Upper bounds for the dimension of tori acting on GKM manifolds

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 $(\Gamma ,\tilde{\alpha },\nabla )$ 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.

22 pages, 9 figures, v2: correct some notations

References in corpus (4)

• How is a graph like a manifold?
• Non-negative curvature and torus actions
• Root systems and symmetries of torus manifolds
• Complex cobordism classes of homogeneous spaces

Cited by in corpus (2)

• Non-negatively curved GKM orbifolds
• Equivariant $K$-theory of even-dimensional complex quadrics