Upper bounds for the dimension of tori acting on GKM manifolds

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.