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 T. In this article, for any closed symplectic four manifold N with b > 1, we show that there is a Hamiltonian circle manifold M fibred over N such that its DuistermaatHeckman function is not log-concave. This allows us to construct simply connected Hamiltonian manifolds which have the Hard Lefschetz property andwhich have a non-log-concave DuistermaatHeckman function. Along the same line, we also give examples of non-Kähler Hamiltonian manifolds which have a log-concave Duistermaat-Heckman function. On the other hand, we prove that if there is a torus action of complexity two such that all the symplectic reduced spaces taken at regular values satisfy the condition b = 1, then its DuistermaatHeckman 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 four dimensional components, or only have four dimensional pieceswith b = 1.