2009-11-20

John Arlo Caine, Notre Dame

A Quadratic Poisson Structure on Toric Varieties

Abstract: I will exhibit the construction of a Poisson structure on smooth toric varieties whose symplectic foliation is precisely the complex torus orbits. This Poisson structure is therefore not symplectic, yet is non-degenerate on an open dense set, just like the Bruhat Poisson structure on the full flag manifold. In the natural holomorphic coordinates, the Poisson structure is real and homogeneous quadratic and generalizes quadratic Poisson structures of elliptic type in the plane. The action of the complex torus is Poisson but no subgroup acts in a globally Hamiltonian way, even though every symplectic leaf admits a Hamiltonian action by a sub-torus of the compact torus. The main result I will present is an estimate for the first Poisson cohomology of the toric variety with this Poisson structure. At the end I will exhibit the relationship between this Poisson structure and the Delzant symplectic structure on CP^n through intriguing formula for the modular vector field in terms of Delzant moment data. The zeros of this modular vector field on CP^n are carried by the Delzant momentum map onto the centroids of the faces of the Delzant moment simplex.