Tuesday, February 19th, 2008
10:00am - 11:00am, in Campus Center 2-2545

Yi Lin

University of Toronto

The log-concavity conjecture of the Duistermaat-Heckman measure

Abstract: The log-concavity property of the Duistermaat-Heckman measure was first established by A. Okounkov for the Hamiltonian action of a compact Lie group on a projective variety in a series of very interesting papers. His results led V. Ginzburg to conjecture that the Duistermaat-Heckman measure is log-concave for Hamiltonian torus actions on symplectic manifolds. However, motivated by a construction of D. McDuff, Y. Karshon found a counter-example to this conjecture. In this talk, I will present a systematic construction of Duistermaat-Heckman measure which is not log-concave. On the other hand, I will explain that if there is a Hamiltonian torus action of complexity two such that all the symplectic reduced spaces taken at regular values satisfy the condition $b^+=1$, then its Duistermaat-Heckman measure has to be log-concave. For instance, this result implies that the log-concavity conjecture holds for Hamiltonian circle actions on six manifolds whose fixed points sets have no four dimensional components, or have only four dimensional components with $b^+=1$.