The log-concavity conjecture for the duistermaat-heckman measure revisited

Karshon constructed the first counterexample to the log-concavity conjecture for the Duistermaat-Heckman measure: a Hamiltonian six-manifold whose fixed-points set is the disjoint union of two copies of T⁴. In this article, for any closed symplectic fourmanifold N with b⁺ > 1, we show that there is a Hamiltonian six-manifold M such that its fixed-points set is the disjoint union of two copies of N and such that its DuistermaatHeckman function is not log-concave. On the other hand, we prove that if there is a torus action of complexity 2 such that all the symplectic reduced spaces taken at regular values satisfy the condition b⁺ = 1, then its Duistermaat-Heckman function has to be log-concave. As a consequence, we prove the log-concavity conjecture for Hamiltonian circle actions on six manifolds such that the fixed-points sets have no 4-dimensional components, or only have 4-dimensional pieces with b⁺ = 1.