Log-concavity and symplectic flows

Abstract

Let (M,ω) be a compact, connected symplectic 2n-dimensional manifold on which an (n− 2)-dimensional torus T acts effectively and Hamiltonianly. Under the assumption that there is an effective complementary 2-torus acting on M with symplectic orbits, we show that the Duistermaat-Heckman measure of the T -action is log-concave. This verifies the logarithmic concavity conjecture for a class of inequivalent T -actions. Then we use this conjecture to prove the following: if there is an effective symplectic action of an (n − 2)-dimensional torus T on a compact, connected symplectic 2n-dimensional manifold that admits an effective complementary symplectic action of a 2-torus with symplectic orbits, then the existence of T -fixed points implies that the T -action is Hamiltonian. As a consequence of this, we give new proofs of a classical theorem by McDuff about S1-actions, and some of its recent extensions.