Circle actions on symplectic manifolds

Abstract

This thesis consists of two parts. The first part is as follows. Let ($\It{M, w}$) be a 6-dimensional closed symplectic semifree $\It{S}^1$-manifold whose fixed point set is a disjoint union of surfaces. Suppose that there is a generalized moment map. We prove that the action is Hamiltonian if and only if $\It{M_red}$ is diffeomorphic to an $\It{S}^2$-bundle over some compact Riemann surface and the fixed point set is not empty. We also show that the number of fixed surfaces of genus > 0 is at most four if the action is Hamiltonian. Moreover, if the minimum and the maximum are 2-spheres, then there is at most one fixed surface of non-zero genus. The second part is about the log-concavity properties on symplectic manifolds. we define a notion “(Strong)Log-concavity property” on symplectic manifolds and prove that for a given symplectic manifold $\It{M}$ satisfying the strong log-concavity property, the symplectic blow-ups and blow-downs along symplectic submanifolds of a small $\epsilon$-amount satisfy the log-concavity property. Moreover, we explain that these properties are closely related to the moduli space of symplectic structures.