# Smooth Toric Varieties as real Poisson Manifolds

## GAP Seminar

## Meeting Details

For more information about this meeting, contact Mathieu Stiénon, Nigel Higson, Ping Xu.

**Speaker:** John Arlo Caine, Notre Dame

**Abstract:** The principal examples of highly symmetric compact symplectic manifolds are the homogeneous spaces of compact Lie groups, such as the flag manifold U/T, equipped with the Kostant-Kirillov-Soriau symplectic structure, and the almost homogeneous spaces known as smooth toric varieties, equipped with Delzant symplectic structures. On the other hand, U/T=G/B also carries the Bruhat Poisson structure, a real quadratic Poisson structure whose symplectic leaves are the left B orbits on G/B and thus is non-degenerate on an open dense subset. In this talk, I will construct an analogous real Poisson structure on smooth toric varieties whose symplectic leaves are precisely the orbits of the complex torus acting on the variety and discuss some of its properties. In particular, I will describe some efforts to compute the Poisson cohomology, which can be done explicitly for the simplest toric variety CP^1 using a result of Nakanishi on quadratic Poisson structures in the plane, and then pose some open problems.

## Room Reservation Information

**Room Number:** 106 McAllister

**Date:** 12/07/2010

**Time:** 2:30pm - 3:30pm