Stratified Gradient Hamiltonian Vector Fields and Collective Integrable Systems
Abstract
We construct completely integrable systems on the dual of the Lie algebra of any compact Lie group K with respect to the standard Lie-Poisson structure. These systems generalize key properties of Gelfand-Zeitlin systems: A) the pullback to any Hamiltonian K-manifold defines a Hamiltonian torus action on an open dense subset, B) if the K-manifold is multiplicity-free, then the resulting torus action is completely integrable, and C) the collective moment map has convexity and fiber connectedness properties. These systems generalize the relationship between Gelfand-Zeitlin systems and Gelfand-Zeitlin canonical bases via geometric quantization by a real polarization. To construct these systems, we generalize Harada and Kaveh's construction of integrable systems by toric degeneration on smooth projective varieties to singular quasi-projective varieties. Under certain conditions, we show that the stratified-gradient Hamiltonian vector field of such a degeneration, which is defined piece-wise, has a flow whose limit exists and defines continuous degeneration map.