Some remarks on symplectic actions of compact groups (Q1202212)

The paper is devoted to studying symplectic, but not necessary Hamiltonian, actions of compact groups. Definition: a symplectic action of a torus \(T\) on a symplectic manifold \((W,\omega)\) is called cohomologically free if for any non-zero element \(\xi\) in the Lie algebra of \(T\) the closed 1-form \(i_ \xi\omega\) is not exact. It is proved that for a symplectic action of a compact connected group \(G\) there exists a finite covering \(\tilde G\) of \(G\) which can be split into the direct product \(G_ 0\times T=\tilde G\) such that the \(G_ 0\)-action is Hamiltonian and the \(T\)-action is cohomologically free. When \(G\) is a torus, one can always take \(\tilde G=G\). The equivariant cohomology \(H^*_ G(W,\mathbb{R})\) is explicitly calculated: \(H^*_ G(W,\mathbb{R})=H^*_ T(W,\mathbb{R})\otimes H^*(BG_ 0,\mathbb{R})\). The author also gives another proof of the result of \textit{D. McDuff} [J. Geom. Phys. 5, No. 2, 149-160 (1988; Zbl 0696.53023)] that a symplectic non-Hamiltonian circle action on a compact 4-manifold does not have fixed points. It is also shown that for a symplectic action of the circle an analog of the Duistermaat-Heckman formula holds. A special attention is paid to the actions on manifolds \((W^{2n},\omega)\) satisfying the Lefschetz condition: \(\cup[\omega]^{n-1}: H^ 1(W)\to H^{2n-1}(W)\) is an isomorphism. It is proved that on such a manifold a cohomologically free action is locally free and \(H^*_ G(W,\mathbb{R})=H^*(W/T,\mathbb{R})\otimes H^*(BG_ 0,\mathbb{R})\). In particular, the action is Hamiltonian if and only if it has fixed points.