turkmath.org

Let $(M^{2n},w)$ be a 2n-dimensional closed symplectic manifold with a symplectic circle action. Many mathematicians tried to find some conditions on M which make a symplectic circle action Hamiltonian. Cho, Hwang and Suh discovered a condition on the 6-dimeonsional symplectic manifolds. In this talk, we will discuss CHS’s theorem: Let $(M, w)$ be a 6-dimensional closed symplectic $S^1$-manifold with generalized moment map $\mu : M \to S^1$. Assume that the fixed point set is not empty and dimension of each component at most is 2. Then the action is Hamiltonian if and only if $b^{+}_{2}(M_{\zeta})=1$ for any regular value $\zeta$ of $\mu$.