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 $\mathcal{A}(\mathrm{\Gamma },\alpha ,\mathrm{\nabla })$, from an (abstract) $(m,n)$-type GKM graph $(\mathrm{\Gamma },\alpha ,\mathrm{\nabla })$. 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 $\mathcal{A}(\mathrm{\Gamma },\alpha ,\mathrm{\nabla })$ has rank $\mathrm{\ell }(>n)$ if and only if there exists an $(m,\mathrm{\ell })$-type GKM graph $(\mathrm{\Gamma },\tilde{\alpha },\mathrm{\nabla })$ which is an extension of $(\mathrm{\Gamma },\alpha ,\mathrm{\nabla })$. Using this necessarily and sufficient condition, we prove that the rank of $\mathcal{A}(\mathrm{\Gamma },\alpha ,\mathrm{\nabla })$ 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 $\mathcal{A}(\mathrm{\Gamma },\alpha ,\mathrm{\nabla })$ 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.