Mathematical Research Letters
Volume 22 (2015)
Log-concavity and symplectic flows
Pages: 501 – 527
The Duistermaat-Heckman measure of a Hamiltonian torus action on a symplectic manifold $(M,\omega)$ 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 $S^1$-action on a compact connected symplectic $4$-manifold is Hamiltonian if and only if it has fixed points.