 Moment map for rotation of the complex projective line

For the symplectic form $\omega =idz\wedge d\bar{z}/(1+|z{|}^2{)}^2$, the circle group rotates $z$ by $e^{i\theta }$. Its real generator is $K=iz{\partial }_z-i\bar{z}{\partial }_{\bar{z}}$. The contraction is ${\iota }_K\omega =-dH$, so $H$ is a Hamiltonian function in this sign convention. In the other chart $w=1/z$, $H=1/(1+|w{|}^2)$, proving smoothness at infinity. Its image is $[0,1]$ with the endpoints at the fixed poles. The symplectic area is $2\pi$; rescaling to the round area $4\pi$ would double this moment function.