Moment maps

Let $G$ be a Lie group, a moment map of a symplectic $G$-action on a symplectic manifold $(M, \omega)$ is a smooth map $\mu: M \to \mathfrak{g}^*$ with the following properties:

1. $\langle d\mu(x)v, \xi \rangle = \omega_x(v, \xi_M(x))$, for all $x \in M$, $v \in T_xM$ and $\xi \in \mathfrak{g}$, where $\xi_M$ is the infinitesimal action. Sometimes this property is written as,

   $d\mu^\xi = -i_{\xi_M}\omega.$

2. $\mu$ is equivariant with respect to the coadjoint action on $\mathfrak{g}^*$:

   $\mu(gx) = g\cdot \mu(x) = Ad^*_{g^{-1}}\mu(x).$

A symplectic Lie group action on a symplectic manifold is called a Hamiltonian group action if a moment map µ exists. $(M,\omega, G, \mu)$ is called a Hamiltonian $G$-space.

Holomorphic and Hamiltonian group actions on the sphere

Let $\rho: U(1) \times \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$, $(g,z) \mapsto g.z$ be a group action of the abelian group of unit complex numbers, $U(1)$, on the Riemann sphere. If $\rho$ is holomorphic, for each $g \in U(1)$, the map $\hat{g} = \rho(g, \cdot)$ is a Möbius transformation, i.e., $\hat{g} \in \mathrm{PSL}(2,\mathbb{C})$. Whence, if $g \neq 1$, $\hat{g}$ has either one or two fixed points.