Log-Concavity and Symplectic Flows

The Duistermaat-Heckman measure of a Hamiltonian torus action on a symplectic manifold (M, ω) is the push forward of the Liouville measure on M by the momentum map of the action. In this paper we prove the logarithmic concavity of the Duistermaat-Heckman measure of a complexity two Hamiltonian torus action, for which there exists an effective commuting symplectic action of a 2-torus with symplectic orbits. Using this, we show that given a complexity two symplectic torus action satisfying the additional 2-torus action condition, if the fixed point set is non-empty, then it has to be Hamiltonian. This implies a classical result of McDuff: a symplectic S¹-action on a compact connected symplectic 4-manifold is Hamiltonian if and only if it has fixed points.