Canonical bases for the equivariant cohomology and K-theory rings of symplectic toric manifolds

TL;DR

This paper constructs explicit canonical bases for the equivariant cohomology and K-theory rings of symplectic toric manifolds, providing combinatorial algorithms for their computation and establishing connections with known classes.

Contribution

It introduces a new combinatorial method to explicitly compute canonical bases for equivariant cohomology and K-theory of symplectic toric manifolds, linking them to existing classes.

Findings

Explicit bases for equivariant K-theory and cohomology rings are constructed.

Algorithms for computing restrictions of basis elements to fixed points are provided.

The bases coincide with known classes such as equivariant Poincaré duals and Goldin-Tolman classes.

Abstract

Let $M$ be a symplectic toric manifold acted on by a torus $\mathbb{T}$. In this work we exhibit an explicit basis for the equivariant K-theory ring $\mathcal{K}_{\mathbb{T}}(M)$ which is canonically associated to a generic component of the moment map. We provide a combinatorial algorithm for computing the restrictions of the elements of this basis to the fixed point set; these, in turn, determine the ring structure of $\mathcal{K}_{\mathbb{T}}(M)$. The construction is based on the notion of local index at a fixed point, similar to that introduced by Guillemin and Kogan in [GK]. We apply the same techniques to exhibit an explicit basis for the equivariant cohomology ring $H_{\mathbb{T}}(M; \mathbb{Z})$ which is canonically associated to a generic component of the moment map. Moreover we prove that the elements of this basis coincide with some well-known sets of classes: the equivariant Poincar\'e duals to the closures of unstable manifolds, and also the canonical classes introduced by Goldin and Tolman in [GT], which exist whenever the moment map is index increasing.