2007-04-06

Yi Lin, University of Toronto

Log-concavity conjecture of Duistermaat-Heckman measure revisited

Karshon constructed the first counterexample to the log-concavity conjecture for Duistermaat-Heckman measures: a Hamiltonian six-manifold whose fixed-point set is the disjoint union of two copies of T^4. In this talk, for any closed symplectic four manifold N with b^+>1, we will show how to construct a Hamiltonian circle manifold M fibred over N such that its Duistermaat-Heckman function is not log-concave. Along the same lines, we will discuss how to construct simply connected Hamiltonian manifolds which have the Hard Lefschetz property and which have a non-log-concave Duistermaat-Heckman function.

On the other hand, we will explain that if there is a six-dimensional Hamiltonian circle manifold 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. For instance, this result implies the log-concavity conjecture for semi-free Hamiltonian circle actions on six manifolds with fixed points set of codimension greater than or equal to 4.

This talk is based on recent work of the speaker.