Example of a non-log-concave Duistermaat-Heckman measure

TL;DR

This paper constructs a specific example of a compact symplectic manifold with a Hamiltonian circle action where the Duistermaat-Heckman function defies log-concavity, challenging previous assumptions.

Contribution

It provides the first explicit example of a non-log-concave Duistermaat-Heckman measure in a compact symplectic setting.

Findings

Demonstrates existence of non-log-concave Duistermaat-Heckman functions

Constructs explicit example of a symplectic manifold with this property

Challenges prior beliefs about measure concavity in symplectic geometry

Abstract

We construct a compact symplectic manifold with a Hamiltonian circle action for which the Duistermaat-Heckman function is not log-concave.