The log-concavity conjecture on semifree symplectic S^1-manifolds with isolated fixed points

TL;DR

This paper proves the log-concavity of the Duistermaat-Heckman measure density function and the Hard Lefschetz property for a class of semifree Hamiltonian S^1-manifolds with isolated fixed points, advancing symplectic geometry theory.

Contribution

It establishes the log-concavity of the Duistermaat-Heckman measure and confirms the Hard Lefschetz property for these specific symplectic manifolds, which was previously conjectural.

Findings

Density function of the Duistermaat-Heckman measure is log-concave.

The manifold and its reductions satisfy the Hard Lefschetz property.

Abstract

Let $(M,\omega)$ be a closed $2n$-dimensional semifree Hamiltonian $S^1$-manifold with only isolated fixed points. We prove that a density function of the Duistermaat-Heckman measure is log-concave. Moreover, we prove that $(M,\omega)$ and any reduced symplectic form satisfy the Hard Lefschetz property.