Notes. Some basic terms

1. Let $R$ be a commutative ring, $S$ be a multiplicatively closed subset in the sense that $a,b\in S \Rightarrow ab \in S$. Then we consider the localization $S^{-1}R$ as the quotient $S\times R/\sim$, where $(r,a)\sim (s,b)$ if $(br-as)t=0$ for some $t\in S$.

Let $f\in R$. We can construct a m.c.subset $S=\{f^n: n\ge 0\}$, and denote the corresponding local ring by $R_f=S^{-1}R$.

Let $p\triangleleft R$ be a prime ideal of $R$. Then $S=R\backslash p$ is m.c. We denote the corresponding local ring by $R_p=S^{-1}R$.

Let $\text{Spec}R$ be the set of all prime ideals of $R$. For each ideal $I\triangleleft R$, let $V_I=\{p\in \text{Spec}R: p\supset I\}$. The Zariski topology on $\text{Spec}R$ is defined that the closed subsets are exactly $\{V_I: I\triangleleft R\}$.

A basis for the Zariski topology on $\text{Spec}R$ can be constructed as follows. For each $f\in R$, let $D_f\subset \text{Spec}R$ to be the set of prime ideals not containing $f$. Then each $D_f= \text{Spec}R\backslash V_{(f)}$ is open.

The points corresponding to maximal ideals $m \triangleleft R$ are closed points in the sense that the singleton $\{m\}=V_m$.

In the case $R=C[x_1, \dots, x_n]$, we see that each maximal ideal $m=\langle x_1-a_1,\dots, x_n-a_n \rangle$ corresponds to a point $(a_1,\dots, a_n)\in C^n$. So one can interprat this as $C^n \subset X= \text{Spec} R$. A non-max prime ideal $p$ (a non-closed point) corresponds an affine variety $P$, which is a closed subset in $C^n$. Then $p$ is called the generic point of the varity $P$.

2. Let $(M,\omega)$ be a symplectic manifold, $G$ be a Lie group acting on $M$ via symplectic diffeomorphisms. Let $\mathfrak{g}$ be the Lie algebra of $G$. Each $\xi \in \mathfrak{g}$ induces a vector field $\rho(\xi):x\in M \mapsto \frac{d}{dt}\Big|_{t=0}\Big(\exp(t\xi)\cdot x\Big)$. Note that $\rho(g^{-1}\xi g)=g_\ast \rho(\xi)$, and $\rho([\xi,\eta])=[\rho(\xi),\rho(\eta)]$.

Consider the 1-form induced by the contraction $\iota_{\rho(\xi)}\omega$. Clearly this 1-form is closed: $d\iota_{\rho(\xi)}\omega=L_{\rho(\xi)}\omega=0$ since $G$ preserves the form $\omega$.

Then the action is called weakly Hamiltonian, if for every $\xi\in \mathfrak{g}$, the one-form $\iota_{\rho(\xi)} \omega$ is exact: $\iota_{\rho(\xi)} \omega=dH_\xi$ for some smooth function $H_{\xi}$ on $M$. Although $H_\xi$ is only determined up to a constant $C_\xi$, the constant $\xi \mapsto C_\xi$ can be chosen such that the map $\xi\mapsto H_\xi$ becomes linear.

The action is called Hamiltonian, if the map $\mathfrak{g} \to C^\infty(M)$, $\xi\mapsto H_\xi$ is a Lie algebra homomorphism with respect to Poisson structure. Then $\rho(\xi)=X_{H_\xi}$ and $H_{g^{-1}\xi g}(x)=H_\xi(gx)$.

A moment map for a Hamiltonian $G$-action on $(M,\omega)$ is a map $\mu: M\to \mathfrak{g}^\ast$ such that $H_\xi(x)=\mu(x)\cdot \xi$ for all $\xi\in \mathfrak{g}$. In other words, for each fixed point $x\in M$, the map $\xi \mapsto H_\xi(x)$ from $\mathfrak{g}$ to $\mathbb{R}$ is a linear functional on $\mathfrak{g}$ and is denoted by $\mu(x)$. Also note that $\mu(gx)\cdot \xi=H_\xi(gx)=H_{g^{-1}\xi g}(x)$. So $\mu(gx)=g\mu(x)g^{-1}$.