[Paper Review] Towards Generalizing Schubert Calculus in the Symplectic Category

Rebecca Goldin, Susan Tolman|ArXiv.org|Apr 8, 2009

Advanced Combinatorial Mathematics5 references3 citations

TL;DR

This paper introduces a new restriction formula for canonical classes in the equivariant cohomology of compact symplectic manifolds with isolated fixed points under Hamiltonian torus actions. By leveraging a generic moment map component and GKM graph data, it provides a manifestly positive, rational formula for restricting these classes to fixed points, generalizing equivariant Schubert calculus and proving the existence of integral canonical classes in the GKM case with index-increasing moment maps.

ABSTRACT

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 of the moment map, we define a canonical class α_p in the equivariant cohomology of the manifold M for each fixed point p of 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 α_p 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 manifold the moment map component is index increasing. In this case, our restriction formula specializes to an easily computable rational sum which depends only on the GKM graph.

Motivation & Objective

• To extend Schubert calculus beyond flag varieties to general compact symplectic manifolds with isolated fixed points under Hamiltonian torus actions.
• To define canonical classes in equivariant cohomology that generalize equivariant Schubert classes.
• To derive a computable, manifestly positive restriction formula for these classes using only moment map values and GKM graph data.
• To prove the existence of integral canonical classes when the manifold is GKM and the moment map component is index-increasing.
• To provide a systematic method for computing structure constants in the equivariant cohomology ring.

Proposed method

• Define canonical classes αₚ ∈ H_T^{2λ(p)}(M;ℚ) as equivariant cohomology classes that restrict to the product of positive weights at p and vanish at all fixed points q with λ(q) ≤ λ(p).
• Use a generic component Ψ = Φ^ξ of the moment map as a Morse function to define the index λ(p) and the negative normal bundle weights.
• Derive a rational restriction formula for αₚ(q) that depends only on Ψ-values and the restrictions α_r(r') for r' of index two higher than r.
• Prove that the restriction formula is manifestly positive when the GKM graph satisfies certain positivity conditions, including in the flag manifold case.
• Establish the existence of integral canonical classes in the GKM case with index-increasing moment map via a lifting construction from a quotient space.
• Specialize the formula to GKM manifolds by expressing the restriction as a rational sum over paths in the GKM graph, depending only on edge weights and moment map values.

Experimental results

Research questions

• RQ1 Can canonical classes in equivariant cohomology be defined for general symplectic manifolds with isolated fixed points, beyond flag varieties?
• RQ2 Is there a restriction formula for canonical classes that depends only on moment map values and GKM graph structure?
• RQ3 Can this restriction formula be made manifestly positive in natural geometric settings?
• RQ4 Under what conditions do integral canonical classes exist in the GKM setting?
• RQ5 How can structure constants in the equivariant cohomology ring be computed from the restriction data?

Key findings

• The restriction of a canonical class αₚ to a fixed point q is given by a rational function depending only on Ψ-values and α_r(r') for r' of index two higher than r.
• The restriction formula is manifestly positive in many cases, including when M is a flag manifold, due to positive moment map differences and positive weights.
• In the GKM case with an index-increasing moment map, integral canonical classes exist and the restriction formula reduces to a rational sum over paths in the GKM graph.
• For the flag manifold Fl(ℂⁿ), the restriction formula yields only positive terms, as all moment map differences and weights are positive.
• There exist GKM manifolds without Kähler structures where the canonical class αₚ restricts negatively to some fixed points, as shown by negative Θ(p,q) values.
• The structure constants c_pq^r in the equivariant cohomology ring can be computed from the restriction data via the formula α_p α_q = ∑_r c_pq^r α_r.