I am little confused about some basic symplectic geometry about Hamiltonian actions on sphere. I appreciate your comments.
Consider sphere $S^2 = \mathbb{C}P^1$ with its standard symplectic (Kaehler) form $\omega$ and usual action of $S^1$ on $\mathbb{C}P^1$ and its usual moment map $\phi: \mathbb{C}P^1 \to \mathbb{R}$ with image $[-1, 1]$.
Also consider the "n-tuple embedding" $\mathbb{C}P^1 \to \mathbb{C}P^n$ by $$(z:w) \mapsto (z^n : z^{n-1}w : \cdots : w^n),$$
and consider the Kaehler form $\omega_n$ on $\mathbb{C}P^1$ which is the pull-back of the Kaehler form on $\mathbb{C}P^n$ (Fubini-Study form). Let $\phi_n$ be the moment map for the Hamiltonian action of $S^1$ with respect to $\omega_n$ (that is the pull-back of $S^1$-moment map on $\mathbb{C}P^n$ to $\mathbb{C}P^1$). Here $t \in S^1$ acts on $\mathbb{C}P^1$ and $\mathbb{C}P^n$ respectively by: $$ t \cdot (z : w) = (z : tw),$$ $$t \cdot (z_0 : \cdots : z_n) = (z_0: tz_1: \cdots t^nz_n),$$
I am pretty sure that $\omega_n$ is a constant multiple of $\omega$ and in fact: $$\omega_n = n \omega. \quad (*)$$
From (*) it should follow that $n\phi(x)$ is a moment map for the $S^1$ action with respect to the symplectic form $\omega_n$. On the other hand moment map should be unique up to a constant and hence $\phi_n = n \phi + const.$ but this seem not to be the case (direct calculation).
What am I missing?
Here are more detail of computing the moment maps:
The moment map of the usual action of $(S^1)^{n+1}$ on $\mathbb{C}P^n$ is given by: $$\Phi(z_0: \cdots :z_n) = (\frac{|z_0|^2}{\sum_{i=0}^n |z_i|^2}, \ldots,\frac{|z_n|^2}{\sum_{i=0}^n |z_i|^2}).$$ Now as in above, under the $n$-uple embedding circle $S^1$ embeds in $(S^1)^{n+1}$ by: $$t \mapsto (1, t, \ldots, t^n).$$ The corresponding map between dual Lie algebras $\mathbb{R}^{n+1}=Lie((S^1)^{n+1})^* \to \mathbb{R}=Lie(S^1)^*$ is: $$(t_0, \ldots, t_n) \mapsto t_1 + 2t_2 + \cdots + nt_n.$$Thus the moment map of $S^1$ acting on $\mathbb{C}P^n$ is: $$\Phi(z_0 : \cdots : z_n) = \sum_{i=0}^n \frac{i|z_i|^2}{\sum_{i=0}^n |z_i|^2}.$$ Restricting to the image of $\mathbb{C}P^1$ in $\mathbb{C}P^n$ we get: $$\phi_n(1:w) =\sum_{i=0}^n \frac{i|w|^{2i}}{\sum_{i=0}^n |w|^{2i}}.$$ For n=1 we have the usual moment map of $\mathbb{C}P^1$ which is: $$\phi(1:w) = \frac{|w|^2}{1 + |w|^2}.$$ $\phi_n$ does not seem to be equal to $n\phi$ ...