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# Mathematical Research Letters

## Volume 22 (2015)

### Number 2

### Log-concavity and symplectic flows

Pages: 501 – 527

DOI: https://dx.doi.org/10.4310/MRL.2015.v22.n2.a9

#### Authors

#### Abstract

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.

Accepted 26 June 2014

Published 16 April 2015