Log-concavity and symplectic flows

TL;DR

This paper proves the log-concavity of the Duistermaat-Heckman measure for certain symplectic torus actions and uses this to establish conditions under which such actions are Hamiltonian, providing new proofs of classical results.

Contribution

It verifies the log-concavity conjecture for specific symplectic torus actions and links fixed points to Hamiltonian properties, offering new proofs of classical theorems.

Findings

Log-concavity of Duistermaat-Heckman measure established for certain torus actions.

Fixed points imply Hamiltonian actions under specified conditions.

New proofs of McDuff's theorem and extensions provided.

Abstract

Let M 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 S^1-actions, and some of its recent extensions.