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

Let $(M,ω)$ 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 $(M,ω)$ and any reduced symplectic form satisfy the Hard Lefschetz property.

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