University of Toronto's Symplectic Geometry Seminar

Monday, 16 March 2009, 2:10pm
BA 6183


George Mason University

Towards generalizing Schubert calculus in the symplectic category


We extend some of the ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. Given a generic component of the moment map, which is a Morse function, we define a canonical class in the equivariant cohomology of the manifold M for each fixed point p in M. When they exist, canonical classes form a natural basis of the equivariant cohomology of M; in particular, when M is a flag variety, these classes are the equivariant Schubert classes. We show that the restriction of a canonical class to a fixed point can be calculated by a rational function which depends only on the value of the moment map, and the restriction of other canonical classes to points of index exactly two higher. Therefore, the structure constants can be calculated by a similar rational function. Our restriction formula is manifestly positive in many cases, including when M is a flag manifold. This is joint work with Susan Tolman of University of IL, Champaign-Urbana.