The log-concavity conjecture on semifree symplectic S¹-manifolds with isolated fixed points

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.