Upper bounds for the dimension of tori acting on GKM manifolds

TL;DR

This paper establishes an upper bound for the dimension of tori acting effectively on GKM manifolds by introducing a new algebraic invariant derived from GKM graphs, and applies it to the complex Grassmannian.

Contribution

It introduces the group alA(\u03b3,\u03b1, abla) from GKM graphs and proves it bounds the torus action dimension, with explicit calculations for Grassmannians.

Findings

The rank of alA provides an upper bound for torus action dimension.

The rank of alA for G_{2}(\u2102^{n+2}) matches the maximal effective torus dimension.

The method applies to specific GKM manifolds like complex Grassmannians.

Abstract

The aim of this paper is to give an upper bound for the dimension of a torus $T$ which acts on a GKM manifold $M$ effectively. In order to do that, we introduce a free abelian group of finite rank, denoted by $\mathcal{A}(\Gamma,\alpha,\nabla)$, from an (abstract) $(m,n)$-type GKM graph $(\Gamma,\alpha,\nabla)$. Here, an $(m,n)$-type GKM graph is the GKM graph induced from a $2m$-dimensional GKM manifold $M^{2m}$ with an effective $n$-dimensional torus $T^{n}$-action, say $(M^{2m},T^{n})$. Then it is shown that $\mathcal{A}(\Gamma,\alpha,\nabla)$ has rank $\ell(> n)$ if and only if there exists an $(m,\ell)$-type GKM graph $(\Gamma,\widetilde{\alpha},\nabla)$ which is an extension of $(\Gamma,\alpha,\nabla)$. Using this necessarily and sufficient condition, we prove that the rank of $\mathcal{A}(\Gamma,\alpha,\nabla)$ for the GKM graph of $(M^{2m},T^{n})$ gives an upper bound for the dimension of a torus which can act on $M^{2m}$ effectively. As an application, we compute the rank of $\mathcal{A}(\Gamma,\alpha,\nabla)$ of the complex Grassmannian of $2$-planes $G_{2}(\mathbb{C}^{n+2})$ with some effective $T^{n+1}$-action, and prove that the $T^{n+1}$-action on $G_{2}(\mathbb{C}^{n+2})$ is the maximal effective torus action.