Lecture 4: Symplectic Group Actions

We set S1 = R/2πZ throughout. Let (M,ω) be a symplectic manifold. A symplectic S1-action on (M,ω) is a smooth family ψt ∈ Symp(M,ω), t ∈ S1, such that ψt+s = ψt ◦ ψs for any t, s ∈ S1. One can easily check that the corresponding vector fields Xt ≡ d dtψt ◦ψ −1 t is time-independent, i.e., Xt = X is constant in t. We call X the associated vector field of the given symplectic S1-action. Note that X is a symplectic vector field, i.e., ι(X)ω is a closed 1-form. When ι(X)ω = dH is an exact 1-form, the corresponding symplectic S1-action is called a Hamiltonian S1-action, and the function H : M → R is called a moment map. Note that H is uniquely determined up to a constant. We point out that a symplectic S1-action on (M,ω) is automatically Hamiltonian if H1(M ;R) = 0. In general, ι(X)ω is only a closed form. In this case, one can perturb ω by adding a sufficiently small S1-invariant harmonic 2-form β so that ω + β is still symplectic and S1-invariant, and furthermore, the deRham cohomology class [ω+β] lies in H2(M ;Q). A positive integral multiple of ω+β, ω′ ≡ N(ω+β), is a S1-invariant symplectic form on M such that the deRham cohomology class of ι(X)ω′ lies in H1(M ;Z). Consequently, there exists a smooth function H : M → R/Z such that ι(X)ω′ = dH. Such a circlevalued smooth function H : M → R/Z is called a generalized moment map. As far as the topology of the S1-action is concerned, one may always assume that there is a generalized moment map. LetH be a moment map or generalized moment map of a given symplectic S1-action. We shall make the following two observations.