Towards generalizing schubert calculus in the symplectic category

Rebecca F. Goldin, Susan Tolman

Research output: Contribution to journalArticlepeer-review


The main purpose of this article is to 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 Ψ of the moment map, which is a Morse function, we define a canonical class αp in the equivariant cohomology of the manifold M for each fixed point p ∈ 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 αp to a fixed point q 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. Finally, we prove the existence of integral canonical classes in the case that M is a GKM space (after Goresky, Kottwitz and MacPherson) and Ψ is index increasing. In this case, our restriction formula specializes to an easily computable rational sum which depends only on the GKM graph.

Original languageEnglish (US)
Pages (from-to)449-473
Number of pages25
JournalJournal of Symplectic Geometry
Issue number4
StatePublished - Dec 2009

ASJC Scopus subject areas

  • Geometry and Topology


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