Document Type

Article

Publication Date

2015

Publication Title

Mathematical Research Letters

DOI

10.4310/MRL.2015.v22.n2.a9

ISSN

1073-2780

Abstract

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.

Comments

This version of the paper was obtained from arXIV.org. In order for the work to be deposited in arXIV.org, the authors must hold the rights or the work must be under Creative Commons Attribution license, Creative Commons Attribution-Noncommercial-ShareAlike license, or Create Commons Public Domain Declaration. The publisher's final edited version of this article is available at Mathematical Research Letters.

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