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}.

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