Abstract:
Symplectic four-manifolds with Hamiltonian actions of a 2-torus or a circle were classified, by Delzant and Karshon, using the image of the moment map, which is a convex polytope in \R^2. Recent works of Karshon, Kessler and Pinsonnault applied Gromov theory of pseudo-holomorphic curves to further characterize Hamiltonian group actions. In this talk I will review basic and recent results. In particular, I will describe my current work in which I apply holomorphic methods to extend combinatorial tools developed for Hamiltonian circle actions to study symplectic actions of finite cyclic groups.