The Log-Concavity Conjecture for the Duistermaat–Heckman Measure Revisited
Department of Mathematics, University of Toronto 40, St. George St., Toronto, Ontario M5S 2E4, Canada
Correspondence: Correspondence to be sent to: ylin{at}math.toronto.edu
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 four-manifold 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 Duistermaat–Heckman 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.