\sqrt{-1}t}z_0,...e^{\sqrt{-1}t}z_n).$$ The resulting Fubini-Study form is equal (up to a coefficient) to $$\omega = \partial\bar\partial \log\sum_{i=0}^{n}z_i\bar z_i$$

One can try to generalize this form to a weighted projective space $\mathbb P(\lambda_0,...\lambda_n)$ in such a way:

$$\omega = \partial\bar\partial \log\sum_{i=0}^{n}z_i^{\mu_i}\bar z_i^{\mu_i},$$ where $\mu_i$ are integers such that $\lambda_i \mu_i = const$.

The problem is that this form is not the form you get by symplectic reduction of $\mathbb C^{n+1}$ by the $S^1$ action

$$t\cdot (z_0,...z_n) = (e^{\sqrt{-1}\lambda_0t}z_0,...e^{\sqrt{-1}\lambda_nt}z_n),$$

although the weighted projective space is exactly the result of this construction (of course, if you choose a non-zero element of the dual $S^1$ Lie algebra).

My question is: what is the right generalization of Fubini-Study form to weighted projective space? By the right generalization I mean the form resulting from symplectic reduction.

• This is considered in Section 3, page 21, of "Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics" by Ross-Thomas. In particular they discuss the different generalisations of the FS metric to orbifolds. – Ruadhaí Dervan Apr 10 '16 at 17:56
• @AnnaAbasheva how does one show that the reduced symplectic form on $\mathbb C P^n$ is equal, up to a coefficient, to $\omega = \partial \bar \partial \log \sum_{i=0}^n z_i\bar z_i$? – Aaron Maroja Jul 8 '19 at 12:49