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-dimensional 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 $2$. Then the action is Hamiltonian if and only if $b_2^+(M_\xi)=1$ for any regular value $\xi$ of $\mu$.